The Elvish language consists of 4 words: "elara", "quen", "silva", and "nore". In a sentence, "elara" cannot come directly before "quen", and "silva" cannot come directly before "nore"; all other word combinations are grammatically correct (including sentences with repeated words). How many valid 3-word sentences are there in Elvish?

Answer: 48


There are three cups of solution with equal weight, and their concentrations are $10 \%, 20 \%, 45 \%$ respectively. If $\frac{1}{2}$, $\frac{1}{4}$, and $\frac{1}{5}$ of the weights from the three cups are poured into a fourth empty cup in order, what is the concentration of the solution in the fourth cup? $\qquad$ $\%$.

Answer: 20


Given a positive real sequence \(a_{1}, a_{2}, \cdots\) such that for any \(k \in \mathbf{Z}_{+}\), it satisfies
\[ a_{k+1} \geqslant \frac{k a_{k}}{a_{k}^{2}+k-1} \text{.} \]

Prove: For every positive integer \(n \geq 2\), we have
\[ \sum_{i=1}^{n} a_{i} \geq n . \]

Answer: \sum_{i=1}^{n}a_{i}\gen


How many whole numbers between 1 and 500 do not contain the digit 2?

Answer: 323


Which of the following numbers is an odd integer, contains the digit 5, is divisible by 3, and lies between \(12^{2}\) and \(13^{2}\)?
(A) 105
(B) 147
(C) 156
(D) 165
(E) 175

Answer: 165


Triangle \(ABC\) has side lengths \(AB=2\sqrt{5}\), \(BC=1\), and \(CA=5\). Point \(D\) is on side \(AC\) such that \(CD=1\), and \(F\) is a point such that \(BF=2\) and \(CF=3\). Let \(E\) be the intersection of lines \(AB\) and \(DF\). Find the area of \(CDEB\).

Answer: \frac{22}{35}


Petya and Vasya are playing a game. Initially, each of the 2022 boxes contains one matchstick. In one move, all matchsticks from any non-empty box can be transferred to any other non-empty box. They take turns, with Petya starting. The winner is the one who, after their move, has at least half of all the matchsticks in one box for the first time. Who will win with the correct play?

Answer: Vasya


Mom made homemade currant juice and poured it into bottles. She had two types of bottles: small bottles with a volume of \(500 \ \mathrm{ml}\) and large bottles with a volume of \(750 \ \mathrm{ml}\). In the end, she had 12 small empty bottles, and the remaining bottles were completely filled. Then Mom realized that she could have poured the juice in such a way that only large bottles would be left empty and all others would be completely filled.

How many empty bottles would she have in that case?

Answer: 8


Given non-negative real numbers \( x_1, x_2, \ldots, x_n \) such that \( x_1 + x_2 + \cdots + x_n = s \), show that:
\[ x_1 x_2 + x_2 x_3 + \cdots + x_{n-1} x_n \leq \frac{s^2}{4}. \]

Answer: \frac{^2}{4}


Schools A and B are having a sports competition with three events. In each event, the winner gets 10 points and the loser gets 0 points, with no draws. The school with the highest total score after the three events wins the championship. It is known that the probabilities of school A winning in the three events are 0.5, 0.4, and 0.8, respectively, and the results of each event are independent.
$(1)$ Find the probability of school A winning the championship;
$(2)$ Let $X$ represent the total score of school B, find the distribution table and expectation of $X$.

Answer: 13


Let a sequence $\left\{a_{n}\right\}_{n=0}^{\infty}$ be defined by $a_{0}=\sqrt{2}, a_{1}=2$, and $a_{n+1}=a_{n} a_{n-1}^{2}$ for $n \geq 1$. The sequence of remainders when $a_{0}, a_{1}, a_{2}, \cdots$ are divided by 2014 is eventually periodic with some minimal period $p$ (meaning that $a_{m}=a_{m+p}$ for all sufficiently large integers $m$, and $p$ is the smallest such positive integer). Find $p$.

Answer: 12


Given the functions $f(x)= \begin{cases} 2^{x-2}-1,x\geqslant 0 \\ x+2,x < 0 \end{cases}$ and $g(x)= \begin{cases} x^{2}-2x,x\geqslant 0 \\ \frac {1}{x},x < 0. \end{cases}$, find the sum of all the zeros of the function $f[g(x)]$.

Answer: \frac{1}{2} + \sqrt{3}


Determine the smallest positive integer \( a \) for which \( 47^n + a \cdot 15^n \) is divisible by 1984 for all odd \( n \).

Answer: 1055


Let $T$ be a positive integer whose only digits are 0s and 1s. If $X = T \div 12$ and $X$ is an integer, what is the smallest possible value of $X$?

Answer: 925


Let  $m\ge 3$  and  $n$  be positive integers such that  $n>m(m-2)$ . Find the largest positive integer  $d$  such that  $d\mid n!$  and  $k\nmid d$  for all  $k\in\{m,m+1,\ldots,n\}$ .

Answer: -1


(2014•Shanghai) In a certain game, the scores are 1, 2, 3, 4, 5. The random variable $\xi$ represents Xiao Bai's score in this game. If $E(\xi) = 4.2$, then the probability that Xiao Bai scores 5 points is at least ___.

Answer: 0.2


Answer: 0.1997


Consider $x^2+px+q=0$, where $p$ and $q$ are positive numbers. If the roots of this equation differ by 1, then $p$ equals

Answer: \sqrt{4q+1}


Let \[P(x) = (3x^5 - 45x^4 + gx^3 + hx^2 + ix + j)(4x^3 - 60x^2 + kx + l),\] where $g, h, i, j, k, l$ are real numbers. Suppose that the set of all complex roots of $P(x)$ includes $\{1, 2, 3, 4, 5, 6\}$. Find $P(7)$.

Answer: 51840


Let \(ABCD\) be a quadrilateral inscribed in the unit circle such that \(\angle BAD\) is 30 degrees. Let \(m\) denote the minimum value of \(CP+PQ+CQ\), where \(P\) and \(Q\) may be any points lying along rays \(AB\) and \(AD\), respectively. Determine the maximum value of \(m\).

