Distribute 5 students into dormitories A, B, and C, with each dormitory having at least 1 and at most 2 students. Among these, the number of different ways to distribute them without student A going to dormitory A is \_\_\_\_\_\_.

Answer: 60


In the trapezoid $ABCD$, the bases $AD$ and $BC$ are 8 and 18, respectively. It is known that the circumcircle of triangle $ABD$ is tangent to the lines $BC$ and $CD$. Find the perimeter of the trapezoid.

Answer: 56


A container in the shape of a right circular cone is 12 inches tall and its base has a 5-inch radius. The liquid that is sealed inside is 9 inches deep when the cone is held with its point down and its base horizontal. When the liquid is held with its point up and its base horizontal, the liquid is  $m-n\sqrt[3]{p},$  where  $m,$   $n,$  and  $p$  are positive integers and  $p$  is not divisible by the cube of any prime number. Find  $m+n+p.$

Answer: 52


Mary thought of a positive two-digit number. She multiplied it by $3$ and added $11$. Then she switched the digits of the result, obtaining a number between $71$ and $75$, inclusive. What was Mary's number?
$\textbf{(A)}\ 11\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 13\qquad\textbf{(D)}\ 14\qquad\textbf{(E)}\ 15$

Answer: \textbf{(B)}\12


It is known that the equation \(x^4 - 8x^3 + ax^2 + bx + 16 = 0\) has (with multiplicity) four positive roots. Find \(a - b\).

Answer: 56


Find the radius of the largest circle that touches internally two intersecting circles with radii \( R \) and \( r \), if the distance between their centers is \( a \) (where \( a < R + r \)).

Answer: \frac{R+r-}{2}


Given that a five-digit palindromic number equals the product of 45 and a four-digit palindromic number (i.e., $\overline{\mathrm{abcba}} = 45 \times \overline{\mathrm{deed}}$), what is the largest possible value of this five-digit palindromic number?

Answer: 59895


Given the equation $3x^{2}-4=-2x$, find the quadratic coefficient, linear coefficient, and constant term.

Answer: -4


Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that for all \( x, y \in \mathbb{R} \),

\[
(x-y) f(x+y)-(x+y) f(x-y)=4 x y\left(x^{2}-y^{2}\right)
\]

Answer: f(x)=x^3+cx


A certain high school has 1000 students in the first year. Their choices of elective subjects are shown in the table below:

| Subject | Physics | Chemistry | Biology | Politics | History | Geography |
|---------|---------|-----------|---------|----------|---------|-----------|
| Number of Students | 300 | 200 | 100 | 200 | 100 | 100 |

From these 1000 students, one student is randomly selected. Let:
- $A=$ "The student chose Physics"
- $B=$ "The student chose Chemistry"
- $C=$ "The student chose Biology"
- $D=$ "The student chose Politics"
- $E=$ "The student chose History"
- $F=$ "The student chose Geography"

$(Ⅰ)$ Find $P(B)$ and $P(DEF)$.

$(Ⅱ)$ Find $P(C \cup E)$ and $P(B \cup F)$.

$(Ⅲ)$ Are events $A$ and $D$ independent? Please explain your reasoning.

Answer: \frac{3}{10}


The center of the upper base of a cube with edge length $a$ is connected to the midpoints of the sides of the lower base, which are also connected in a sequential order. Calculate the total surface area of the resulting pyramid.

Answer: 2a^2


Acute triangle \( A B C \) has circumcenter \( O \). The bisector of \( \angle A B C \) and the altitude from \( C \) to side \( A B \) intersect at \( X \). Suppose that there is a circle passing through \( B, O, X, \) and \( C \). If \( \angle B A C = n^{\circ} \), where \( n \) is a positive integer, compute the largest possible value of \( n \).

Answer: 67


Given the sequence \( a_{n}=\left\lfloor(\sqrt{2}+1)^{n}+\left(\frac{1}{2}\right)^{n}\right\rfloor \) for \( n \geq 0 \), where \(\left\lfloor x \right\rfloor\) denotes the greatest integer less than or equal to \( x \), compute the sum:
\[ \sum_{n=1}^{\infty} \frac{1}{a_{n-1} a_{n+1}} \]

Answer: \frac{1}{8}


The lateral edges of a triangular pyramid are pairwise perpendicular, and the areas of the lateral faces are \( S \), \( P \), and \( Q \). Find the radius of the inscribed sphere. Also, find the radius of the sphere that touches the base and the extensions of the lateral faces of the pyramid.

Answer: \frac{\sqrt{2SPQ}}{S+P+Q-\sqrt{S^2+P^2+Q^2}}


On a chessboard, the centers of some squares are marked in such a way that no triangle with marked vertices is a right triangle. What is the maximum number of points that could be marked?

Answer: 14


Given an ellipse $C$: $\frac{x^2}{16}+\frac{y^2}{4}=1$, with the left and right foci being $F_{1}$ and $F_{2}$, respectively. Line $l$ intersects the ellipse at points $A$ and $B$, where the chord $AB$ is bisected by the point $(\sqrt{3},\frac{\sqrt{3}}{2})$.
$(1)$ Find the equation of line $l$;
$(2)$ Find the area of $\triangle F_{1}AB$.

Answer: 2\sqrt{15}


For non-negative numbers \(a, b, c, d\), the following equalities are satisfied:

\[
\sqrt{a+b}+\sqrt{c+d}=\sqrt{a+c}+\sqrt{b+d}=\sqrt{a+d}+\sqrt{b+c}
\]

What is the maximum number of distinct values that can be among the numbers \(a, b, c, d\)?

Answer: 2


A juggler throws a ball upward every second; with what initial speed must the balls be thrown so that the first ball falls back into his hand when

a) the second ball is thrown,

b) the third ball is thrown?

Answer: 9.8\,


[Diagonal](https://artofproblemsolving.com/wiki/index.php/Diagonal) $DB$ of [rectangle](https://artofproblemsolving.com/wiki/index.php/Rectangle) $ABCD$ is divided into three segments of length $1$ by [parallel](https://artofproblemsolving.com/wiki/index.php/Parallel) lines $L$ and $L'$ that pass through $A$ and $C$ and are [perpendicular](https://artofproblemsolving.com/wiki/index.php/Perpendicular) to $DB$. The area of $ABCD$, rounded to the one decimal place, is

$\mathrm{(A)\ } 4.1 \qquad \mathrm{(B) \ }4.2 \qquad \mathrm{(C) \  } 4.3 \qquad \mathrm{(D) \  } 4.4 \qquad \mathrm{(E) \  }4.5$

Answer: (B)4.2


$k_1, k_2, k_3$  are three circles.  $k_2$  and  $k_3$  touch externally at  $P$ ,  $k_3$  and  $k_1$  touch externally at  $Q$ , and  $k_1$  and  $k_2$  touch externally at  $R$ . The line  $PQ$  meets  $k_1$  again at  $S$ , the line  $PR$  meets  $k_1$  again at  $T$ . The line  $RS$  meets  $k_2$  again at  $U$ , and the line  $QT$  meets  $k_3$  again at  $V$ . Show that  $P, U, V$  are collinear.

Answer: P,U,V


In triangle \(ABC\), \(AC = 1\), \(AB = 2\), and \(O\) is the point where the angle bisectors intersect. A segment passing through point \(O\) parallel to side \(BC\) intersects sides \(AC\) and \(AB\) at points \(K\) and \(M\) respectively. Find the perimeter of triangle \(AKM\).

Answer: 3


Each row of a seating arrangement seats either 9 or 10 people. A total of 100 people are to be seated. How many rows seat exactly 10 people if every seat is occupied?

Answer: 10


Real numbers $x$, $y$, and $z$ satisfy the inequalities
$0<x<1$, $-1<y<0$, and $1<z<2$.
Which of the following numbers is necessarily positive?
$\textbf{(A)}\ y+x^2\qquad\textbf{(B)}\ y+xz\qquad\textbf{(C)}\ y+y^2\qquad\textbf{(D)}\ y+2y^2\qquad\textbf{(E)}\ y+z$

Answer: \textbf{(E)}\y+z


Answer: 2 \sqrt{2} + 5^{2/3}


In $\triangle ABC$, the angle bisector of $\angle A$ intersects $BC$ at $D$ and intersects the circumcircle of $\triangle ABC$ at $P$. A circle with chord $AD$ intersects $AB$ at $M$ and extends to intersect the extension of $AC$ at $N$. Also, $PM$ intersects $BC$ at $G$. If the area of $\triangle BMG$ is $S_{1}$, and the area of quadrilateral $PNCG$ is $S_{2}$, compare the sizes of $S_{1}$ and $S_{2}$.

Answer: S_1=S_2


Let \( x \) and \( y \) be positive integers such that \( 27x + 35y \leq 945 \). Find the largest possible value of \( xy \).

Answer: 234


Given the set of positive integers
$$
A=\left\{a_{1}, a_{2}, \cdots, a_{1000}\right\},
$$
where $ a_{1} < a_{2} < \cdots < a_{1000} \leq 2017 $. If for any $ 1 \leq i, j \leq 1000$, $ i+j \in A $ implies $ a_{i} + a_{j} \in A $, determine the number of sets $ A $ that satisfy these conditions.

Answer: 2^{17}


There are 5 students on a team for a math competition. The math competition has 5 subject tests. Each student on the team must choose 2 distinct tests, and each test must be taken by exactly two people. In how many ways can this be done?

Answer: 2040


A positive number $x$ has the property that $x\%$ of $x$ is $4$.  What is $x$?
$\textbf{(A) }\ 2      \qquad \textbf{(B) }\ 4      \qquad \textbf{(C) }\ 10      \qquad \textbf{(D) }\ 20      \qquad \textbf{(E) }\ 40$

Answer: \textbf{(D)}20


Given triangle  $ ABC$  with sidelengths  $ a,b,c$ . Tangents to incircle of  $ ABC$  that parallel with triangle's sides form three small triangle (each small triangle has 1 vertex of  $ ABC$ ). Prove that the sum of area of incircles of these three small triangles and the area of incircle of triangle  $ ABC$  is equal to
 $ \frac{\pi (a^{2}\plus{}b^{2}\plus{}c^{2})(b\plus{}c\minus{}a)(c\plus{}a\minus{}b)(a\plus{}b\minus{}c)}{(a\plus{}b\plus{}c)^{3}}$

(hmm,, looks familiar, isn't it? :wink: )

Answer: \frac{\pi(^{2}+b^{2}+^{2})()(b)()}{()^{3}}


Five children \( A \), \( B \), \( C \), \( D \), and \( E \) participate in the "Happy Pinyin Reading" competition. When they come to the stage, they stand in a line. Each has a player number tag on their chest, and the sum of the 5 player numbers is 35. It is known that the sum of the numbers of the players standing to the right of \( E \), \( D \), \( A \), and \( C \) are 13, 31, 21, and 7 respectively. What is the sum of the player numbers for \( A \), \( C \), and \( E \)?

Answer: 24


In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively. Given that $\cos(B-C) - 2\sin B\sin C = -\frac{1}{2}$.
1. Find the measure of angle $A$.
2. When $a = 5$ and $b = 4$, find the area of $\triangle ABC$.

Answer: 2\sqrt{3} + \sqrt{39}


Arrange $\frac{131}{250}, \frac{21}{40}, 0.5 \dot{2} \dot{3}, 0.52 \dot{3}, 0.5 \dot{2}$ in ascending order. The third number is $\qquad$.

Answer: 0.52\dot{3}


Given positive numbers $a$ and $b$ satisfying $\log _{6}(2a+3b)=\log _{3}b+\log _{6}9-1=\log _{2}a+\log _{6}9-\log _{2}3$, find $\lg \left(2a+3b\right)-\lg \left(10a\right)-\lg \left(10b\right)=\_\_\_\_\_\_$.

Answer: -2


Let \(a,\) \(b,\) \(c\) be distinct real numbers such that
\[\frac{a}{1 + b} = \frac{b}{1 + c} = \frac{c}{1 + a} = k.\] Find the product of all possible values of \(k.\)

Answer: -1


Yasmine makes her own chocolate beverage by mixing volumes of milk and syrup in the ratio \(5: 2\). Milk comes in \(2 \text{ L}\) bottles and syrup comes in \(1.4 \text{ L}\) bottles. Yasmine has a limitless supply of full bottles of milk and of syrup. Determine the smallest volume of chocolate beverage that Yasmine can make that uses only whole bottles of both milk and syrup.

Answer: 19.6


Let $\alpha$ be the remainder of $2^{1000}$ divided by 13, determine the value of $\alpha$.

Determine the value of $\beta=\frac{(7+4 \sqrt{\alpha})^{\frac{1}{2}}-(7-4 \sqrt{\alpha})^{\frac{1}{2}}}{\sqrt{\alpha}}$.

If $f(a)=a-\beta$ and $F(a, b)=b^{2}+a$, determine the value of $\gamma=F(3, f(4))$.

If $\delta$ is the product of all real roots of $x^{\log _{7} x}=10$, determine the value of $\delta$.

Answer: 1


A circle $\omega$ with center $O$ touches the sides \(BA\) and \(BC\) of the acute angle \(ABC\) at points \(M\) and \(N\) respectively. A line passing through point \(M\) parallel to \(BC\) intersects ray \(BO\) at point \(K\). On ray \(MN\), a point \(T\) is chosen such that \(\angle MTK = \frac{1}{2} \angle ABC\). It turns out that line \(KT\) is tangent to \(\omega\). Find the area of triangle \(OKT\) if \(BM = a\).