Answer: 2


Given that there are 4 tiers of horses, with Tian Ji's top-tier horse being better than King Qi's middle-tier horse, worse than King Qi's top-tier horse, Tian Ji's middle-tier horse being better than King Qi's bottom-tier horse, worse than King Qi's middle-tier horse, and Tian Ji's bottom-tier horse being worse than King Qi's bottom-tier horse, determine the probability that Tian Ji's selected horse wins the race.

Answer: \frac{1}{3}


The orchestra has more than 150 members but fewer than 300 members. When they line up in rows of 6 there are two extra people; when they line up in rows of 8 there are four extra people; and when they line up in rows of 9 there are six extra people. How many members are in the orchestra?

Answer: 212


Let $s$ be the limiting sum of the geometric series $4- \frac83 + \frac{16}{9} - \dots$, as the number of terms increases without bound. Then $s$ equals:
$\textbf{(A)}\ \text{a number between 0 and 1}\qquad\textbf{(B)}\ 2.4\qquad\textbf{(C)}\ 2.5\qquad\textbf{(D)}\ 3.6\qquad\textbf{(E)}\ 12$

Answer: 2.4\textbf{(B)}


Given a sequence of $n$ numbers $a_{1}, a_{2}, \cdots, a_{n}$, they are transformed as follows:
$$
\left|a_{1}-\alpha\right|,\left|a_{2}-\alpha\right|,\left|a_{3}-\alpha\right|, \cdots,\left|a_{n}-\alpha\right|,
$$

where $\alpha$ is any arbitrary number. This transformation can be performed multiple times, with $\alpha$ being different each time.

1. Prove that it is possible to transform the sequence such that all $n$ numbers become zero after a finite number of transformations.
2. Determine the minimum number of transformations needed to achieve this goal for any initial sequence.

Answer: n


Let $a=\sqrt{17}$ and $b=i \sqrt{19}$, where $i=\sqrt{-1}$. Find the maximum possible value of the ratio $|a-z| /|b-z|$ over all complex numbers $z$ of magnitude 1 (i.e. over the unit circle $|z|=1$ ).

Answer: \frac{4}{3}


Answer: 1242_6


How many numbers are in the list $-50, -44, -38, \ldots, 68, 74$?

Answer: 22


Stan has a stack of 100 blocks and starts with a score of 0, and plays a game in which he iterates the following two-step procedure: (a) Stan picks a stack of blocks and splits it into 2 smaller stacks each with a positive number of blocks, say $a$ and $b$. (The order in which the new piles are placed does not matter.) (b) Stan adds the product of the two piles' sizes, $a b$, to his score. The game ends when there are only 1-block stacks left. What is the expected value of Stan's score at the end of the game?

Answer: 4950


Consider triangle \(ABC\) with angles \(\angle BAC = 24^\circ\) and \(\angle ACB = 28^\circ\). Point \(D\) is constructed such that \(AB \parallel CD\), \(AD = BC\), and \(AD\) and \(BC\) are not parallel. Similarly, point \(E\) is constructed such that \(AE \parallel BC\), \(AB = CE\), and \(AB\) and \(CE\) are not parallel. Lines \(DE\) and \(AC\) intersect at point \(P\). Determine angle \(\angle CPE\) (in degrees).

Answer: 4


A cylindrical vessel has a base radius of \( R = 4 \) cm. A sphere with radius \( r = 3 \) cm is placed inside the vessel. Water is then added to the vessel such that its free surface touches the surface of the sphere (the sphere does not float). Determine the thickness of the water layer that will result if the sphere is removed from the vessel.

Answer: 3.75


Given a triangle \( \triangle ABC \) and a point \( D \) on its side \( BC \) such that \( AB = 2BD \) and \( AC = 3CD \), and an ellipse \( \Gamma \) can be constructed with \( A \) and \( D \) as foci passing through points \( B \) and \( C \), find the eccentricity of \( \Gamma \).

Answer: \frac{\sqrt{21}}{7}


Let \( s \) be the set of all rational numbers \( r \) that satisfy the following conditions:
\[
(1) \quad 0<r<1 ;
\]
(2) \( r=0.abcabcabc\cdots=0.ab\dot{c} \), where \( a, b, c \) are not necessarily distinct.

When the numbers \( r \) in \( s \) are written as irreducible fractions, how many different numerators are there?

Answer: 660


The ferry boat begins transporting tourists to an island every hour starting at 9 AM until its last trip, which starts at 4 PM. On the first trip at 9 AM, there were 120 tourists, and on each successive trip, there were 2 fewer tourists than on the previous trip. Determine the total number of tourists the ferry transported to the island that day.

Answer: 904


The number $2013$ has the property that its units digit is the sum of its other digits, that is $2+0+1=3$. How many integers less than $2013$ but greater than $1000$ have this property?
$\textbf{(A)}\ 33\qquad\textbf{(B)}\ 34\qquad\textbf{(C)}\ 45\qquad\textbf{(D)}\ 46\qquad\textbf{(E)}\ 58$

Answer: \textbf{(D)}\46


Given that $x_{1}$ is a root of the one-variable quadratic equation about $x$, $\frac{1}{2}m{x^2}+\sqrt{2}x+{m^2}=0$, and ${x_1}=\sqrt{a+2}-\sqrt{8-a}+\sqrt{-{a^2}}$ (where $a$ is a real number), find the values of $m$ and the other root of the equation.

Answer: 2\sqrt{2}


Unit square  $ABCD$  is divided into four rectangles by  $EF$  and  $GH$ , with  $BF = \frac14$ .  $EF$  is parallel to  $AB$  and  $GH$  parallel to  $BC$ .  $EF$  and  $GH$  meet at point  $P$ . Suppose  $BF + DH = FH$ , calculate the nearest integer to the degree of  $\angle FAH$ .
[asy]
size(100); defaultpen(linewidth(0.7)+fontsize(10));
pair D2(pair P) {
dot(P,linewidth(3)); return P;
}
// NOTE: I've tampered with the angles to make the diagram not-to-scale. The correct numbers should be 72 instead of 76, and 45 instead of 55.
pair A=(0,1), B=(0,0), C=(1,0), D=(1,1), F=intersectionpoints(A--A+2*dir(-76),B--C)[0], H=intersectionpoints(A--A+2*dir(-76+55),D--C)[0], E=F+(0,1), G=H-(1,0), P=intersectionpoints(E--F,G--H)[0];
draw(A--B--C--D--cycle);
draw(F--A--H); draw(E--F); draw(G--H);
label(" $A$ ",D2(A),NW);
label(" $B$ ",D2(B),SW);
label(" $C$ ",D2(C),SE);
label(" $D$ ",D2(D),NE);
label(" $E$ ",D2(E),plain.N);
label(" $F$ ",D2(F),S);
label(" $G$ ",D2(G),W);
label(" $H$ ",D2(H),plain.E);
label(" $P$ ",D2(P),SE);
[/asy]

Answer: 45


Find all functions \( f: \mathbb{Z} \rightarrow \mathbb{Z} \) such that for all pairs of integers \((x, y)\),

\[
f(x + f(y)) = f(x) - y
\]

Answer: f(x)=-x


The value of $\frac{x}{2}$ is less than the value of $x^{2}$. The value of $x^{2}$ is less than the value of $x$. Which of the following could be a value of $x$?

Answer: \frac{3}{4}


Let $g(x, y)$ be the function for the set of ordered pairs of positive coprime integers such that:
\begin{align*}
g(x, x) &= x, \\
g(x, y) &= g(y, x), \quad \text{and} \\
(x + y) g(x, y) &= y g(x, x + y).
\end{align*}
Calculate $g(15, 33)$.

Answer: 165


Find the smallest integer $k$ for which the conditions
(1) $a_1,a_2,a_3\cdots$ is a nondecreasing sequence of positive integers
(2) $a_n=a_{n-1}+a_{n-2}$ for all $n>2$
(3) $a_9=k$
are satisfied by more than one sequence.

Answer: 748


Find the largest real number $k$ such that there exists a sequence of positive reals $\left\{a_{i}\right\}$ for which $\sum_{n=1}^{\infty} a_{n}$ converges but $\sum_{n=1}^{\infty} \frac{\sqrt{a_{n}}}{n^{k}}$ does not.

Answer: \frac{1}{2}


A circle $\omega$ with center $O$ is tangent to 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 the ray $BO$ at point $K$. A point $T$ is chosen on the ray $MN$ such that $\angle MTK = \frac{1}{2} \angle ABC$. It turns out that the line $KT$ is tangent to $\omega$. Find the area of triangle $OKT$ if $BM = a$.

Answer: \frac{^2}{2}


What is the following value when expressed as a common fraction: $$\frac{1}{3^{1}}+\frac{1}{3^{2}}+\frac{1}{3^{3}}+\frac{1}{3^{4}}+\frac{1}{3^{5}}+\frac{1}{3^{6}}?$$

Answer: \frac{364}{729}


Given a positive integer  $n$ . One of the roots of a quadratic equation  $x^{2}-ax +2 n = 0$  is equal to $\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+...+\frac{1}{\sqrt{n}}$ . Prove that  $2\sqrt{2n}\le a\le 3\sqrt{n}$

Answer: 2\sqrt{2n}\le\le3\sqrt{n}


Let \( P_{1}, P_{2}, \ldots, P_{6} \) be points in the complex plane, which are also roots of the equation \( x^{6}+6 x^{3}-216=0 \). Given that \( P_{1} P_{2} P_{3} P_{4} P_{5} P_{6} \) is a convex hexagon, determine the area of this hexagon.

Answer: 9\sqrt{3}


$M$ is the midpoint of the side $AB$ in an equilateral triangle $\triangle ABC.$ The point $D$ on the side $BC$ is such that $BD : DC = 3 : 1.$ On the line passing through $C$ and parallel to $MD$ there is a point $T$ inside the triangle $\triangle ABC$ such that $\angle CTA = 150.$ Find the $\angle MT D.$
[i](K. Ivanov )[/i]

Answer: 120^\circ


$ABCD$, a rectangle with $AB = 12$ and $BC = 16$, is the base of pyramid $P$, which has a height of $24$. A plane parallel to $ABCD$ is passed through $P$, dividing $P$ into a frustum $F$ and a smaller pyramid $P'$. Let $X$ denote the center of the circumsphere of $F$, and let $T$ denote the apex of $P$. If the volume of $P$ is eight times that of $P'$, then the value of $XT$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute the value of $m + n$.

Answer: 177


Inside the cube \( ABCD A_1 B_1 C_1 D_1 \) is located the center \( O \) of a sphere with a radius of 10. The sphere intersects the face \( A A_1 D_1 D \) in a circle with a radius of 1, the face \( A_1 B_1 C_1 D_1 \) in a circle with a radius of 1, and the face \( C D D_1 C_1 \) in a circle with a radius of 3. Find the length of the segment \( O D_1 \).

Answer: 17


Given Madeline has 80 fair coins. She flips all the coins. Any coin that lands on tails is tossed again. Additionally, any coin that lands on heads in the first two tosses is also tossed again, but only once. What is the expected number of coins that are heads after these conditions?

Answer: 40


To obtain the graph of the function $y=2\cos \left(2x-\frac{\pi }{6}\right)$, all points on the graph of the function $y=2\sin 2x$ need to be translated $\frac{\pi }{6}$ units to the left.

Answer: \frac{\pi }{6}


Anna and Béla arrange a meeting between 5 PM and $\frac{1}{2} 6$ PM. What is the probability that the one who arrives earlier does not wait more than 10 minutes for the other, given that both of them adhere to the agreed time frame?