Answer: \frac{^2}{2}


Use the method of random simulation to estimate the probability that it will rain on exactly two of the three days. Using a calculator, generate random integer values between 0 and 9, where 1, 2, 3, and 4 represent raining days and 5, 6, 7, 8, 9, and 0 signify non-raining days. Then, group every three random numbers to represent the weather for these three days. After conducting the random simulation, the following 20 groups of random numbers were produced:
907    966    191    925    271    932    812    458    569    683
431    257    393    027    556    488    730    113    537    989
Estimate the probability that exactly two days out of three will have rain.

Answer: 0.25


Find the sum of all integer values of \( c \) for which the equation \( 15|p-1| + |3p - |p + c|| = 4 \) has at least one solution for \( p \).

Answer: -2


Suppose  $ 2015= a_1 <a_2 < a_3<\cdots <a_k $  be a finite sequence of positive integers, and for all  $ m, n \in \mathbb{N} $  and  $1\le m,n \le k $ ,   $$  a_m+a_n\ge a_{m+n}+|m-n|  $$   Determine the largest possible value  $ k $  can obtain.

Answer: 2016


The pentagon $ABCDE$ is obtained by rotating the pentagon $HICBG$ around point $C$, and the pentagon $FGBAE$ is obtained by rotating the pentagon $ABCDE$ around point $E$, as shown in the sketch. The length of the segment $AB$ is $7 \text{ cm}$.

What is the total length of the 11 segments?

Answer: 77\,


(a) In how many ways is it possible to write the number 105 as the difference of two perfect squares?

(b) Show that it is not possible to write the number 106 as the difference of two perfect squares.

Answer: 4


I is the incenter of the triangle ABC. A circle with center I meets the side BC at D and P, with D nearer to B. Similarly, it meets the side CA at E and Q, with E nearer to C, and it meets AB at F and R, with F nearer to A. The lines EF and QR meet at S, the lines FD and RP meet at T, and the lines DE and PQ meet at U. Show that the circumcircles of DUP, ESQ and FTR have a single point in common.

Answer: I


Find the natural numbers  $ n\ge 2 $  which have the property that the ring of integers modulo  $ n $  has exactly an element that is not a sum of two squares.

Answer: 4


Let \( X = \{1, 2, \ldots, 10\} \). The mapping \( f: X \rightarrow X \) satisfies:
1. \( f \circ f = I_X \), where \( f \circ f \) is the composition of mappings, and \( I_X \) is the identity mapping on \( X \);
2. \( |f(i) - i| \leqslant 2 \) for any \( i \in X \).

Determine the number of such mappings \( f \).

Answer: 401


The numbers 1, 2, 3, 4, 5, 6, 7 are written in a circle in some order. A number is called "good" if it is equal to the sum of the two numbers written next to it. What is the maximum possible number of "good" numbers among those written?

Answer: 3


Let \(CD\) be a chord of a circle \(\Gamma_{1}\) and \(AB\) a diameter of \(\Gamma_{1}\) perpendicular to \(CD\) at \(N\) with \(AN > NB\). A circle \(\Gamma_{2}\) centered at \(C\) with radius \(CN\) intersects \(\Gamma_{1}\) at points \(P\) and \(Q\), and the segments \(PQ\) and \(CD\) intersect at \(M\). Given that the radii of \(\Gamma_{1}\) and \(\Gamma_{2}\) are 61 and 60 respectively, find the length of \(AM\).

Answer: 78


Given the hyperbola $x^{2}-4y^{2}=4$ with left and right foci $F_{1}$ and $F_{2}$. A line passing through $F_{1}$ intersects the left branch at points $A$ and $B$. If $|AB|=3$, find the perimeter of $\triangle AF_{2}B$.

Answer: 14


Answer: 20


For how many integers $n$ is $\dfrac n{20-n}$ the [square](https://artofproblemsolving.com/wiki/index.php/Perfect_square) of an integer?
$\mathrm{(A)}\ 1 \qquad\mathrm{(B)}\ 2 \qquad\mathrm{(C)}\ 3 \qquad\mathrm{(D)}\ 4 \qquad\mathrm{(E)}\ 10$

Answer: \mathrm{(D)}\4


If the sum of all binomial coefficients in the expansion of $(2x- \frac {1}{x^{2}})^{n}$ is $64$, then $n=$ \_\_\_\_\_\_; the constant term in the expansion is \_\_\_\_\_\_.

Answer: 240


Find the smallest positive number \( r \) for which the following statement is true: For every positive \( a \), there exists \( x \) such that \( 2 - a r \leq x \leq 2 \) and \( a x^{3} + x^{2} - 4 = 0 \).

Answer: 2


Each rational number is painted either white or red. Call such a coloring of the rationals *sanferminera* if for any distinct rationals numbers  $x$  and  $y$  satisfying one of the following three conditions: [list=1][*] $xy=1$ ,
[*] $x+y=0$ ,
[*] $x+y=1$ ,[/list]we have  $x$  and  $y$  painted different colors. How many sanferminera colorings are there?

Answer: 2


In the sum shown, each letter represents a different digit with $T \neq 0$ and $W \neq 0$. How many different values of $U$ are possible?

\begin{tabular}{rrrrr}
& $W$ & $X$ & $Y$ & $Z$ \\
+ & $W$ & $X$ & $Y$ & $Z$ \\
\hline & $W$ & $U$ & $Y$ & $V$
\end{tabular}

Answer: 3


King Qi and Tian Ji are competing in a horse race. Tian Ji's top horse is better than King Qi's middle horse, worse than King Qi's top horse; Tian Ji's middle horse is better than King Qi's bottom horse, worse than King Qi's middle horse; Tian Ji's bottom horse is worse than King Qi's bottom horse. Now, each side sends one top, one middle, and one bottom horse, forming 3 groups for separate races. The side that wins 2 or more races wins. If both sides do not know the order of the opponent's horses, the probability of Tian Ji winning is ____; if it is known that Tian Ji's top horse and King Qi's middle horse are in the same group, the probability of Tian Ji winning is ____.

Answer: \frac{1}{2}


A church rings its bells every 15 minutes, the school rings its bells every 20 minutes and the day care center rings its bells every 25 minutes. If they all ring their bells at noon on the same day, at what time will they next all ring their bells together?  (Answer in the form AB:CD without am or pm, such as 08:00)

Answer: 05\!:\!00


A block of iron solidifies from molten iron, and its volume reduces by $\frac{1}{34}$. Then, if this block of iron melts back into molten iron (with no loss in volume), by how much does its volume increase?

Answer: \frac{1}{33}


Given the polar coordinate system with the origin $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis. Point $A(4, \frac{5\pi}{4})$ is known. The polar coordinate equation of curve $E$ is $ρ=ρcos^2θ+\sqrt{2}acosθ (a > 0)$. A perpendicular line $l$ is drawn through point $A$ intersecting curve $E$ at points $B$ and $C$ when $θ=\frac{3\pi}{4} (\rho \in R)$.
$(1)$ Write down the rectangular coordinate equations of curve $E$ and line $l$.
$(2)$ If $|AB|$, $|BC|$, and $|AC|$ form a geometric progression, find the value of the real number $a$.

Answer: a = 1 + \sqrt{5}


Let  $x_1,\ldots,x_n$  be a sequence with each term equal to  $0$  or  $1$ . Form a triangle as follows: its first row is  $x_1,\ldots,x_n$  and if a row if  $a_1, a_2, \ldots, a_m$ , then the next row is  $a_1 + a_2, a_2 + a_3, \ldots, a_{m-1} + a_m$  where the addition is performed modulo  $2$  (so  $1+1=0$ ). For example, starting from  $1$ ,  $0$ ,  $1$ ,  $0$ , the second row is  $1$ ,  $1$ ,  $1$ , the third one is  $0$ ,  $0$  and the fourth one is  $0$ .

A sequence is called good it is the same as the sequence formed by taking the last element of each row, starting from the last row (so in the above example, the sequence is  $1010$  and the corresponding sequence from last terms is  $0010$  and they are not equal in this case). How many possibilities are there for the sequence formed by taking the first element of each row, starting from the last row, which arise from a good sequence?

Answer: 2^{\lceiln/2\rceil}


In a triangle, let \( r \) and \( R \) be the radii of the inscribed and circumscribed circles, respectively. Prove that \( R \geqslant 2r \). When does equality hold?

Answer: R\ge2r


Define a sequence  $(a_n)$  by  $a_0 =0$  and  $a_n = 1 +\sin(a_{n-1}-1)$  for  $n\geq 1$ . Evaluate  $$ \lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^{n} a_k. $$

Answer: 1


Given triangle ABC, where a, b, and c are the sides opposite to angles A, B, and C respectively, sin(2C - $\frac {π}{2}$) = $\frac {1}{2}$, and a2 + b2 < c2.
(1) Find the measure of angle C.
(2) Find the value of $\frac {a + b}{c}$.

Answer: \frac {2 \sqrt{3}}{3}


The differentiable functions \( f \) and \( g \), defined on the interval \([0 ; 1]\), are such that the function \( f(0) = f(1) = 1 \) and the function \( 19 f^{\prime} g + 93 f g^{\prime} \) is nonnegative. Prove that \( g(1) \geq g(0) \).

Answer: (1)\ge(0)


Find the sum of digits of all the numbers in the sequence $1,2,3,4,\cdots ,10000$.
$\text{(A) } 180001\quad \text{(B) } 154756\quad \text{(C) } 45001\quad \text{(D) } 154755\quad \text{(E) } 270001$

Answer: (A)180001


Andrea flips a fair coin repeatedly, continuing until she either flips two heads in a row (the sequence \(HH\)) or flips tails followed by heads (the sequence \(TH\)). What is the probability that she will stop after flipping \(HH\)?

Answer: \frac{1}{4}


Let \( d = \overline{xyz} \) be a three-digit number that cannot be divisible by 10. If the sum of \( \overline{xyz} \) and \( \overline{zyx} \) is divisible by \( c \), find the largest possible value of this integer \( d \).

Answer: 979


Two cyclists started a trip at the same time from the same location. They traveled the same route and returned together. Both rested along the way. The first cyclist rode twice as long as the second cyclist rested. The second cyclist rode four times as long as the first cyclist rested. Who rides their bicycle faster and by how many times?

Answer: 1.5


As shown in the diagram, the area of parallelogram \(ABCD\) is 60. The ratio of the areas of \(\triangle ADE\) and \(\triangle AEB\) is 2:3. Find the area of \(\triangle BEF\).

Answer: 12


I have two cents and Bill has \( n \) cents. Bill wants to buy some pencils, which come in two different packages. One package of pencils costs 6 cents for 7 pencils, and the other package of pencils costs a dime for a dozen pencils (i.e., 10 cents for 12 pencils). Bill notes that he can spend all \( n \) of his cents on some combination of pencil packages to get \( P \) pencils. However, if I give my two cents to Bill, he then notes that he can instead spend all \( n+2 \) of his cents on some combination of pencil packages to get fewer than \( P \) pencils. What is the smallest value of \( n \) for which this is possible?

Note: Both times Bill must spend all of his cents on pencil packages, i.e., have zero cents after either purchase.

Answer: 100


Three cones with the vertex $A$ touch each other externally, where the first two cones are identical, and the third cone has a vertex angle of $\frac{\pi}{2}$. All the cones are also tangent to a plane passing through the point $A$ and lie on one side of it. Find the vertex angle of the first two cones. (The vertex angle of a cone is defined as the angle between its generatrices in an axial cross-section.)

Answer: 2\arctan\frac{4}{5}


When drawing a histogram of the lifespans of 1000 people, if the class interval is uniformly 20, and the height of the vertical axis for the age range 60 to 80 years is 0.03, calculate the number of people aged 60 to 80.

Answer: 600


In the expression $1 \square 2 \square 3 \square 6 \square 12$, where $\square$ represents either a '+' or '-', how many different natural number results can be obtained?

Answer: 8


For positive integers $n$, let $c_{n}$ be the smallest positive integer for which $n^{c_{n}}-1$ is divisible by 210, if such a positive integer exists, and $c_{n}=0$ otherwise. What is $c_{1}+c_{2}+\cdots+c_{210}$?

Answer: 329


A museum is organizing a quiz where participants must guess the year a certain artifact was created. Clues given: the year uses each of the digits: 1, 2, 2, 5, 5, 9 exactly once, and the year must start with a prime digit.

How many different possible years could be guessed based on these clues?

Answer: 120


Determine all functions \( f \) defined on the set of positive real numbers that satisfy the equation \( f(x) + 2 f\left(\frac{1}{x}\right) = 3x + 6 \) for all \( x \) in their domain.

Answer: f(x)=\frac{2}{x}-x+2


Given several numbers, one of them,  $a$ , is chosen and replaced by the three numbers  $\frac{a}{3}, \frac{a}{3}, \frac{a}{3}$ . This process is repeated with the new set of numbers, and so on. Originally, there are  $1000$  ones, and we apply the process several times. A number  $m$  is called *good* if there are  $m$  or more numbers that are the same after each iteration, no matter how many or what operations are performed. Find the largest possible good number.