Answer: \frac{5}{9}


Compute \( t(0)-t\left(\frac{\pi}{5}\right)+t\left(\frac{2\pi}{5}\right)-t\left(\frac{3\pi}{5}\right)+\ldots+t\left(\frac{8\pi}{5}\right)-t\left(\frac{9\pi}{5}\right) \), where \( t(x) = \cos 5x + * \cos 4x + * \cos 3x + * \cos 2x + *^2 \cos x + * \).

The coefficients indicated by * are missing. A math student claimed that he could compute the sum without knowing the values of the missing coefficients. Is he right?

Answer: 10


Given 100 lines on a plane, let $T$ be the set of right triangles formed by any three of these lines. Find the maximum value of $|T|$.

Answer: 62500


Let \( S \) be a set of size 11. A random 12-tuple \((s_1, s_2, \ldots, s_{12})\) of elements of \( S \) is chosen uniformly at random. Moreover, let \(\pi: S \rightarrow S\) be a permutation of \( S \) chosen uniformly at random. The probability that \( s_{i+1} \neq \pi(s_i) \) for all \( 1 \leq i \leq 12 \) (where \( s_{13} = s_{1} \)) can be written as \(\frac{a}{b}\) where \( a \) and \( b \) are relatively prime positive integers. Compute \( a \).

Answer: 1000000000004


The sequence $\{a\_n\}$ satisfies $a\_1=1$, $a\_2=3$, and $a_{n+2}=|a_{n+1}|-a_{n}$, where $n∈N^{*}$. Let $S_{n}$ denote the sum of the first $n$ terms of the sequence. Find $S_{100}$.

Answer: 89


In $\triangle ABC$, it is known that $\cos C+(\cos A- \sqrt {3}\sin A)\cos B=0$.
(1) Find the measure of angle $B$;
(2) If $\sin (A- \frac {π}{3})= \frac {3}{5}$, find $\sin 2C$.

Answer: \frac {24+7 \sqrt {3}}{50}


Among the values of the function

$$
f(x) = \frac{\left(x^{2} + 1\right)^{2}}{x\left(x^{2} - 1\right)}
$$

find those values which the function takes the most number of times.

Answer: (-\infty,-4)\cup(4,+\infty)


Given an ellipse $T$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with eccentricity $\frac{\sqrt{3}}{2}$, a line passing through the right focus $F$ with slope $k (k > 0)$ intersects $T$ at points $A$ and $B$. If $\overline{AF} = 3\overline{FB}$, determine the value of $k$.

Answer: \sqrt{2}


In a certain World Cup, the Brazil team's probabilities of outcomes against each opponent are as follows:
- Winning: $\frac{1}{2}$
- Drawing: $\frac{1}{3}$
- Losing: $\frac{1}{6}$

Points awarded are:
- Win: 3 points
- Draw: 1 point
- Loss: 0 points

During the group stage, each team plays three matches, and a team needs to earn at least 4 points to advance to the knockout stage. In the knockout stage, if a match ends in a draw, it goes to a penalty shootout. The probability of Brazil winning a penalty shootout is $\frac{3}{5}$. What is the probability that Brazil wins the championship with exactly one of the four knockout stage matches decided by a penalty shootout?

Answer: \frac{1}{12}


The ancient Chinese mathematical classic "The Nine Chapters on the Mathematical Art" contains a problem called "Rice and Grain Separation". During the collection of grain in a granary, 1524 "shi" (a unit of weight) of rice was received, but it was found to contain grains of another type mixed in. A sample of rice was taken and it was found that out of 254 grains, 28 were not rice. Approximately how much of this batch of rice is not rice?

Answer: 168


The circle inscribed in triangle \(ABC\) touches its sides \(AB\), \(BC\), and \(AC\) at points \(K\), \(M\), and \(N\) respectively. Find the angle \(\angle KMN\) if \(\angle A = 70^\circ\).

Answer: 55


Let
\[\mathbf{M} = \begin{pmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b \end{pmatrix}.\]If $\mathbf{M} \mathbf{M}^T = 9 \mathbf{I},$ then enter the ordered pair $(a,b).$

Note: For a matrix $\mathbf{A},$ $\mathbf{A}^T$ is the transpose of $\mathbf{A},$ which is generated by reflecting the matrix $\mathbf{A}$ over the main diagonal, going from the upper-left to the lower-right.  So here,
\[\mathbf{M}^T = \begin{pmatrix} 1 & 2 & a \\ 2 & 1 & 2 \\ 2 & -2 & b \end{pmatrix}.\]

Answer: (-2,-1)


For how many ordered pairs of positive integers $(x, y)$ with $x < y$ is the harmonic mean of $x$ and $y$ equal to $4^{15}$?

Answer: 29


For positive integers $n$, let $f(n)$ be the product of the digits of $n$. Find the largest positive integer $m$ such that $$\sum_{n=1}^{\infty} \frac{f(n)}{m\left\lfloor\log _{10} n\right\rfloor}$$ is an integer.

Answer: 2070


Let $ABC$ be a triangle and let $J$ be the center of the $A$-excircle. The reflection of $J$ in $BC$ is $K$. The points $E$ and $F$ are on $BJ$ and $CJ$, respectively, such that $\angle EAB = \angle CAF = 90^\circ$. Prove that $\angle FKE + \angle FJE = 180^\circ$.

Answer: 180


Given a vertical wooden pillar, a rope is tied to its top, with 4 feet of the rope hanging down to the ground. Additionally, when pulling the rope, it runs out when 8 feet away from the base of the pillar. Determine the length of the rope.

Answer: 10


Tanya took a list of one hundred numbers $1, 2, 3, \ldots, 100$ and crossed out some of them. It turned out that for any two numbers $a$ and $b$ from the remaining numbers, the equation $x^2 + ax + b = 0$ has at least one real root. What is the maximum number of numbers that could remain uncrossed?

Answer: 81


Equilateral triangle $T$ is inscribed in circle $A$, which has radius $10$. Circle $B$ with radius $3$ is internally tangent to circle $A$ at one vertex of $T$. Circles $C$ and $D$, both with radius $2$, are internally tangent to circle $A$ at the other two vertices of $T$. Circles $B$, $C$, and $D$ are all externally tangent to circle $E$, which has radius $\dfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[asy] unitsize(3mm); defaultpen(linewidth(.8pt)); dotfactor=4; pair A=(0,0), D=8*dir(330), C=8*dir(210), B=7*dir(90); pair Ep=(0,4-27/5); pair[] dotted={A,B,C,D,Ep}; draw(Circle(A,10)); draw(Circle(B,3)); draw(Circle(C,2)); draw(Circle(D,2)); draw(Circle(Ep,27/5)); dot(dotted); label("$E$",Ep,E); label("$A$",A,W); label("$B$",B,W); label("$C$",C,W); label("$D$",D,E); [/asy]

Answer: 32


Piglet ran down a moving escalator and counted 66 steps. Then he ran up the same escalator at the same speed relative to the escalator and counted 198 steps. How many steps would he count if he went down a stationary escalator?

Answer: 99


Suppose that  $a_1, a_2, a_3, \ldots$  is an infinite geometric sequence such that for all  $i \ge 1$ ,  $a_i$  is a positive integer. Suppose furthermore that  $a_{20} + a_{21} = 20^{21}$ . If the minimum possible value of  $a_1$  can be expressed as  $2^a 5^b$  for positive integers  $a$  and  $b$ , find  $a + b$ .

*Proposed by Andrew Wu*

Answer: 24


A line with slope $k$ intersects the parabola $y^{2}=4x$ at points $A$ and $B$, and is tangent to the circle $(x-5)^{2}+y^{2}=9$ at point $M$, where $M$ is the midpoint of segment $AB$. Find the value of $k$.

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


One of Euler's conjectures was disproved in the 1960s by three American mathematicians when they showed there was a positive integer such that $133^5+110^5+84^5+27^5=n^{5}$. Find the value of $n$.

Answer: 144


Given 9 points in space, where no 4 points lie on the same plane. Connect some of these points with line segments while ensuring that no tetrahedron is formed. How many triangles can the graph contain at most?

Answer: 27


Suppose that $p$ is prime and $1007_p+306_p+113_p+125_p+6_p=142_p+271_p+360_p$. How many possible values of $p$ are there?

Answer: 0


Someone bought 5 consecutive train ticket numbers, and the sum of these 5 ticket numbers is 120. What is the product of these 5 ticket numbers?

Answer: 7893600


There are three committees of bureaucrats. It is known that for each pair of bureaucrats from different committees, among the members of the remaining committee, there are exactly 10 bureaucrats who know both of them, and exactly 10 bureaucrats who do not know both of them. Find the total number of bureaucrats in the committees.

Answer: 120


234. Through the point \( M \) of intersection of the diagonals \( AC \) and \( BD \) of the quadrilateral \( ABCD \), a secant line is drawn parallel to the side \( AB \) and intersecting the other three sides or their extensions respectively at points \( P, Q, R \) (point \( R \) lies on the line \( CD \)). Prove that the segments formed on the secant line satisfy the equality: \( RM^2 = RP \cdot RQ \).

Answer: RM^{2}=RP\cdotRQ


Find the coordinates of all points on the plane that are equidistant from all the points of intersection of the parabolas given in the Cartesian coordinate system by the equations \( y = 2x^2 - 1 \) and \( x = 4y^2 - 2 \).

Answer: (\frac{1}{8},\frac{1}{4})


On the board, there are three two-digit numbers. One starts with 5, another starts with 6, and the third one starts with 7. The teacher asked three students to each choose any two of these numbers and add them together. The first student got 147, and the results of the second and third students were different three-digit numbers starting with 12. What could be the number starting with 7? If there are multiple answers, list them all.

Answer: 78


For how many positive numbers less than 1000 is it true that among the numbers $2, 3, 4, 5, 6, 7, 8,$ and $9$, there is exactly one that is not its divisor?

Answer: 4


The sequence $\{a_n\}$ satisfies $a_{n+1} + (-1)^n a_n = 2n - 1$. Find the sum of the first 60 terms of $\{a_n\}$.

Answer: 1830


Find the last three digits of \(1 \times 3 \times 5 \times \cdots \times 1997\).

Answer: 375


After the 8th round of the game, Zhuzhuxia has defeated 20 monsters. How many times has Zhuzhuxia been defeated by the monsters?

Answer: 8


A fair coin is flipped $7$ times. What is the probability that at least $5$ consecutive flips come up heads?

Answer: \frac{1}{16}


Point $A$ lies on the line $y=\frac{8}{15} x-6$, and point $B$ on the parabola $y=x^{2}$. What is the minimum length of the segment $AB$?

Answer: 1334/255


Given that \( 2^{a} \times 3^{b} \times 5^{c} \times 7^{d} = 252000 \), what is the probability that a three-digit number formed by any 3 of the natural numbers \( a, b, c, d \) is divisible by 3 and less than 250?

Answer: 1/4


Let \( O \) be the point \( (0,0) \). Let \( A, B, C \) be three points in the plane such that \( AO=15 \), \( BO=15 \), and \( CO=7 \), and such that the area of triangle \( ABC \) is maximal. What is the length of the shortest side of \( ABC \)?

Answer: 20


In the geometric sequence $\{a_{n}\}$, $a_{3}$ and $a_{7}$ are two distinct extreme points of the function $f\left(x\right)=\frac{1}{3}x^{3}+4x^{2}+9x-1$. Find $a_{5}$.

Answer: -3


In triangle \( \triangle ABC \), \( a, b, c \) are the sides opposite to angles \( \angle A, \angle B, \angle C \) respectively. If \( \frac{a}{b}+\frac{b}{a}=4 \cos C \), and \( \cos (A-B)=\frac{1}{6} \), then \( \cos C \) is?