Answer: 667


In triangle  $ABC$ ,  $D$  is a point on  $AB$  between  $A$  and  $B$ ,  $E$  is a point on  $AC$  between  $A$  and  $C$ , and  $F$  is a point on  $BC$  between  $B$  and  $C$  such that  $AF$ ,  $BE$ , and  $CD$  all meet inside  $\triangle ABC$  at a point  $G$ . Given that the area of  $\triangle ABC$  is  $15$ , the area of  $\triangle ABE$  is  $5$ , and the area of  $\triangle ACD$  is  $10$ , compute the area of  $\triangle ABF$ .

Answer: 3


Let \(A \cup B \cup C = \{1, 2, 3, 4, 5, 6\}\), and \(A \cap B = \{1, 2\}\). Additionally, \(\{1, 2, 3, 4\} \subseteq B \cup C\). Determine the number of distinct groups \((A, B, C)\) that meet these conditions. (Note: The order of \(A\), \(B\), and \(C\) matters, so different orderings are considered different groups.)

Answer: 1600


Given $k > 1$, and non-negative real numbers $a$, $b$, and $c$ such that $a \leq k c$, $b \leq k c$, and $a b \leq c^2$, prove that $a + b \leq \left(k + \frac{1}{k}\right) c$.

Answer: b\le(k+\frac{1}{k})


The length of the midsegment $E F$ of trapezoid $A B C D$ is 15 cm. Given that $\angle A B C = \angle A E F = 90^{\circ}$, and $G$ is a point on $E F$, if the area of triangle $A B G$ is $\frac{1}{5}$ of the area of trapezoid $A B C D$, what is the length of $E G$ in centimeters?

Answer: 6


In the Cartesian coordinate system Oxyz, given points A(2, 0, 0), B(2, 2, 0), C(0, 2, 0), and D(1, 1, $\sqrt{2}$), calculate the relationship between the areas of the orthogonal projections of the tetrahedron DABC onto the xOy, yOz, and zOx coordinate planes.

Answer: \sqrt{2}


Huahua is writing letters to Yuanyuan with a pen. When she finishes the 3rd pen refill, she is working on the 4th letter; when she finishes the 5th letter, the 4th pen refill is not yet used up. If Huahua uses the same amount of ink for each letter, how many pen refills does she need to write 16 letters?

Answer: 13


In triangle \( \triangle ABC \), \(\sin A : \sin B : \sin C = 2 : 3 : 4\). Determine \(\angle ABC\) (express the result using inverse trigonometric functions).

Answer: \arccos\frac{11}{16}


Given three non-zero vectors $\bar{a}, \bar{b}, \bar{c}$, each pair of which is non-collinear. Find their sum, if $(\bar{a}+\bar{b}) \| \bar{c}$ and $(\bar{b}+\bar{c}) \| \bar{a}$.

Answer: \overline{0}


Let  $a$ ,  $b$  be positive integers such that  $b^n+n$  is a multiple of  $a^n+n$  for all positive integers  $n$ . Prove that  $a=b$ .

*Proposed by Mohsen Jamali, Iran*

Answer: b


If $x=t^{1/(t-1)}$ and  $y=t^{t/(t-1)},t>0,t \ne 1$, a relation between $x$ and $y$ is:
$\text{(A) } y^x=x^{1/y}\quad \text{(B) } y^{1/x}=x^{y}\quad \text{(C) } y^x=x^y\quad \text{(D) } x^x=y^y\quad \text{(E) none of these}$

Answer: \textbf{(C)}y^x=x^y


The Intermediate Maths Challenge has 25 questions with the following scoring rules:
5 marks are awarded for each correct answer to Questions 1-15;
6 marks are awarded for each correct answer to Questions 16-25;
Each incorrect answer to Questions 16-20 loses 1 mark;
Each incorrect answer to Questions 21-25 loses 2 marks.
Where no answer is given 0 marks are scored.
Fiona scored 80 marks in total. What possible answers are there to the number of questions Fiona answered correctly?

Answer: 16


At a conference, there are several scientists, some of whom speak English, some French, and some German. The organizers noticed the following distribution:
- Among those who speak English, exactly \( \frac{1}{5} \) speak French, and exactly \( \frac{1}{3} \) speak German.
- Among those who speak French, exactly \( \frac{1}{8} \) speak English, and exactly \( \frac{1}{2} \) speak German.
- Among those who speak German, exactly \( \frac{1}{6} \) speak English.

What fraction of those who speak German also speak French?

Answer: \frac{2}{5}


Let \( n \) be a fixed positive integer. Find the maximum possible value of
\[
\sum_{1 \leq r < s \leq 2n} (s - r - n) x_r x_s,
\]
where \( -1 \leq x_i \leq 1 \) for all \( i = 1, 2, \ldots, 2n \).

Answer: n(n-1)


Carlos took $70\%$ of a whole pie. Maria took one third of the remainder. What portion of the whole pie was left?
$\textbf{(A)}\ 10\%\qquad\textbf{(B)}\ 15\%\qquad\textbf{(C)}\ 20\%\qquad\textbf{(D)}\ 30\%\qquad\textbf{(E)}\ 35\%$

Answer: \textbf{(C)}\20


Which of the following is closest to the product $(.48017)(.48017)(.48017)$?
$\text{(A)}\ 0.011 \qquad \text{(B)}\ 0.110 \qquad \text{(C)}\ 1.10 \qquad \text{(D)}\ 11.0 \qquad \text{(E)}\ 110$

Answer: \textbf{(B)}\0.110


Determine how many roots of the equation

$$
4 \sin 2x + 3 \cos 2x - 2 \sin x - 4 \cos x + 1 = 0
$$

are located in the interval \(\left[10^{2014!} \pi, 10^{2014!+2015} \pi\right]\). In the answer, write down the sum of all the digits of the found number.

Answer: 18135


Two congruent right circular cones each with base radius $3$ and height $8$ have the axes of symmetry that intersect at right angles at a point in the interior of the cones a distance $3$ from the base of each cone. A sphere with radius $r$ lies withing both cones. The maximum possible value of $r^2$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Answer: 298


Given that in triangle $\triangle ABC$, $\angle A=120^{\circ}$, let $\alpha=\frac{\overrightarrow{BA}}{|\overrightarrow{BA}| \cos A}+\frac{\overrightarrow{BC}}{|\overrightarrow{BC}| \cos C}$ and $\beta=\frac{\overrightarrow{CA}}{|\overrightarrow{CA}| \cos A}+\frac{\overrightarrow{CB}}{|\overrightarrow{CB}| \cos B}$. Find the angle between vectors $\alpha$ and $\beta$.

Answer: 60


Given that point $P$ is any point on the curve $(x-1)^2+(y-2)^2=9$ with $y \geq 2$, find the minimum value of $x+ \sqrt {3}y$.

Answer: 2\sqrt{3} - 2


Prove that if the midline of a quadrilateral passes through the intersection point of the diagonals and is bisected by this point, then the quadrilateral is a parallelogram.

Answer: Thequadrilateralisparallelogram.


George, Jeff, Brian, and Travis decide to play a game of hot potato. They begin by arranging themselves clockwise in a circle in that order. George and Jeff both start with a hot potato. On his turn, a player gives a hot potato (if he has one) to a randomly chosen player among the other three (if a player has two hot potatoes on his turn, he only passes one). If George goes first, and play proceeds clockwise, what is the probability that Travis has a hot potato after each player takes one turn?

Answer: \frac{5}{27}


Consider a circle centered at  $O$ . Parallel chords  $AB$  of length  $8$  and  $CD$  of length  $10$  are of distance  $2$  apart such that  $AC < AD$ . We can write  $\tan \angle BOD =\frac{a}{b}$  , where  $a, b$  are positive integers such that gcd  $(a, b) = 1$ . Compute  $a + b$ .

Answer: 113


Each of two baskets contains white and black balls such that the total number of balls in both baskets is 25. One ball is randomly drawn from each basket. It is known that the probability that both drawn balls are white is 0.54. Find the probability that both drawn balls are black.

Answer: 0.04


$M$ is the midpoint of side $BC$ in $\triangle ABC$, and $\angle MAC = 15^\circ$. Find the maximum value of $\angle B$.

Answer: 105


For each real number $x$, let
\[
f(x) = \sum_{n\in S_x} \frac{1}{2^n},
\]
where $S_x$ is the set of positive integers $n$ for which $\lfloor nx \rfloor$ is even. What is the largest real number $L$ such that $f(x) \geq L$ for all $x \in [0,1)$? (As usual, $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$.)

Answer: 4/7


The equation $\frac{2x^2}{x-1}-\frac{2x+7}{3}+\frac{4-6x}{x-1}+1=0$ can be transformed by eliminating fractions to the equation $x^2-5x+4=0$.
The roots of the latter equation are $4$ and $1$. Then the roots of the first equation are:
$\textbf{(A)}\ 4 \text{ and }1 \qquad \textbf{(B)}\ \text{only }1 \qquad \textbf{(C)}\ \text{only }4 \qquad \textbf{(D)}\ \text{neither 4 nor 1}\qquad\textbf{(E)}\ \text{4 and some other root}$

Answer: \textbf{(C)}\only4


Given the numbers 1, 2, 3, 4, find the probability that $\frac{a}{b}$ is not an integer, where $a$ and $b$ are randomly selected numbers from the set $\{1, 2, 3, 4\}$.

Answer: \frac{2}{3}


Quadrilateral \( KLMN \) is inscribed in a circle of radius \( R \). Given \( LM = n \) and diagonals \( KM \) and \( LN \) are perpendicular, find \( KN \).

Answer: \sqrt{4R^2-n^2}


In rectangle \(ABCD\), \(AB = 2\) and \(AD = 1\), point \(P\) is a moving point on side \(DC\) (including points \(D\) and \(C\)), and point \(Q\) is a moving point on the extension line of \(CB\) (including point \(B\)), such that \(|\overrightarrow{DP}| = |\overrightarrow{BQ}|\). Determine the minimum value of the dot product \(\overrightarrow{PA} \cdot \overrightarrow{PQ}\).

Answer: 3/4


Pentagon \(ABCDE\) is described around a circle \(\Omega\). Side \(BC\) touches the circle at point \(K\). It is known that \(AB = BC = CD\).

Prove that \(\angle EKB = 90^{\circ}\).

Answer: 90


A courier departs from point $A$ towards point $B$, and at the same time, another courier departs from point $B$ towards point $A$. They meet after eight hours. How many kilometers does each courier travel per hour if the distance between $A$ and $B$ is 176 km, and one courier takes 1 hour longer than the other to travel 60 km?

Answer: 12


A power boat and a raft both left dock $A$ on a river and headed downstream. The raft drifted at the speed of the river current. The power boat maintained a constant speed with respect to the river. The power boat reached dock $B$ downriver, then immediately turned and traveled back upriver. It eventually met the raft on the river 9 hours after leaving dock $A.$ How many hours did it take the power boat to go from $A$ to $B$?
$\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 3.5 \qquad \textbf{(C)}\  4 \qquad \textbf{(D)}\ 4.5 \qquad \textbf{(E)}\ 5$

Answer: \textbf{D}


155 birds are sitting on a circle \( C \). If the arc \( \overparen{P_{i} P_{j}} \leqslant 10^\circ \), then \( P_{i} \) and \( P_{j} \) are considered to be a pair of mutually visible birds. Find the minimum number of pairs of mutually visible birds (assuming multiple birds can be at the same position).

Answer: 270


Armen paid \$190 to buy movie tickets for a group of \( t \) people, consisting of some adults and some children. Movie tickets cost \$5 for children and \$9 for adults. How many children's tickets did he buy?

Answer: 20


Compute  $e^{\pi}+\pi^e$ . If your answer is  $A$  and the correct answer is  $C$ , then your score on this problem will be  $\frac{4}{\pi}\arctan\left(\frac{1}{\left|C-A\right|}\right)$  (note that the closer you are to the right answer, the higher your score is).

*2017 CCA Math Bonanza Lightning Round #5.2*

Answer: 45.5999


Two spheres are inscribed in a dihedral angle such that they touch each other. The radius of one sphere is 4 times that of the other, and the line connecting the centers of the spheres forms an angle of \(60^\circ\) with the edge of the dihedral angle. Find the measure of the dihedral angle. Provide the cosine of this angle, rounded to two decimal places if necessary.

Answer: 0.04


On the edges \( DA \), \( DB \), and \( DC \) of pyramid \( ABCD \), points \( K \), \( L \), and \( M \) are taken respectively, such that \( DK = \frac{1}{2} DA \), \( DL = \frac{2}{5} DB \), and \( DM = \frac{3}{4} DC \). Point \( G \) is the centroid of triangle \( ABC \). In what ratio does plane \( KLM \) divide segment \( DG \)?

Answer: \frac{18}{17}


Let  $T_k = \frac{k(k+1)}{2}$  be the  $k$ -th triangular number. The infinite series  $$ \sum_{k=4}^{\infty}\frac{1}{(T_{k-1} - 1)(Tk - 1)(T_{k+1} - 1)} $$  has the value  $\frac{m}{n}$  , where  $m$  and  $n$  are relatively prime positive integers. Find  $m + n$ .

Answer: 451


Given that $α \in (0,π)$, if $\sin α + \cos α = \frac{\sqrt{3}}{3}$, find the value of $\cos^2 α - \sin^2 α$.