Answer: \frac{2}{3}


Find all natural numbers  $n$  for which equality holds  $n + d (n) + d (d (n)) +... = 2021$ , where  $d (0) = d (1) = 0$  and for  $k> 1$ ,  $ d (k)$  is the *superdivisor* of the number  $k$  (i.e. its largest divisor of  $d$  with property  $d <k$ ).

(Tomáš Bárta)

Answer: 1919


A positive integer  $n$  is called *Olympic*, if there exists a quadratic trinomial with integer coeffecients  $f(x)$  satisfying  $f(f(\sqrt{n}))=0$ . Determine, with proof, the largest Olympic number not exceeding  $2015$ .

*A. Khrabrov*

Answer: 2010


Given two sets
$$
\begin{array}{l}
A=\{(x, y) \mid |x|+|y|=a, a>0\}, \\
B=\{(x, y) \mid |xy|+1=|x|+|y|\}.
\end{array}
$$

If \( A \cap B \) is the set of vertices of a regular octagon in the plane, determine the value of \( a \).

Answer: 2 + \sqrt{2}


Elmer the emu takes $44$ equal strides to walk between consecutive telephone poles on a rural road. Oscar the ostrich can cover the same distance in $12$ equal leaps. The telephone poles are evenly spaced, and the $41$st pole along this road is exactly one mile ($5280$ feet) from the first pole. How much longer, in feet, is Oscar's leap than Elmer's stride?
$\textbf{(A) }6\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }11\qquad\textbf{(E) }15$

Answer: \textbf{(B)}8


Show that there are no two consecutive even integers in the Fibonacci sequence.

Answer: 2


The function \( f \) maps the set of positive integers onto itself and satisfies the equation

\[
f(f(n)) + f(n) = 2n + 6
\]

What could this function be?

Answer: f(n)=n+2


Given 2016 points in the plane, with no three points collinear, demonstrate that it is possible to construct 504 pairwise disjoint quadrilaterals, not necessarily convex, whose vertices are the given points.

Answer: 504


The photographer wants to arrange three boys and three girls in a row such that a boy or a girl could be at each end, and the rest alternate in the middle, calculate the total number of possible arrangements.

Answer: 72


Manya has a stack of blocks comprised of 4 layers, where the \( k \)-th layer from the top has \( 4^{k-1} \) blocks. Each block rests on 4 smaller blocks, each with dimensions half those of the larger block. Laura removes blocks one at a time from this stack, removing only blocks that currently have no blocks on top of them. Find the number of ways Laura can remove precisely 5 blocks from Manya's stack. Note that the order in which the blocks are removed matters.

Answer: 3384


Given that $\alpha \in \left( 0, \pi \right)$ and $3\cos 2\alpha = \sin \left( \frac{\pi}{4} - \alpha \right)$, find the value of $\sin 2\alpha$.

Answer: -\frac{17}{18}


Answer: 8


The slope of the tangent line to the curve $y=\frac{1}{3}{x^3}-\frac{2}{x}$ at $x=1$ is $\alpha$. Find $\frac{{sin\alpha cos2\alpha}}{{sin\alpha+cos\alpha}}$.

Answer: -\frac{3}{5}


Answer: 1


Given that the sum of the first $n$ terms of the sequence ${a_n}$ is $S_n$, and the sum of the first $n$ terms of the sequence ${b_n}$ is $T_n$. It is known that $a_1=2$, $3S_n=(n+m)a_n$, ($m\in R$), and $a_nb_n=\frac{1}{2}$. If for any $n\in N^*$, $\lambda>T_n$ always holds true, then the minimum value of the real number $\lambda$ is $\_\_\_\_\_\_$.

Answer: \frac{1}{2}


Draw the perpendicular to the tangent line of the ellipse  $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\ (a>0,\ b>0)$  from the origin  $O(0,\ 0).$

Let  $\theta$  be the angle between the perpendicular and the positive direction of  $x$  axis. Denote the length of the perpendicular by  $r(\theta).$

Calculate  $\int_{0}^{2\pi}r(\theta )^{2}\ d\theta.$

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


At the first site, higher-class equipment was used, while at the second site, first-class equipment was used, with higher-class being less than first-class. Initially, 30% of the equipment from the first site was transferred to the second site. Then, 10% of the equipment at the second site was transferred to the first site, with half of the transferred equipment being first-class. After this, the amount of higher-class equipment at the first site exceeded that at the second site by 6 units, and the total amount of equipment at the second site increased by more than 2% compared to the initial amount. Find the total amount of first-class equipment.

Answer: 17


A settlement is built in the shape of a square consisting of 3 blocks by 3 blocks (each block is a square with a side length $b$, for a total of 9 blocks). What is the shortest path that a paver must take to pave all the streets if they start and end their path at a corner point $A$? (The sides of the square are also streets).

Answer: 28b


Find the area of the polygon whose vertex coordinates are the solutions of the system of equations

\[
\left\{
\begin{array}{l}
x^{4}+\frac{7}{2} x^{2} y+2 y^{3}=0 \\
4 x^{2}+7 x y+2 y^{3}=0
\end{array}
\right.
\]

Answer: 8.25


A shape is created by joining seven unit cubes, as shown. What is the ratio of the volume in cubic units to the surface area in square units?
[asy] import three; defaultpen(linewidth(0.8)); real r=0.5; currentprojection=orthographic(1,1/2,1/4); draw(unitcube, white, thick(), nolight); draw(shift(1,0,0)*unitcube, white, thick(), nolight); draw(shift(1,-1,0)*unitcube, white, thick(), nolight); draw(shift(1,0,-1)*unitcube, white, thick(), nolight); draw(shift(2,0,0)*unitcube, white, thick(), nolight); draw(shift(1,1,0)*unitcube, white, thick(), nolight); draw(shift(1,0,1)*unitcube, white, thick(), nolight);[/asy]
$\textbf{(A)} \:1 : 6 \qquad\textbf{ (B)}\: 7 : 36 \qquad\textbf{(C)}\: 1 : 5 \qquad\textbf{(D)}\: 7 : 30\qquad\textbf{ (E)}\: 6 : 25$

Answer: \textbf{(D)}\:7 :30


An alloy consists of zinc and copper in the ratio of $1:2$, and another alloy contains the same metals in the ratio of $2:3$. How many parts of the two alloys can be combined to obtain a third alloy containing the same metals in the ratio of $17:27$?

Answer: 9/35


In triangle \(ABC\), where all sides are different, the angle bisector of \(\angle A\) intersects side \(BC\) at point \(D\). It is known that \( |AB| - |BD| = a \) and \( |AC| + |CD| = b \). Find \( |AD| \).

Answer: \sqrt{}


Two cars, Car A and Car B, simultaneously depart from point $A$ to point $B$. At the start, Car A’s speed is 2.5 km/h faster than Car B’s speed. After 10 minutes, Car A slows down; another 5 minutes later, Car B also slows down, and at this time, Car B is 0.5 km/h slower than Car A. After another 25 minutes, both cars arrive at point $B$ at the same time. By how many kilometers per hour did Car A's speed decrease?

Answer: 10


The sum of a set of numbers is the sum of all its elements. Let \( S \) be a set consisting of positive integers not exceeding 15, where the sums of any two disjoint subsets of \( S \) are not equal. Among all sets with this property, find the \( S \) with the maximum sum, and determine the sum of this set.

Answer: 61


It is known that the line \( l \) passes through the point \( P(3,0) \) with a slope \( k \), and intersects the right branch of the hyperbola \( C: x^{2}-\frac{y^{2}}{3}=1 \) at points \( A \) and \( B \). The focus \( F \) of the hyperbola \( C \) is on the right, and \( |AF| + |BF| = 16 \). Find the value of \( k \).

Answer: \3


In the diagram, the largest circle has a radius of 10 meters. Seven congruent smaller circles are symmetrically aligned in such a way that in an east-to-west and north-to-south orientation, the diameter of four smaller circles equals the diameter of the largest circle. What is the radius in meters of one of the seven smaller circles?

Answer: 2.5


Given \( x_{i} \geq 0 \) for \( i = 1, 2, \cdots, n \) and \( \sum_{i=1}^{n} x_{i} = 1 \) with \( n \geq 2 \), find the maximum value of \( \sum_{1 \leq i \leq j \leq n} x_{i} x_{j} (x_{i} + x_{j}) \).

Answer: \frac{1}{4}


A positive integer will be called "sparkly" if its smallest (positive) divisor, other than 1, equals the total number of divisors (including 1). How many of the numbers $2,3, \ldots, 2003$ are sparkly?

Answer: 3


Determine the number of ways to arrange the letters of the word "PERCEPTION".

Answer: 907200


Answer: 5 \sqrt[4]{5}


All segments joining  $n$  points (no three of which are collinear) are coloured in one of  $k$  colours. What is the smallest  $k$  for which there always exists a closed polygonal line with the vertices at some of the  $n$  points, whose sides are all of the same colour?

Answer: \lceil\frac{n}{2}\rceil-1


The cost of a 3 minute cell phone call is $0.18. At the same rate per minute, what is the cost of a 10 minute call?
(A) $0.30
(B) $0.60
(C) $1.80
(D) $5.40
(E) $6.00

Answer: 0.60


Arsenius has 2018 buckets, each containing 1, 2, 3, ... 2017, 2018 liters of water respectively. Arsenius is allowed to take any two buckets and pour into the second bucket exactly as much water as is already in the second bucket from the first bucket. Can Arsenius collect all the water into one bucket? All buckets are large enough to hold all the water they get.

Answer: No


Suppose
\[\frac{1}{x^3 - 3x^2 - 13x + 15} = \frac{A}{x+3} + \frac{B}{x-1} + \frac{C}{(x-1)^2}\]
where $A$, $B$, and $C$ are real constants. What is $A$?

Answer: \frac{1}{16}


Given a function $f(x)$ defined on $R$ such that $f(x) + f(x+4) = 23$. When $x \in (0,4]$, $f(x) = x^2 - 2^x$. Find the number of zeros of the function $f(x)$ on the interval $(-4,2023]$.

Answer: 506


Find the three-digit integer in the decimal system that satisfies the following properties:
1. When the digits in the tens and units places are swapped, the resulting number can be represented in the octal system as the original number.
2. When the digits in the hundreds and tens places are swapped, the resulting number is 16 less than the original number when read in the hexadecimal system.
3. When the digits in the hundreds and units places are swapped, the resulting number is 18 more than the original number when read in the quaternary system.

Answer: 139


In triangle \(A B C, A B=6, B C=7\) and \(C A=8\). Let \(D, E, F\) be the midpoints of sides \(B C\), \(A C, A B\), respectively. Also let \(O_{A}, O_{B}, O_{C}\) be the circumcenters of triangles \(A F D, B D E\), and \(C E F\), respectively. Find the area of triangle \(O_{A} O_{B} O_{C}\).

Answer: \frac{21 \sqrt{15}}{16}


Find $B^2$, where $B$ is the sum of the absolute values of all roots of the following equation:
\[ x = \sqrt{29} + \frac{121}{{\sqrt{29}+\frac{121}{{\sqrt{29}+\frac{121}{{\sqrt{29}+\frac{121}{{\sqrt{29}+\frac{121}{x}}}}}}}}}.\]

Answer: 513


Triangle ABC is right-angled at A. D is a point on AB such that CD = 1. AE is the altitude from A to BC. If BD = BE = 1, what is the length of AD?

Answer: \sqrt[3]{2}-1


Let \( p(x) \) be the product of the digits of the decimal number \( x \). Find all positive numbers \( x \) for which \( p(x) = x^2 - 10x - 22 \).

Answer: 12


In a certain exam, students' math scores follow a normal distribution $N(100,100)$. It is known that there are 1000 students taking this exam. Then the number of students whose math scores are between 70 and 110 points is approximately ____.