Answer: \frac{\sqrt{5}}{3}


How many solutions does the equation $\tan x = \tan (\tan x)$ have in the interval $0 \le x \le \tan^{-1} 1000$? Assume $\tan \theta > \theta$ for $0 < \theta < \frac{\pi}{2}$.

Answer: 318


Suppose the three sides of a triangular field are all integers, and its area is equal to the perimeter. What is the largest possible area of the field?

Answer: 60


On the sides \( AB \) and \( BC \) of an equilateral triangle \( ABC \), points \( L \) and \( K \) are marked, respectively. \( M \) is the intersection point of segments \( AK \) and \( CL \). It is known that the area of triangle \( AMC \) is equal to the area of quadrilateral \( LBKM \). Find the angle \( \angle AMC \).

Answer: 120


What time is it 2017 minutes after 20:17?

Answer: 05:54


The range of the function \( f(x) = \sin^4(x) \tan(x) + \cos^4(x) \cot(x) \) is ____.

Answer: (-\infty,-\frac{1}{2}]\cup[\frac{1}{2},+\infty)


Answer: \frac {1}{4030}


Point \( X \) is located on the diameter \( AB \) of a circle with radius \( R \). Points \( K \) and \( N \) lie on the circle in the same half-plane with respect to \( AB \), and \(\angle KXA = \angle NXB = 60^\circ\). Find the length of the segment \( KN \).

Answer: KN=R


In $\Delta ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$. If $a^{2}=b^{2}+4bc\sin A$ and $\tan A \cdot \tan B=2$, then $\tan B-\tan A=$ ______.

Answer: -8


In a right triangle \(ABC\), the altitude dropped to the hypotenuse \(AB\) is \(a\), and the bisector of the right angle is \(b\). Find the area of triangle \(ABC\).

Answer: \frac{^2b^2}{2a^2-b^2}


Let the polynomial \( p(x) = x^{3} + a x^{2} + b x + c \) (where \( a, b, c \in \mathbf{Z} \)) have a root that equals the product of the other two roots. Prove that \( 2 p(-1) \) is divisible by \( p(1) + p(-1) - 2[1 + p(0)] \).

Answer: 2p(-1)isdivisiblep(1)+p(-1)-2[1+p(0)]


We have seen in Problem 1109 that if a hexagon is both inscribed and circumscribed and symmetric about one of its diagonals, the relationship between the radius \( R \) of the circumscribed circle, the radius \( r \) of the inscribed circle, and the distance \( c \) between the centers of the circles is given by:

$$
3\left(R^{2}-c^{2}\right)^{4}-4 r^{2}\left(R^{2}-c^{2}\right)^{2}\left(R^{2}+c^{2}\right)-16 R^{2} c^{2} r^{4}=0
$$

Show that this relationship also holds for hexagons that are both inscribed and circumscribed and are symmetric about the perpendicular bisector of one of their sides.

Answer: 3(R^{2}-^{2})^{4}-4r^{2}(R^{2}-^{2})^{2}(R^{2}+^{2})-16R^{2}^{2}r^{4}=0


In an exam, there are 6 questions, and each question is solved by exactly 100 people. Each pair of examinees has at least one question that neither of them has solved. What is the minimum number of participants in the exam?

Answer: 200


For any quadratic polynomial $f(x)$, do there exist distinct numbers $a, b, c, d$ such that $f(a) = b$, $f(b) = c$, $f(c) = d$, and $f(d) = a$?

Answer: No


Given that \( z \in \mathbf{C} \) and \(\left|z+\frac{1}{z}\right|=1\), prove that \(\frac{\sqrt{5}-1}{2} \leqslant|z| \leqslant \frac{\sqrt{5}+1}{2}\).

Answer: \frac{\sqrt{5}-1}{2}\leslant|z|\leslant\frac{\sqrt{5}+1}{2}


Given an acute triangle \( \triangle ABC \) with circumcircle \( \odot O \). The feet of the altitudes from vertices \( B \), \( C \), and \( A \) to the opposite sides \( BC \), \( CA \), and \( AB \) are \( D \), \( E \), and \( F \) respectively. The line \( EF \) intersects the arcs \( \overparen{AB} \) and \( \overparen{AC} \) of \( \odot O \) not containing \( C \) and \( B \) at points \( G \) and \( H \) respectively. The line \( DF \) intersects \( BG \) and \( BH \) at points \( K \) and \( L \) respectively. The line \( DE \) intersects \( CG \) and \( CH \) at points \( M \) and \( N \) respectively. Prove that \( K \), \( L \), \( M \), and \( N \) are concyclic, and the diameter of the circle passing through these four points is \( \sqrt{2(b^2 + c^2 - a^2)} \), where \( BC = a \), \( CA = b \), and \( AB = c \).

Answer: \sqrt{2(b^2+^2-^2)}


In space, there are \( n \) (\( n \geqslant 3 \)) planes, where any three planes do not share a common perpendicular plane. There are the following four assertions:
(1) No two planes are parallel to each other;
(2) No three planes intersect in a single line;
(3) Any two intersection lines between the planes are not parallel;
(4) Each intersection line between the planes intersects with \( n-2 \) other planes.

Determine how many of these assertions are correct.

Answer: 4


Calculate the number of multiples of 4 that are between 100 and 500.

Answer: 99


Given that \( A, B, \) and \( C \) are points on the parabola \( y^{2} = x \), and \((2, \frac{4}{3})\) is the centroid of triangle \( \triangle ABC \). What is the maximum area of \( \triangle ABC \)?

Answer: \frac{\sqrt{3}}{9}


Let \( n \) be a natural number, and \( a \) and \( b \) be positive real numbers such that \( a+b=2 \). Find the minimum value of \( \frac{1}{1+a^{n}}+\frac{1}{1+b^{n}} \).

Answer: 1


Given $f(x)=\cos x(\sqrt{3}\sin x-\cos x)+\frac{3}{2}$.
$(1)$ Find the interval on which $f(x)$ is monotonically decreasing on $[0,\pi]$.
$(2)$ If $f(\alpha)=\frac{2}{5}$ and $\alpha\in(\frac{\pi}{3},\frac{5\pi}{6})$, find the value of $\sin 2\alpha$.

Answer: \frac{-3\sqrt{3} - 4}{10}


Two congruent parallelograms have equal sides parallel to each other. In both parallelograms, we mark the vertices, the midpoints of the sides, and the intersection point of the diagonals. We choose one point from the marked points of the first parallelogram and another point from the marked points of the second parallelogram in every possible way, and construct the midpoint of the segment connecting the two chosen points.

How many different midpoints do we get in this way?

Answer: 25


Answer: [0,1]


Given that \( n \) is a positive integer and \( n \leq 144 \), ten questions of the type "Is \( n \) smaller than \( a \)?" are allowed. The answers are provided with a delay such that the answer to the \( i \)-th question is given only after the \((i+1)\)-st question is asked for \( i = 1, 2, \ldots, 9 \). The answer to the tenth question is given immediately after it is asked. Find a strategy for identifying \( n \).

Answer: 144


In the parallelepiped \(ABCD A_1B_1C_1D_1\), points \(K\) and \(M\) are taken on the lines \(AC\) and \(BA_1\) respectively, with \(KM \parallel DB_1\). Find the ratio \(KM: DB_1\).

Answer: 1:3


When $x=$____, the expressions $\frac{x-1}{2}$ and $\frac{x-2}{3}$ are opposite in sign.

Answer: \frac{7}{5}


Triangle $PQR$ is a right triangle with legs $PQ$ and $PR$. Points $U$ and $V$ are on legs $PQ$ and $PR$, respectively so that $PU:UQ = PV:VR = 1:3$. If $QU = 18$ units, and $RV = 45$ units, what is the length of hypotenuse $PQ$? Express your answer in simplest radical form.

Answer: 12\sqrt{29}


Find the number of positive integers \( n \) that satisfy
\[
(n - 1)(n - 3)(n - 5) \dotsm (n - 99) < 0.
\]

Answer: 25


Answer: 94


Anna and Berta are playing a game where they take turns removing marbles from the table. Anna makes the first move. If at the beginning of a turn there are \( n \geq 1 \) marbles on the table, then the player whose turn it is takes \( k \) marbles, where \( k \geq 1 \) is either an even number with \( k \leq \frac{n}{2} \) or an odd number with \( \frac{n}{2} \leq k \leq n \). A player wins the game if she takes the last marble from the table.

Determine the smallest number \( N \geq 100000 \) such that Berta can force a win if there are initially exactly \( N \) marbles on the table.

Answer: 131070


Four people sit around a circular table, and each person will roll a standard six-sided die. What is the probability that no two people sitting next to each other will roll the same number after they each roll the die once? Express your answer as a common fraction.

Answer: \frac{35}{72}


When the natural numbers $P$ and $P'$, with $P>P'$, are divided by the natural number $D$, the remainders are $R$ and $R'$, respectively. When $PP'$ and $RR'$ are divided by $D$, the remainders are $r$ and $r'$, respectively. Then:
$\text{(A) } r>r' \text{  always}\quad \text{(B) } r<r' \text{  always}\quad\\ \text{(C) } r>r' \text{  sometimes and } r<r' \text{  sometimes}\quad\\ \text{(D) } r>r' \text{  sometimes and } r=r' \text{  sometimes}\quad\\ \text{(E) } r=r' \text{  always}$

Answer: \textbf{(E)}r=r'


In a right trapezoid, the shorter diagonal is equal to the longer leg.

Find the longer diagonal if the longer leg is equal to \(a\) and the shorter base is equal to \(b\).

Answer: \sqrt{^2+3b^2}


What fraction of the area of a regular hexagon of side length 1 is within distance $\frac{1}{2}$ of at least one of the vertices?

Answer: \pi \sqrt{3} / 9


Find the positive integer $N$, such that numbers $N$ and $N^2$ end in the same sequence of four digits $abcd$ where $a$ is not zero, under the modulus $8000$.

Answer: 625


Circles $P$, $Q$, and $R$ are externally tangent to each other and internally tangent to circle $S$. Circles $Q$ and $R$ are congruent. Circle $P$ has radius 2 and passes through the center of $S$. What is the radius of circle $Q$?

Answer: \frac{16}{9}


What is the minimum number of factors that need to be crossed out from the number \( 99! = 1 \cdot 2 \cdot \ldots \cdot 99 \) so that the product of the remaining factors ends in \( 2 \)?

Answer: 20


A circle intersects the  $y$ -axis at two points  $(0, a)$  and  $(0, b)$  and is tangent to the line  $x+100y = 100$  at  $(100, 0)$ . Compute the sum of all possible values of  $ab - a - b$ .

Answer: 10000


Two runners started simultaneously in the same direction from the same point on a circular track. The first runner, moving ahead, caught up with the second runner at the moment when the second runner had only run half a lap. From that moment, the second runner doubled their speed. Will the first runner catch up with the second runner again? If so, how many laps will the second runner complete by that time?

Answer: 2.5


The trinomial \(a x^{2}+b x+c\) is a perfect square for all integer values of \(x\). Prove that \(a x^{2}+b x+c = (d x + e)^{2}\).

Answer: ^2++(+e)^2


Two sides of a triangle are $8 \mathrm{dm}$ and $5 \mathrm{dm}$; the angle opposite to the first side is twice as large as the angle opposite to the second side. What is the length of the third side of the triangle?

Answer: 7.8


Point $P$ lies on the diagonal $AC$ of square $ABCD$ with $AP > CP$. Let $O_{1}$ and $O_{2}$ be the circumcenters of triangles $ABP$ and $CDP$ respectively. Given that $AB = 10$ and $\angle O_{1}PO_{2} = 90^{\circ}$, then $AP = \sqrt{c} + \sqrt{d}$, where $c$ and $d$ are positive integers. Find $c + d$.

Answer: 100


Given a plane \( S \) and a triangle \( ABC \) that does not lie on \( S \), is there a point \( P \) in space such that projecting \( ABC \) onto \( S \) from \( P \) results in an isosceles right triangle?

Answer: 0


There is a four-digit number. By rearranging the digits of this four-digit number (none of which are zero), the largest possible new number is greater than the original number by 5562, and the smallest possible new number is less than the original number by 2700. What is the original four-digit number?

Answer: 4179


Vanya wrote the number 1 on the board, and then several more numbers. Each time Vanya writes a new number, Mitya calculates the median of the existing set of numbers and writes it down in his notebook. At a certain moment, the numbers in Mitya's notebook are: $1 ; 2 ; 3 ; 2.5 ; 3 ; 2.5 ; 2 ; 2 ; 2.5$.

a) What is the fourth number written on the board?

b) What is the eighth number written on the board?

Answer: 2


A cylindrical vessel is filled with water. The height of the water layer is \( h \). Water flows out of an orifice (its size can be neglected) located at a height \( z \). At what \( z \) will the range of the water stream be the greatest?

Answer: \frac{}{2}


Given an arbitrary tetrahedron, prove that there exists a triangle whose side lengths are numerically equal to the products of the lengths of opposite edges of the tetrahedron. Let $S$ be the area of this triangle, $V$ be the volume of the tetrahedron, and $R$ be the radius of the circumscribed sphere. Then the equality $S = 6VR$ holds.

Answer: 6VR


A woman buys a property for $150,000 with a goal to achieve a $7\%$ annual return on her investment. She sets aside $15\%$ of each month's rent for maintenance costs, and pays property taxes at $0.75\%$ of the property's value each year. Calculate the monthly rent she needs to charge to meet her financial goals.

Answer: 1139.71


Among five externally identical coins, 3 are genuine and 2 are counterfeit, with the counterfeits weighing the same but it is unknown whether they are heavier or lighter than the genuine coins. How can you find at least one genuine coin with the fewest possible weighings?

Answer: 2


 $99$  identical balls lie on a table.  $50$  balls are made of copper, and  $49$  balls are made of zinc. The assistant numbered the balls. Once spectrometer test is applied to  $2$  balls and allows to determine whether they are made of the same metal or not. However, the results of the test can be obtained only the next day. What minimum number of tests is required to determine the material of each ball if all the tests should be performed today?

*Proposed by N. Vlasova, S. Berlov*

Answer: 98


Chicks hatch on the night from Sunday to Monday. For two weeks, a chick sits with its beak open, during the third week it silently grows feathers, and during the fourth week it flies out of the nest. Last week, there were 20 chicks in the nest sitting with their beaks open, and 14 growing feathers, while this week 15 chicks were sitting with their beaks open and 11 were growing feathers.

a) How many chicks were sitting with their beaks open two weeks ago?

b) How many chicks will be growing feathers next week?

Record the product of these numbers as the answer.

Answer: 165


For a positive integer  $n$ , let  $s(n)$  and  $c(n)$  be the number of divisors of  $n$  that are perfect squares and perfect cubes respectively. A positive integer  $n$  is called fair if  $s(n)=c(n)>1$ . Find the number of fair integers less than  $100$ .

Answer: 7


Zeus starts at the origin \((0,0)\) and can make repeated moves of one unit either up, down, left or right, but cannot make a move in the same direction twice in a row. What is the smallest number of moves that he can make to get to the point \((1056,1007)\)?

Answer: 2111


Inside a circle, there is a convex pentagon (its vertices can lie either inside or on the circle). Prove that at least one of its sides is not longer than a side of a regular pentagon inscribed in this circle.

Answer: 1


A dandelion blooms in the morning, stays yellow for that day and the next day, then turns white on the morning of the third day, and disperses by the evening. Yesterday, in the meadow, there were 20 yellow and 14 white dandelions during the day, and today there are 15 yellow and 11 white dandelions. How many yellow dandelions were in the meadow the day before yesterday? List all possible answers and prove that there are no other solutions.

Answer: 25


Three sectors of a circle are removed from a regular hexagon to form the shaded shape shown. Each sector has a perimeter of 18 mm. What is the perimeter, in mm, of the shaded shape formed?

Answer: 54


Let $AD, BE$, and $CF$ be segments sharing a common midpoint, with $AB < AE$ and $BC < BF$. Suppose that each pair of segments forms a $60^{\circ}$ angle, and that $AD=7, BE=10$, and $CF=18$. Let $K$ denote the sum of the areas of the six triangles $\triangle ABC, \triangle BCD, \triangle CDE, \triangle DEF, \triangle EFA$, and $\triangle FAB$. Compute $K \sqrt{3}$.

Answer: 141


The landlord of an apartment building needs to purchase enough digits to label all of the apartments from 100 through 125 on the first floor and 200 through 225 on the second floor. The digits can only be purchased in a package that contains one of each digit 0 through 9. How many packages must the landlord purchase?

Answer: 52


There are 35 apples in a basket: 8 green, 11 yellow, and 16 red. Alena sequentially removes one apple at a time from the basket. If at any point she has removed fewer green apples than yellow apples, and fewer yellow apples than red apples, she will stop removing more apples from the basket.

(a) What is the maximum number of yellow apples Alena can remove from the basket?

(b) What is the maximum number of apples Alena can remove from the basket?

Answer: 33


It takes person A 1 minute and 20 seconds to complete a lap, and person B meets person A every 30 seconds. Determine the time it takes for person B to complete a lap.

Answer: 48


Given an ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ $(a>b>0)$ and a hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ sharing the common vertices $A$ and $B$, let $P$ and $Q$ be points on the hyperbola and the ellipse respectively, different from $A$ and $B$, and satisfying the condition:
$$
\overrightarrow{A P}+\overrightarrow{B P}=\lambda(\overrightarrow{A Q}+\overrightarrow{B Q})(\lambda \in \mathbf{R},|\lambda|>1).
$$
Prove:
1. Points $O$, $P$, and $Q$ are collinear.
2. If the slopes of lines $A P$, $B P$, $A Q$, and $B Q$ are $k_{1}$, $k_{2}$, $k_{3}$, and $k_{4}$ respectively, then $k_{1}+k_{2}+k_{3}+k_{4}$ is a constant.

Answer: 0


Let  $a,b,c$  be positive integers such that  $a,b,c,a+b-c,a+c-b,b+c-a,a+b+c$  are  $7$  distinct primes. The sum of two of  $a,b,c$  is  $800$ . If  $d$  be the difference of the largest prime and the least prime among those  $7$  primes, find the maximum value of  $d$ .

Answer: 1594


Answer: 1


Let $Q$ be a $(2n+1) \times (2n+1)$ board. Some of its cells are colored black in such a way that every $2 \times 2$ board of $Q$ has at most $2$ black cells. Find the maximum amount of black cells that the board may have.

Answer: (2n+1)(n+1)


There is an unlimited number of chips in six colors. What is the minimum number of chips that need to be arranged in a row so that for any two different colors in the row, there are two adjacent chips of those colors?

Answer: 18


A classified area is a rectangle of $200 \times 300$ meters. There is one guard at each of the four corners outside the area. An intruder approaches the perimeter of the classified area from the outside, and all the guards run to the intruder by the shortest routes along the outside perimeter (the intruder remains stationary). The total distance covered by three of the guards to reach the intruder is 850 meters. How many meters did the fourth guard run to reach the intruder?

Answer: 150


The four positive integers $a,$ $b,$ $c,$ $d$ satisfy
\[a \times b \times c \times d = 10!.\]Find the smallest possible value of $a + b + c + d.$

Answer: 175


Real numbers  $a, b, c$  are not equal  $0$  and are solution of the system: $\begin{cases} a^2 + a =  b^2  b^2 + b = c^2  c^2  +c = a^2 \end{cases}$ Prove that  $(a - b)(b - c)(c - a) = 1$ .

Answer: (b)()()=1


A \( 5 \times 5 \) grid of squares is filled with integers. Call a rectangle corner-odd if its sides are grid lines and the sum of the integers in its four corners is an odd number. What is the maximum possible number of corner-odd rectangles within the grid?

Note: A rectangle must have four distinct corners to be considered corner-odd; i.e. no \( 1 \times k \) rectangle can be corner-odd for any positive integer \( k \).

Answer: 60


Answer: 2


In a candy store, a salesperson placed 91 candies of several kinds in a row on the counter. It turned out that between any two candies of the same kind, there is an even number of candies. What is the minimum number of kinds of candies that could be?

Answer: 46


(1) Given $\cos(15°+\alpha) = \frac{15}{17}$, with $\alpha \in (0°, 90°)$, find the value of $\sin(15°-\alpha)$.
(2) Given $\cos\alpha = \frac{1}{7}$, $\cos(\alpha-\beta) = \frac{13}{14}$, and $0 < \beta < \alpha < \frac{\pi}{2}$, find the value of $\beta$.

Answer: \frac{\pi}{3}


Let  $a,b,c$  be distinct positive integers such that  $b+c-a$ ,  $c+a-b$  and  $a+b-c$  are all perfect squares. What is the largest possible value of  $a+b+c$  smaller than  $100$  ?

Answer: 91


Answer: (99!)^{100!}\cdot(100!)^{99!}


Let $S$ be the set of points whose coordinates $x,$ $y,$ and $z$ are integers satisfying $0 \leq x \leq 1$, $0 \leq y \leq 3$, and $0 \leq z \leq 5$. Two distinct points are chosen at random from $S$. The probability that the midpoint of the segment connecting them also belongs to $S$ is $\frac{p}{q}$, where $p$ and $q$ are relatively prime. Find $p + q$.

Answer: 52


Find the least positive integer of the form a b a a b a, where a and b are distinct digits, such that the integer can be written as a product of six distinct primes

Answer: 282282


In a sequence of positive integers, an inversion is a pair of positions where the element at the leftmost position is greater than the element at the rightmost position. For example, the sequence 2, 5, 3, 1, 3 has 5 inversions: between the first and the fourth position, between the second position and all the positions to its right, and finally between the third and the fourth position. Among all sequences of positive integers whose elements sum to $n$, what is the maximum possible number of inversions if

a) $n=7$ ?

b) $n=2019$ ?

Note: The sequences of positive integers considered in this problem can have more than 5 elements.

Answer: 509545


A large supermarket has recorded the number of customers during eight holidays (unit: hundreds of people) as $29$, $30$, $38$, $25$, $37$, $40$, $42$, $32$. What is the $75$th percentile of this data set?

Answer: 39


Given a class with 21 students, such that at least two of any three students are friends, determine the largest possible value of k.

Answer: 10


In the ellipse $\dfrac {x^{2}}{36}+ \dfrac {y^{2}}{9}=1$, there are two moving points $M$ and $N$, and $K(2,0)$ is a fixed point. If $\overrightarrow{KM} \cdot \overrightarrow{KN} = 0$, find the minimum value of $\overrightarrow{KM} \cdot \overrightarrow{NM}$.

Answer: \dfrac{23}{3}


Find all pairs of positive integers \(x, y\) which satisfy the equation

\[
2x^2 + 5y^2 = 11(xy - 11).
\]

Answer: (14,27)


From the center \( O \) of the inscribed circle of a right triangle, the half of the hypotenuse that is closer to \( O \) appears at a right angle. What is the ratio of the sides of the triangle?

Answer: 3 : 4 : 5


Find all triplets  $(a, b, c)$  of positive integers, such that  $a+bc, b+ac, c+ab$  are primes and all divide  $(a^2+1)(b^2+1)(c^2+1)$ .

Answer: (1,1,1)


Given that a high school senior year has 12 classes, with exactly 8 classes to be proctored by their own homeroom teachers, find the number of different proctoring arrangements for the math exam.

Answer: 4455


Alex wrote all natural divisors of a natural number \( n \) on the board in ascending order. Dima erased several of the first and several of the last numbers of the resulting sequence so that 151 numbers remained. What is the maximum number of these 151 divisors that could be fifth powers of natural numbers?

Answer: 31


Let the sequence \( a_1, a_2, \cdots \) have the sum of its first \( n \) terms, \( S_n = a_1 + a_2 + \cdots + a_n \), satisfying:
\[ S_1 = 1, \quad S_{n+1} = \frac{(2 + S_n)^2}{4 + S_n} \text{ for } n \geq 1. \]

Prove that for every positive integer \( n \), \( a_n \geq \frac{4}{\sqrt{9n + 7}} \).

Answer: a_n\ge\frac{4}{\sqrt{9n+7}}


Let $A B C D$ be an isosceles trapezoid with $A D=B C=255$ and $A B=128$. Let $M$ be the midpoint of $C D$ and let $N$ be the foot of the perpendicular from $A$ to $C D$. If $\angle M B C=90^{\circ}$, compute $\tan \angle N B M$.

Answer: \frac{120}{353}


Given a right triangle \( \triangle ABC \) with \(\angle C = 90^\circ\) and circumcenter \( O \). Draw a perpendicular from point \( C \) to \( AB \) with foot \( D \). Define circle \( \odot O_1 \) that is tangent to arc \( \overparen{BC} \), segment \( CD \), and segment \( DB \) at points \( E \), \( F \), and \( G \) respectively. Similarly, define circle \( \odot O_2 \) that is tangent to arc \( \overparen{AC} \), segment \( CD \), and segment \( DA \). Let the radii of \( \odot O_1 \) and \( \odot O_2 \) be \( r_1 \) and \( r_2 \) respectively, and let the inradius of \( \triangle ABC \) be \( r \). Prove that \( r = \frac{r_1 + r_2}{2} \).

Answer: \frac{r_1+r_2}{2}


9 pairs of table tennis players participate in a doubles match, their jersey numbers are 1, 2, …, 18. The referee is surprised to find that the sum of the jersey numbers of each pair of players is exactly a perfect square. The player paired with player number 1 is     .

Answer: 15


Let $n$ be a positive integer. Find the number of permutations $a_1$, $a_2$, $\dots a_n$ of the
sequence $1$, $2$, $\dots$ , $n$ satisfying
$$a_1 \le 2a_2\le 3a_3 \le \dots \le na_n$$.

Answer: F_{n+1}


The angle bisectors $\mathrm{AD}$ and $\mathrm{BE}$ of the triangle $\mathrm{ABC}$ intersect at point I. It turns out that the area of triangle $\mathrm{ABI}$ is equal to the area of quadrilateral $\mathrm{CDIE}$. Find the maximum possible value of angle $\mathrm{ACB}$.

Answer: 60


Solve the equation \(16^{x^{2}+y}+16^{y^{2}+x}=1\). Write the value of the variable \(x\) in the answer.

Answer: -0.5


Write the four numbers \(1, 9, 8, 8\) in a row and perform the following operation: for each pair of adjacent numbers, subtract the left number from the right number, then write the difference between these two numbers in between them. This completes one operation. Then, perform the same operation on the new series of 7 numbers. Continue doing this a total of 100 times. Find the sum of the numbers in the final row.

Answer: 726


Masha, while bored during a math lesson, performed the following operation on a certain 2015-digit natural number: she removed the last digit from its decimal notation, multiplied the resulting number by 3, and added twice the digit she removed. She repeated this operation with the resulting number and so on. After many applications of this operation, the resulting numbers stopped changing, and she then stopped.

a) What number did Masha end up with in the end?

b) What is the smallest number that Masha could have started with (indicate its last two digits)?

Answer: 9


From the set of three-digit numbers that do not contain the digits $0,1,2,3,4,5$, several numbers were written down in such a way that no two numbers could be obtained from each other by swapping two adjacent digits. What is the maximum number of such numbers that could have been written?

Answer: 40


If the solution set of the inequality $a x^{2} - |x + 1| + 2a < 0$ with respect to $x$ is an empty set, then the range of values of $a$ is

Answer: [\frac{\sqrt{3}+1}{4},+\infty)


Answer: 0.8


Calculate the length of the curve

$$
y = \frac{1 - e^x - e^{-x}}{2}, \quad 0 \leq x \leq 3
$$

Answer: \sinh3


Two circles with radii \(R_{1}\) and \(R_{2}\) are externally tangent at point \(E\). External tangents \(AB\) and \(CD\) are drawn to these circles. Prove that a circle can be inscribed in quadrilateral \(ABCD\). Find the radius of this inscribed circle.

Answer: \frac{2R_1R_2}{R_1+R_2}


Answer: 51


Determine all positive integers  $n$  such that the following statement holds: If a convex polygon with with  $2n$  sides  $A_1 A_2 \ldots A_{2n}$  is inscribed in a circle and  $n-1$  of its  $n$  pairs of opposite sides are parallel, which means if the pairs of opposite sides

\[(A_1 A_2, A_{n+1} A_{n+2}), (A_2 A_3, A_{n+2} A_{n+3}), \ldots , (A_{n-1} A_n, A_{2n-1} A_{2n})\]

are parallel, then the sides \[ A_n A_{n+1}, A_{2n} A_1\] are parallel as well.

Answer: n


For each natural number from 1 to 999, Damir subtracted the last digit from the first digit and wrote all the resulting 1000 differences on the board. For example, for the number 7, Damir wrote 0; for the number 105, he wrote (-4); for the number 61, he wrote 5.

What is the sum of all the numbers on the board?

Answer: 495


Write either  $1$  or  $-1$  in each of the cells of a  $(2n) \times (2n)$ -table, in such a way that there are exactly  $2n^2$  entries of each kind. Let the minimum of the absolute values of all row sums and all column sums be  $M$ . Determine the largest possible value of  $M$ .

Answer: 2


The numbers $1000^{2}, 1001^{2}, 1002^{2}, \ldots$ have their last two digits discarded. How many of the first terms of the resulting sequence form an arithmetic progression?

Answer: 10


Given that \( a, b, c \) are all positive numbers, find the maximum value of \( y = \frac{ab + 2bc}{a^{2} + b^{2} + c^{2}} \).

Answer: \frac{\sqrt{5}}{2}


In triangle \( \triangle ABC \), \( M \) is the midpoint of side \( BC \), and \( N \) is the midpoint of segment \( BM \). Given \( \angle A = \frac{\pi}{3} \) and the area of \( \triangle ABC \) is \( \sqrt{3} \), find the minimum value of \( \overrightarrow{AM} \cdot \overrightarrow{AN} \).

Answer: \sqrt{3} + 1


For two perpendicular lines \(a\) and \(b\) in a plane, \((a, b)\) is called a "perpendicular pair", and both \(a\) and \(b\) are lines belonging to this "perpendicular pair". Given that there are twenty lines in the plane, what is the maximum number of "perpendicular pairs" that can be formed?

Answer: 100


Mindy is attempting to solve the quadratic equation by completing the square: $$100x^2+80x-144 = 0.$$ She rewrites the given quadratic equation in the form $$(dx + e)^2 = f,$$ where \(d\), \(e\), and \(f\) are integers and \(d > 0\). What are the values of \(d + e + f\)?

Answer: 174


Let  $p,q$  be prime numbers such that  $n^{3pq}-n$  is a multiple of  $3pq$  for **all** positive integers  $n$ . Find the least possible value of  $p+q$ .

Answer: 28


The slope angle of the tangent line to the curve $f\left(x\right)=- \frac{ \sqrt{3}}{3}{x}^{3}+2$ at $x=1$ is $\tan^{-1}\left( \frac{f'\left(1\right)}{\mid f'\left(1\right) \mid} \right)$, where $f'\left(x\right)$ is the derivative of $f\left(x\right)$.

Answer: \frac{2\pi}{3}


Given that \(a\) is a natural number, and there is a quadratic trinomial with integer coefficients using \(a\) as the leading coefficient that has two distinct positive roots less than 1. Find the smallest value of \(a\).

Answer: 5


Let  $\alpha\geq 1$  be a real number. Define the set  $$ A(\alpha)=\{\lfloor \alpha\rfloor,\lfloor 2\alpha\rfloor, \lfloor 3\alpha\rfloor,\dots\} $$  Suppose that all the positive integers that **does not belong** to the  $A(\alpha)$  are exactly the positive integers that have the same remainder  $r$  in the division by  $2021$  with  $0\leq r<2021$ . Determine all the possible values of  $\alpha$ .

Answer: \frac{2021}{2020}


Answer: 82^\circ


When plotted in the standard rectangular coordinate system, trapezoid $PQRS$ has vertices $P(2, -4)$, $Q(2, 3)$, $R(7, 10)$, and $S(7, 2)$. What is the area of trapezoid $PQRS$?

Answer: 37.5


Given a permutation \(\sigma\) of \(\{1, 2, \ldots, 2013\}\), let \(f(\sigma)\) be the number of fixed points of \(\sigma\) — that is, the number of \(k \in \{1, 2, \ldots, 2013\}\) such that \(\sigma(k) = k\). If \(S\) is the set of all possible permutations \(\sigma\), compute
\[ \sum_{\sigma \in S} f(\sigma)^{4}. \]

(Here, a permutation \(\sigma\) is a bijective mapping from \(\{1, 2, \ldots, 2013\}\) to \(\{1, 2, \ldots, 2013\}\).)

Answer: 15(2013!)


In triangle \(ABC\), the angle bisector \(AP\) is drawn. It is known that \(BP = 16\), \(PC = 20\), and the circumcenter of triangle \(ABP\) lies on segment \(AC\). Find side \(AB\).

Answer: \frac{144\sqrt{5}}{5}


Two types of cast iron with different percentages of chromium were fused together. If one type is taken in 5 times the amount of the other type, the percentage of chromium in the alloy will be twice the percentage of chromium in the smaller portion. If equal amounts of both types are taken, the alloy will contain 8% chromium. Determine the percentage of chromium in each type of cast iron.

Answer: 511


A rook has toured a chessboard, visiting each square at least once. What is the minimum number of turns it could have made in the process?

Answer: 14


Petya approaches the entrance door with a combination lock, which has buttons numbered from 0 to 9. To open the door, three correct buttons need to be pressed simultaneously. Petya does not remember the code and tries combinations one by one. Each attempt takes Petya 2 seconds.

a) How much time will Petya need to definitely get inside?

b) On average, how much time will Petya need?

c) What is the probability that Petya will get inside in less than a minute?

Answer: \frac{29}{120}


The bisector of angle \(ABC\) forms with its sides an angle that is three times smaller than the adjacent angle to \(ABC\). Find the measure of angle \(ABC\).

Answer: 72


In an acute-angled triangle \( ABC \), \( AB \neq AC \), \( H \) and \( G \) are the orthocenter and centroid of the triangle, respectively. It is known that \(\frac{1}{S_{\triangle HAB}} + \frac{1}{S_{\triangle HAC}} = \frac{1}{S_{\triangle HBC}}\). Prove that \(\angle AGH = 90^\circ\).

Answer: 90


Given there are ten steps from the first floor to the second floor, calculate the total number of ways Xiao Ming can go from the first floor to the second floor.

Answer: 89


In a triangle with sides \( a, b, c \), the side \( a \) is the arithmetic mean of \( b \) and \( c \). Prove:

a) \( 0^\circ \leq A \leq 60^\circ \).

b) The altitude relative to side \( a \) is three times the inradius \( r \).

c) The distance from the circumcenter to side \( a \) is \( R - r \).

Answer: R-r


Let $a, b, c,$ and $d$ be real numbers that satisfy the system of equations \begin{align*} a + b &= -3, \\ ab + bc + ca &= -4, \\ abc + bcd + cda + dab &= 14, \\ abcd &= 30. \end{align*} There exist relatively prime positive integers $m$ and $n$ such that \[a^2 + b^2 + c^2 + d^2 = \frac{m}{n}.\]Find $m + n$.

Answer: 145


If the integer solutions to the system of inequalities
\[
\begin{cases}
9x - a \geq 0, \\
8x - b < 0
\end{cases}
\]
are only 1, 2, and 3, how many ordered pairs \((a, b)\) of integers satisfy this system?

Answer: 72


What digits should replace the asterisks to make the number 454** divisible by 2, 7, and 9?

Answer: 45486


Jun Jun sets off at a constant speed from point A to point B, while Ping sets off at the same time from point B to point A at a constant speed. They meet at point C. After meeting, Jun Jun walks 100 meters further and then turns back to chase Ping. When Jun Jun catches up with Ping, they are 360 meters away from point C. Jun Jun then immediately turns around to head to point B, and when Jun Jun reaches point B, Ping also reaches point A. What is the distance between points A and B in meters?

Answer: 1656\,


Find the pattern that governs the numerical sequence: $1, 2, 3, 4, 5, 8, 7, 16, 9, \ldots$, and write down the next five numbers.

Answer: 32,11,64,13,128


Centered at each lattice point in the coordinate plane are a circle radius $\frac{1}{10}$ and a square with sides of length $\frac{1}{5}$ whose sides are parallel to the coordinate axes. The line segment from $(0,0)$ to $(1001, 429)$ intersects $m$ of the squares and $n$ of the circles. Find $m + n$.

Answer: 574


The length of edge PQ of a tetrahedron PQRS measures 51 units, and the lengths of the other edges are 12, 19, 24, 33, and 42 units. Determine the length of edge RS.

Answer: 24


Algebraists have invented a new operation $*$ which satisfies the conditions: $a \leqslant a=0$ and $a *(b * c) = (a *) b) + c. Calculate 2015 2014. (The symbol "+" denotes addition in its usual sense, and parentheses indicate the order of operations.)

Answer: 1


If a podcast series that lasts for 837 minutes needs to be stored on CDs and each CD can hold up to 75 minutes of audio, determine the number of minutes of audio that each CD will contain.

Answer: 69.75


Sherry starts at the number 1.  Whenever she's at 1, she moves one step up (to 2).  Whenever she's at a number strictly between 1 and 10, she moves one step up or one step down, each with probability  $\frac{1}{2}$ .  When she reaches 10, she stops.  What is the expected number (average number) of steps that Sherry will take?

Answer: 81


Let \( A, B, C \), and \( D \) be points on a circle such that \( AB = 11 \) and \( CD = 19 \). Point \( P \) is on segment \( AB \) with \( AP = 6 \), and \( Q \) is on segment \( CD \) with \( CQ = 7 \). The line through \( P \) and \( Q \) intersects the circle at \( X \) and \( Y \). If \( PQ = 27 \), find \( XY \).

Answer: 31


A certain project needs 100 days to complete. Initially, 10 people worked for 30 days and completed $\frac{1}{5}$ of the entire project. Then, an additional 10 people were brought in to complete the project. How many days in advance can the project be completed?

Answer: 10


Given a linear function \( f(x)=ax+b \) that satisfies \( |f(x)+f(y)-xy| \leq \frac{1}{4} \) for any \( x, y \in [0,1] \), determine the function \( f(x) \).

Answer: f(x)=\frac{1}{2}x-\frac{1}{8}


Let's define the distance between numbers as the absolute value of their difference. It is known that the sum of the distances from sixteen consecutive natural numbers to a certain number \(a\) is 636, and the sum of the distances from these same sixteen numbers to the number \(a^2\) is 591. Find all possible values of \(a\).

Answer: -\frac{5}{4}


In a town of \( n \) people, a governing council is elected as follows: each person casts one vote for some person in the town, and anyone that receives at least five votes is elected to council. Let \( c(n) \) denote the average number of people elected to council if everyone votes randomly. Find \( \lim _{n \rightarrow \infty} \frac{c(n)}{n} \).

Answer: 1-\frac{65}{24e}


Let \( \triangle ABC \) be a triangle with \(BA > BC\). Let \( M \) be the midpoint of \( AC \), and the internal bisector of angle \( \widehat{B} \) intersects \( AC \) at \( I \). The line parallel to \( BC \) passing through \( I \) intersects \( AM \) at \( E \). The line parallel to \( BA \) passing through \( M \) intersects \( AI \) at \( D \). What is the measure of the angle \( \widehat{EDI} \)?

Answer: 90


Let $N$ be the number of positive integers that are less than or equal to $2003$ and whose base-$2$ representation has more $1$'s than $0$'s. Find the remainder when $N$ is divided by $1000$.

Answer: 155


Answer: a_n=n


In the triangle \(ABC\), a circle passes through the point \(A\), the midpoint \(E\) of \(AC\), the midpoint \(F\) of \(AB\), and is tangent to the side \(BC\) at \(D\). Suppose
\[ \frac{AB}{AC} + \frac{AC}{AB} = 4. \]
Determine the size of \(\angle EDF\) in degrees.

Answer: 120


What is the volume of the region in three-dimensional space defined by the inequalities $|x|+|y|+|z|\le2$ and $|x|+|y|+|z-2|\le2$?

Answer: \frac{2}{3}


Let $x_1\leq x_2\leq \cdots\leq x_{100}$ be real numbers such that $|x_1| + |x_2| + \cdots + |x_{100}| = 1$ and $x_1 + x_2 + \cdots + x_{100} = 0$. Among all such $100$-tuples of numbers, the greatest value that $x_{76} - x_{16}$ can achieve is $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Answer: 841


The real numbers \( x_1, x_2, x_3, \ldots \) satisfy \( |x_{m+n} - x_m - x_n| \leq 1 \) for all \( m, n \). Show that \( \left| \frac{x_m}{m} - \frac{x_n}{n} \right| < \frac{1}{m} + \frac{1}{n} \) for all \( m, n \).

Answer: |\frac{x_m}{}-\frac{x_n}{n}|<\frac{1}{}+\frac{1}{n}


Given \( c > 0 \), let \( p \) be the statement: the function \( y = c^x \) is monotonically decreasing on \( \mathbf{R} \), and \( q \) be the statement: the solution set of the inequality \( x + |x - 2c| > 1 \) is \(\mathbf{R}\). If exactly one of the statements \( p \) and \( q \) is true, find the range of values for \( c \).

Answer: (0,\frac{1}{2}]\cup[1,\infty)


Alice and Bob live $10$ miles apart. One day Alice looks due north from her house and sees an airplane. At the same time Bob looks due west from his house and sees the same airplane. The angle of elevation of the airplane is $30^\circ$ from Alice's position and $60^\circ$ from Bob's position. Which of the following is closest to the airplane's altitude, in miles?
$\textbf{(A)}\ 3.5 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 4.5 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ 5.5$

Answer: \textbf{(E)}\5.5


The height of the isosceles trapezoid $ABCD$ with bases $AD$ and $BC$ is $4 \sqrt{3}$. The diagonals of the trapezoid intersect at point $O$, and $\angle AOD = 120^{\circ}$. Find the median of the trapezoid.

Answer: 12


Given $a, b, c, d \in \mathbf{N}$ such that $342(abcd + ab + ad + cd + 1) = 379(bcd + b + d)$, determine the value of $M$ where $M = a \cdot 10^{3} + b \cdot 10^{2} + c \cdot 10 + d$.

Answer: 1949


An iterative average of the numbers 2, 3, 4, 6, and 7 is computed by arranging the numbers in some order. Find the difference between the largest and smallest possible values that can be obtained using this procedure.

Answer: \frac{11}{4}


Answer: 0.239


The positive integers are grouped as follows:

\( A_1 = \{1\}, A_2 = \{2, 3, 4\}, A_3 = \{5, 6, 7, 8, 9\} \), and so on.

In which group does 2009 belong?

Answer: 45


Ilya Muromets encounters the three-headed dragon Zmey Gorynych and the battle begins. Every minute, Ilya cuts off one of the dragon's heads. With probability \( \frac{1}{4} \), two new heads grow in place of the severed head, with probability \( \frac{1}{3} \), only one new head grows, and with probability \( \frac{5}{12} \), no heads grow back. The dragon is considered defeated if it has no heads left. Find the probability that Ilya will eventually defeat the dragon.

Answer: 1


In rectangle $ABCD$, angle $C$ is trisected by $\overline{CF}$ and $\overline{CE}$, where $E$ is on $\overline{AB}$, $F$ is on $\overline{AD}$, $BE=6$, and $AF=2$. Find the area of $ABCD$.

[asy]
import olympiad; import geometry; size(150); defaultpen(linewidth(0.8)); dotfactor=4;
real length = 2 * (6*sqrt(3) - 2), width = 6*sqrt(3);
draw(origin--(length,0)--(length,width)--(0,width)--cycle);
draw((length,width)--(0,2)^^(length,width)--(length - 6,0));
dot("$A$",origin,SW); dot("$B$",(length,0),SE); dot("$C$",(length,width),NE); dot("$D$",(0,width),NW); dot("$F$",(0,2),W); dot("$E$",(length - 6,0),S);
[/asy]

Answer: 108\sqrt{3}-36


Several circles are inscribed in a corner, and their radii increase such that each subsequent circle touches the previous one. Find the sum of the circumferences of the second and fourth circles, given that the circumference of the third circle is \(18 \pi\), and the area of the circle bounded by the first circle is \(\pi\).

Answer: 60\pi


Complex numbers $p,$ $q,$ and $r$ are zeros of a polynomial $Q(z) = z^3 + sz^2 + tz + u,$ and $|p|^2 + |q|^2 + |r|^2 = 360.$ The points corresponding to $p,$ $q,$ and $r$ in the complex plane are the vertices of a right triangle with hypotenuse $k.$ Find $k^2.$

Answer: 540


Given \( m = n^{4} + x \), where \( n \) is a natural number and \( x \) is a two-digit positive integer, what value of \( x \) will make \( m \) a composite number?

Answer: 64


The year "2021" possesses the following property: $2021 = 43 \times 47$, meaning it can be expressed as the product of two distinct prime numbers less than 100. We call such a number a "bull number." In the set $M = \{1, 2, \cdots, 2021\}$ consisting of the first 2021 positive integers, how many bull numbers are there?

Answer: 201


Quadrilateral \(ABCD\), whose diagonals are mutually perpendicular, is inscribed in a circle with center \(O\). Prove that the broken line \(AOC\) divides it into two equal parts.

Answer: 2


A circle constructed on the smaller leg of an isosceles trapezoid as its diameter touches the longer leg, which is equal to \(a\).

Find the midline of the trapezoid.

Answer: \frac{}{2}


On a sheet of paper, Isabella draws a circle of radius $2$, a circle of radius $3$, and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly $k \ge  0$ lines. How many different values of $k$ are possible?
$\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6$

Answer: \textbf{(D)}\5


What is the maximum number of kings that can be placed on a chessboard such that no two kings can attack each other?

Answer: 16


What is the maximum number of strings connecting adjacent nodes of a volleyball net with square cells that can be cut so that the net does not fall into separate pieces? The dimensions of the net are $10 \times 100$ cells.

Answer: 1000


A word is any finite sequence of letters from the Russian alphabet. Determine how many distinct words can be formed from the words:

a) ВЕКТОР;

б) ЛИНИЯ;

в) ПАРАБОЛА;

г) БИССЕКТРИСА;

д) МАТЕМАТИКА.

Answer: 151200


Answer: 8210


The cells of a $9 \times 9$ board are painted in black and white in a checkerboard pattern. How many ways are there to place 9 rooks on cells of the same color on the board such that no two rooks attack each other? (A rook attacks any cell that is in the same row or column as it.)

Answer: 2880


Find the number of integers $c$ such that the equation \[\left||20|x|-x^2|-c\right|=21\]has $12$ distinct real solutions.

Answer: 57


It is known that the sum of the absolute values of the pairwise differences of five nonnegative numbers is equal to one. Find the smallest possible sum of these numbers.

Answer: \frac{1}{4}


If 1000 were expressed as a sum of at least three distinct powers of 2, what would be the least possible sum of the exponents of these powers?

Answer: 38


In the coordinate plane, the points $A(0, 2)$, $B(1, 7)$, $C(10, 7)$, and $D(7, 1)$ are given. Find the area of the pentagon $ABCDE$, where $E$ is the intersection point of the lines $AC$ and $BD$.

Answer: 36


The incircle of  $\triangle ABC$  touch  $AB$ , $BC$ , $CA$  at  $K$ , $L$ , $M$ .  The common external tangents to the incircles of  $\triangle AMK$ , $\triangle BKL$ , $\triangle CLM$ , distinct from the sides of  $\triangle ABC$ , are drawn.  Show that these three lines are concurrent.

Answer: 3


Let  $AXYZB$  be a regular pentagon with area  $5$  inscribed in a circle with center  $O$ .  Let  $Y'$  denote the reflection of  $Y$  over  $\overline{AB}$  and suppose  $C$  is the center of a circle passing through  $A$ ,  $Y'$  and  $B$ . Compute the area of triangle  $ABC$ .

*Proposed by Evan Chen*

Answer: 1


Find the minimum of the expression   $\frac{a + b + c}{2} -\frac{[a, b] + [b, c] + [c, a]}{a + b + c}$  where the variables  $a, b, c$  are any integers greater than  $1$  and  $[x, y]$  denotes the least common multiple of numbers  $x, y$ .

Answer: \frac{3}{2}


Given that the greatest common divisor (GCD) of \(a\) and \(b\) is 4, the least common multiple (LCM) of \(a\) and \(c\) as well as the LCM of \(b\) and \(c\) are both 100, and \(a \leqslant b\), how many sets of natural numbers \(a\), \(b\), and \(c\) satisfy these conditions?

Answer: 9


A quadrilateral is inscribed in a circle. If an angle is inscribed into each of the four segments outside the quadrilateral, the sum of these four angles, expressed in degrees, is:

Answer: 540


From 1 to 100, take a pair of integers (repetitions allowed) so that their sum is greater than 100. How many ways are there to pick such pairs?

Answer: 5050


Calculate the value of the function $f(x)=3x^{6}-2x^{5}+x^{3}+1$ at $x=2$ using the Horner's method (also known as the Qin Jiushao algorithm) to determine the value of $v_{4}$.

Answer: 34


Two circles intersect at points \(A\) and \(B\). The center \(O\) of the first circle lies on the second circle. A secant line passing through \(O\) intersects the second circle at points \(D\) and \(C\). Prove that point \(D\) is the incenter of triangle \(A B C\).

Answer: 0


Let  $k$  be a positive integer. Marco and Vera play a game on an infinite grid of square cells. At the beginning, only one cell is black and the rest are white.
	A turn in this game consists of the following. Marco moves first, and for every move he must choose a cell which is black and which has more than two white neighbors. (Two cells are neighbors if they share an edge, so every cell has exactly four neighbors.) His move consists of making the chosen black cell white and turning all of its neighbors black if they are not already. Vera then performs the following action exactly  $k$  times: she chooses two cells that are neighbors to each other and swaps their colors (she is allowed to swap the colors of two white or of two black cells, though doing so has no effect). This, in totality, is a single turn. If Vera leaves the board so that Marco cannot choose a cell that is black and has more than two white neighbors, then Vera wins; otherwise, another turn occurs.
	Let  $m$  be the minimal  $k$  value such that Vera can guarantee that she wins no matter what Marco does. For  $k=m$ , let  $t$  be the smallest positive integer such that Vera can guarantee, no matter what Marco does, that she wins after at most  $t$  turns. Compute  $100m + t$ .

*Proposed by Ashwin Sah*

Answer: 203


In a class, there are 15 boys and 15 girls. On Women's Day, some boys called some girls to congratulate them (no boy called the same girl more than once). It turned out that the children can be uniquely divided into 15 pairs, such that each pair consists of a boy and a girl whom he called. What is the maximum number of calls that could have been made?

Answer: 120


On a circular track with a perimeter of 360 meters, three individuals A, B, and C start from the same point: A starts first, running counterclockwise. Before A completes one lap, B and C start simultaneously, running clockwise. When A and B meet for the first time, C is exactly halfway between them. After some time, when A and C meet for the first time, B is also exactly halfway between them. If B's speed is four times that of A's, how many meters has A run when B and C started?

Answer: 90


Let \( G \) be a graph with \( n \) vertices and \( m \) edges. Let \( T \) be the number of triangles in this graph. Show that:

\[
T \geq \frac{m\left(4 m - n^{2}\right)}{3 n}
\]

Answer: T\ge\frac{(4m-n^2)}{3n}


Given \( x, y \in \mathbb{R}^{+} \), find the minimum value of the function:
\[ f(x, y) = \sqrt{x^{2} - 3x + 3} + \sqrt{y^{2} - 3y + 3} + \sqrt{x^{2} - \sqrt{3}xy + y^{2}}. \]

Answer: \sqrt{6}


Let  $\mathcal{P}$  be the set of all polynomials  $p(x)=x^4+2x^2+mx+n$ , where  $m$  and  $n$  range over the positive reals. There exists a unique  $p(x) \in \mathcal{P}$  such that  $p(x)$  has a real root,  $m$  is minimized, and  $p(1)=99$ . Find  $n$ .

*Proposed by **AOPS12142015***

Answer: 56


Let  $a$  be a positive real number such that  $\tfrac{a^2}{a^4-a^2+1}=\tfrac{4}{37}$ .  Then  $\tfrac{a^3}{a^6-a^3+1}=\tfrac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers.  Find  $m+n$ .

Answer: 259


Given that person A traveled from point X to point Y in an integer number of hours, and the number of kilometers A walks per hour is equal to the amount of time taken. Person B walks from point X to point Y at a rate of 2 kilometers per hour, taking a 1-hour rest after every 4 kilometers. B took a total of 11 hours for the trip. What is the distance between points X and Y in kilometers?

Answer: 16


A natural number $N$ has exactly 12 positive divisors (including 1 and $N$). When these divisors are arranged in ascending order: $d_{1}<d_{2}<\cdots<d_{12}$, it is known that the divisor with index $d_{4}-1$ is equal to $\left(d_{1}+d_{2}+d_{4}\right) \cdot d_{8}$. Find the natural number $N$.

Answer: 1989


For how many integer values of $x$ will the following polynomial be a perfect square?

$$
4 x^{4}-12 x^{3}+17 x^{2}-6 x-14
$$

Answer: 2


An $n$-string is a string of digits formed by writing the numbers $1,2, \ldots, n$ in some order (in base ten). For example, one possible 10-string is $$35728910461$$ What is the smallest $n>1$ such that there exists a palindromic $n$-string?

Answer: 19


Let  $\mathcal P$  denote the set of planes in three-dimensional space with positive  $x$ ,  $y$ , and  $z$  intercepts summing to one. A point  $(x,y,z)$  with  $\min \{x,y,z\} > 0$  lies on exactly one plane in  $\mathcal P$ . What is the maximum possible integer value of  $\left(\frac{1}{4} x^2 + 2y^2 + 16z^2\right)^{-1}$ ?

*Proposed by Sammy Luo*

Answer: 21


Let \( b \) be a positive number and \( n > 2 \) be a natural number, and

$$
d=\frac{([b]+1-b) \cdot[b]}{[b]+1}
$$

where \( [b] \) denotes the integer part of \( b \) (i.e., the largest integer not greater than \( b \)). Prove that

$$
\frac{d+n-2}{[b]+n-2}>\frac{[b]+n-1-b}{[b]+n-1}
$$

Answer: \frac{+n-2}{[b]+n-2}>\frac{[b]+n-1-b}{[b]+n-1}


Ten spheres of the same radius are stacked in the form of a triangular pyramid so that each sphere touches at least three others. Find the radius of the circumscribed sphere for the pyramid of spheres, given that the radius of the sphere inscribed in the center of the pyramid of spheres, touching six identical spheres, is $\sqrt{6}-1$.

Answer: 5(\sqrt{2}+1)


Given \(a, b > 0\), satisfy the equation for \(x\):
\[
\sqrt{|x|} + \sqrt{|x + a|} = b
\]

There are exactly three distinct real solutions \(x_1, x_2, x_3\), where \(x_1 < x_2 < x_3 = b\). Find the value of \(a + b\).

Answer: 144


In the number $2 * 0 * 1 * 6 * 0 * 2 *$, each of the 6 asterisks must be replaced with any of the digits $0, 2, 4, 5, 7, 9$ (digits may be repeated) so that the resulting 12-digit number is divisible by 12. How many ways can this be done?

Answer: 5184


Compute the number of ordered pairs of positive integers $(a, b)$ satisfying the equation $\operatorname{gcd}(a, b) \cdot a+b^{2}=10000$

Answer: 99


After expanding and combining like terms in the expression \((x+y+z)^{2028} + (x-y-z)^{2028}\), how many monomials of the form \(x^{a} y^{b} z^{c}\) have a non-zero coefficient?

Answer: 1030225


A truck has new tires fitted on all four wheels. A tire is considered completely worn out if it has traveled $15000 \mathrm{~km}$ on the rear wheel or $25000 \mathrm{~km}$ on the front wheel. How far can the truck travel before all four tires are completely worn out if the front and rear pairs of tires are swapped at suitable intervals?

Answer: 18750


Given a sequence of 15 zeros and ones, determine the number of sequences where all the zeros are consecutive.

Answer: 121


Indicate the integer closest to the number: \(\sqrt{2012-\sqrt{2013 \cdot 2011}}+\sqrt{2010-\sqrt{2011 \cdot 2009}}+\ldots+\sqrt{2-\sqrt{3 \cdot 1}}\).

Answer: 31


Given  $10$  points in the space such that each  $4$  points are not lie on a plane. Connect some points with some segments such that there are no triangles or quadrangles. Find the maximum number of the segments.

Answer: 25


In triangle \( ABC \), a point \( K \) is taken on side \( AB \), such that \( AK: BK = 1:2 \). Another point \( L \) is taken on side \( BC \), such that \( CL: BL = 2:1 \). Point \( Q \) is the intersection of lines \( AL \) and \( CK \). Find the area of triangle \( ABC \) if it is known that \( S_{BQC} = 1 \).

Answer: \frac{7}{4}


There are 99 positive integers, and their sum is 101101. Find the greatest possible value of the greatest common divisor of these 99 positive integers.

Answer: 101


Triangle $ABC$ is inscribed in circle $\omega$ with $AB=5$, $BC=7$, and $AC=3$. The bisector of angle $A$ meets side $\overline{BC}$ at $D$ and circle $\omega$ at a second point $E$. Let $\gamma$ be the circle with diameter $\overline{DE}$. Circles $\omega$ and $\gamma$ meet at $E$ and a second point $F$. Then $AF^2 = \frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Answer: 919


In space, there are 4 pairwise skew lines \( l_{1}, l_{2}, l_{3}, l_{4} \), such that no three of them are parallel to the same plane. Draw a plane \( P \) such that the points \( A_{1}, A_{2}, A_{3}, A_{4} \) at the intersections of these lines with \( P \) form a parallelogram. How many lines do the centers of such parallelograms trace out?

Answer: 3


A person named Jia and their four colleagues each own a car with license plates ending in 9, 0, 2, 1, and 5, respectively. To comply with the local traffic restriction rules from the 5th to the 9th day of a certain month (allowing cars with odd-ending numbers on odd days and even-ending numbers on even days), they agreed to carpool. Each day they can pick any car that meets the restriction, but Jia’s car can be used for one day at most. The number of different carpooling arrangements is __________.

Answer: 80


Triangle  $ABC$  has side lengths  $AB=13, BC=14,$  and  $CA=15$ . Let  $\Gamma$  denote the circumcircle of  $\triangle ABC$ . Let  $H$  be the orthocenter of  $\triangle ABC$ . Let  $AH$  intersect  $\Gamma$  at a point  $D$  other than  $A$ . Let  $BH$  intersect  $AC$  at  $F$  and  $\Gamma$  at point  $G$  other than  $B$ . Suppose  $DG$  intersects  $AC$  at  $X$ . Compute the greatest integer less than or equal to the area of quadrilateral  $HDXF$ .

*Proposed by Kenan Hasanaliyev (claserken)*

Answer: 24


Determine how many positive integer multiples of $2002$ can be represented in the form $10^{j} - 10^{i}$, where $i$ and $j$ are integers and $0 \leq i < j \leq 150$.

Answer: 1825


Let  $k$  and  $n$  be positive integers and let  $x_1, x_2, \cdots, x_k, y_1, y_2, \cdots, y_n$  be distinct integers. A polynomial  $P$  with integer coefficients satisfies
 \[P(x_1)=P(x_2)= \cdots = P(x_k)=54\]
\[P(y_1)=P(y_2)= \cdots = P(y_n)=2013.\]

Determine the maximal value of  $kn$ .

Answer: 6


Here is a fairly simple puzzle: EH is four times greater than OY. AY is four times greater than OH. Find the sum of all four.

Answer: 150


What is the value of $[\sqrt{1}] + [\sqrt{2}] + [\sqrt{3}] + \cdots + [\sqrt{1989 \cdot 1990}] + [-\sqrt{1}] + [-\sqrt{2}] + [-\sqrt{3}] + \cdots + [-\sqrt{1989 \cdot 1990}]$?
(The 1st "Hope Cup" Mathematics Contest, 1990)

Answer: -3956121


A convex equilateral pentagon with side length  $2$  has two right angles. The greatest possible area of the pentagon is  $m+\sqrt{n}$ , where  $m$  and  $n$  are positive integers. Find  $100m+n$ .

*Proposed by Yannick Yao*

Answer: 407


Answer: a^{-2}+b^{-2}


Through the origin, lines are drawn (including the coordinate axes) that divide the coordinate plane into angles of $1^{\circ}$.

Find the sum of the x-coordinates of the points where these lines intersect the line $y=100-x$.

Answer: 8950


The digits 2, 4, 6, and 8 are each used once to create two 2-digit numbers. Find the smallest possible difference between the two 2-digit numbers.

Answer: 14


Point  $P$  lies outside a circle, and two rays are drawn from  $P$  that intersect the circle as shown. One ray intersects the circle at points  $A$  and  $B$  while the other ray intersects the circle at  $M$  and  $N$ .  $AN$  and  $MB$  intersect at  $X$ . Given that  $\angle AXB$  measures  $127^{\circ}$  and the minor arc  $AM$  measures  $14^{\circ}$ , compute the measure of the angle at  $P$ .

[asy]
size(200);
defaultpen(fontsize(10pt));
pair P=(40,10),C=(-20,10),K=(-20,-10);
path CC=circle((0,0),20), PC=P--C, PK=P--K;
pair A=intersectionpoints(CC,PC)[0],
B=intersectionpoints(CC,PC)[1],
M=intersectionpoints(CC,PK)[0],
N=intersectionpoints(CC,PK)[1],
X=intersectionpoint(A--N,B--M);
draw(CC);draw(PC);draw(PK);draw(A--N);draw(B--M);
label(" $A$ ",A,plain.NE);label(" $B$ ",B,plain.NW);label(" $M$ ",M,SE);
label(" $P$ ",P,E);label(" $N$ ",N,dir(250));label(" $X$ ",X,plain.N);[/asy]

Answer: 39


A metal bar at a temperature of $20^{\circ} \mathrm{C}$ is placed in water at a temperature of $100^{\circ} \mathrm{C}$. After thermal equilibrium is established, the temperature becomes $80^{\circ} \mathrm{C}$. Then, without removing the first bar, another identical metal bar also at $20^{\circ} \mathrm{C}$ is placed in the water. What will be the temperature of the water after thermal equilibrium is established?

Answer: 68


There are a total of 2008 black triangles, marked as "▲" and "△", arranged in a certain pattern as follows: ▲▲△△▲△▲▲△△▲△▲▲…, then there are     black triangles.

Answer: 1004


In triangle $PQR$, $PQ = 12$, $QR = 16$, and $PR = 20$. Point $X$ is on $\overline{PQ}$, $Y$ is on $\overline{QR}$, and $Z$ is on $\overline{PR}$. Let $PX = u \cdot PQ$, $QY = v \cdot QR$, and $RZ = w \cdot PR$, where $u$, $v$, and $w$ are positive and satisfy $u+v+w=3/4$ and $u^2+v^2+w^2=1/2$. The ratio of the area of triangle $XYZ$ to the area of triangle $PQR$ can be written in the form $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Answer: 41


What is the largest positive integer $n$ for which there is a unique integer $k$ such that $\frac{8}{15} < \frac{n}{n + k} < \frac{7}{13}$?

Answer: 112


Given a tetrahedron \(ABCD\), a sphere with a radius of 1 is drawn through vertex \(D\). This sphere is tangent to the circumsphere of the tetrahedron at point \(D\) and also tangent to the face \(ABC\). Given that \(AD=2\sqrt{3}\), \(\angle BAD = \angle CAD = 45^\circ\), and \(\angle BAC = 60^\circ\), find the radius \(r\) of the circumsphere of the tetrahedron.

Answer: 3


On the New Year’s table, four glasses are arranged in a row: the first and third contain orange juice, while the second and fourth are empty. While waiting for guests, Valya absentmindedly and randomly pours juice from one glass to another. Each time, she can take a full glass and pour all its contents into one of the two empty glasses.

Find the expected number of pourings required for the situation to be completely reversed: the first and third glasses empty, while the second and fourth glasses are full.

Answer: 6


Convert 2 meters plus 3 centimeters plus 5 millimeters into meters.

Answer: 2.035 \text{ m}


A set of several distinct numbers possesses the following property: the arithmetic mean of some two numbers from this set is equal to the arithmetic mean of some three numbers from the set and is also equal to the arithmetic mean of some four numbers from the set. What is the smallest possible number of numbers in such a set?

Answer: 5


Given the left focus $F$ of the ellipse $C$: $$\frac {x^{2}}{a^{2}}$$+ $$\frac {y^{2}}{b^{2}}$$\=1 ($a>b>0$), a line is drawn through the upper endpoint $B$ of $C$ and intersects the ellipse at another point $A$. If $|BF|=3|AF|$, find the eccentricity of $C$.

Answer: \frac{\sqrt{2}}{2}


Given a triangular prism $S-ABC$, where the base is an isosceles right triangle with $AB$ as the hypotenuse, $SA = SB = SC = 2$, and $AB = 2$, and points $S$, $A$, $B$, and $C$ all lie on a sphere centered at point $O$, find the distance from point $O$ to the plane $ABC$.

Answer: \frac{\sqrt{3}}{3}


Given  $u_0,u_1$  with  $0<u_0,u_1<1$ , define the sequence  $(u_n)$  recurrently by the formula  $$ u_{n+2}=\frac12\left(\sqrt{u_{n+1}}+\sqrt{u_n}\right). $$  (a) Prove that the sequence  $u_n$  is convergent and find its limit.
(b) Prove that, starting from some index  $n_0$ , the sequence  $u_n$  is monotonous.

Answer: 1


A table of numbers consists of several rows. Starting from the second row, each number in a row is the sum of the two numbers directly above it. The last row contains only one number. The first row is the first 100 positive integers arranged in ascending order. What is the number in the last row? (You may express the answer using exponents).

Answer: 101\cdot2^{98}