(Reference data: $P(\mu -\sigma < X < \mu +\sigma )\approx 0.6827, P(\mu -3\sigma < X < \mu +3\sigma )\approx 0.9973$)

Answer: 840


Rectangle $ABCD$ has side lengths $AB=84$ and $AD=42$. Point $M$ is the midpoint of $\overline{AD}$, point $N$ is the trisection point of $\overline{AB}$ closer to $A$, and point $O$ is the intersection of $\overline{CM}$ and $\overline{DN}$. Point $P$ lies on the quadrilateral $BCON$, and $\overline{BP}$ bisects the area of $BCON$. Find the area of $\triangle CDP$.

Answer: 546


On a board, there are written four non-zero numbers such that the sum of any three of them is less than the fourth number. What is the minimum number of negative numbers that can be written on the board? Justify your answer.

Answer: 3


Ed and Ann both have lemonade with their lunch. Ed orders the regular size. Ann gets the large lemonade, which is 50% more than the regular. After both consume $\frac{3}{4}$ of their drinks, Ann gives Ed a third of what she has left, and 2 additional ounces. When they finish their lemonades they realize that they both drank the same amount. How many ounces of lemonade did they drink together?

Answer: 40


In triangle $ABC,\,$ angle $C$ is a right angle and the altitude from $C\,$ meets $\overline{AB}\,$ at $D.\,$ The lengths of the sides of $\triangle ABC\,$ are integers, $BD=29^3,\,$ and $\cos B=m/n\,$, where $m\,$ and $n\,$ are relatively prime positive integers. Find $m+n.\,$

Answer: 450


Robert has 4 indistinguishable gold coins and 4 indistinguishable silver coins. Each coin has an engraving of one face on one side, but not on the other. He wants to stack the eight coins on a table into a single stack so that no two adjacent coins are face to face. Find the number of possible distinguishable arrangements of the 8 coins.

Answer: 630


Let $S$ denote the set of all triples $(i, j, k)$ of positive integers where $i+j+k=17$. Compute $$\sum_{(i, j, k) \in S} i j k$$

Answer: 11628


Given a tetrahedron \( ABCD \) where \( AB = \sqrt{3}, AD = BC = \sqrt{10}, \) and \( AC = CD = BD = \sqrt{7} \). What is the volume of this tetrahedron?

Answer: \sqrt{2}


Prove the following inequality:

$$
(a+b)(a+b-2c) + (b+c)(b+c-2a) + (c+a)(c+a-2b) \geqq 0.
$$

Answer: (+b)(+b-2c)+(b+)(b+-2a)+(+)(+-2b)\ge0


Given a triangle \( \triangle ABC \) with internal angles \( A, B, \) and \( C \), what is the sum of the numerator and denominator when the maximum value of \( (\sin A \cos B + \sin B \cos C + \sin C \cos A)^{2} \) is expressed in its simplest fractional form?

Answer: 43


Katie has a fair 2019-sided die with sides labeled $1,2, \ldots, 2019$. After each roll, she replaces her $n$-sided die with an $(n+1)$-sided die having the $n$ sides of her previous die and an additional side with the number she just rolled. What is the probability that Katie's $2019^{\text {th }}$ roll is a 2019?

Answer: \frac{1}{2019}


Write any natural number on a piece of paper, and rotate the paper 180 degrees. If the value remains the same, such as $0$, $11$, $96$, $888$, etc., we call such numbers "神马数" (magical numbers). Among all five-digit numbers, how many different "magical numbers" are there?

Answer: 60


Given the function $f(x)=\sqrt{3}\sin^{2}(x+\frac{\pi}{4})-\cos^{2}x-\frac{1+\sqrt{3}}{2} (x\in\mathbb{R})$.
(1) Find the minimum value and the minimum positive period of the function $f(x)$;
(2) If $A$ is an acute angle, and vector $\overrightarrow{m}=(1,5)$ is perpendicular to vector $\overrightarrow{n}=(1,f(\frac{\pi}{4}-A))$, find $\cos 2A$.

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


Given that the function \( y = (a \cos^{2} x - 3) \sin x \) has a minimum value of -3, determine the range of the real number \( a \).

Answer: [-\frac{3}{2},12]


Find all integer values of the parameter \(a\) for which the system
\[
\begin{cases}
x - 2y = y^2 + 2, \\
ax - 2y = y^2 + x^2 + 0.25a^2
\end{cases}
\]
has at least one solution. In the answer, indicate the sum of the found values of the parameter \(a\).

Answer: 10


Given is a triangle  $ABC$ , the inscribed circle  $G$  of which has radius  $r$ . Let  $r_a$  be the radius of the circle touching  $AB$ ,  $AC$  and  $G$ . [This circle lies inside triangle  $ABC$ .] Define  $r_b$  and  $r_c$  similarly. Prove that  $r_a + r_b + r_c \geq r$  and find all cases in which equality occurs.

*Bosnia - Herzegovina Mathematical Olympiad 2002*

Answer: r_a+r_b+r_c\ger


Given $a \gt 0$, $b\in R$, if the inequality $\left(ax-2\right)(-x^{2}-bx+4)\leqslant 0$ holds for all $x \gt 0$, then the minimum value of $b+\frac{3}{a}$ is ______.

Answer: 2\sqrt{2}


In an isosceles triangle \( ABC \) with \( AB = BC \), the altitude \( CD \) is drawn. The angle \( BAC \) is \( \alpha \). The radius of the circle passing through the points \( A, C, \) and \( D \) is \( R \). Find the area of the triangle \( ABC \).

Answer: R^2\tan(\alpha)


The number 1000 can be written as the product of two positive integers, neither of which contains zeros. The sum of these two integers is:

(A) 65
(B) 110
(C) 133
(D) 205
(E) 1001

Answer: 133


At the height \(BH\) of triangle \(ABC\), a certain point \(D\) is marked. Line \(AD\) intersects side \(BC\) at point \(E\), and line \(CD\) intersects side \(AB\) at point \(F\). Points \(G\) and \(J\) are the projections of points \(F\) and \(E\) respectively onto side \(AC\). The area of triangle \(H E J\) is twice the area of triangle \(H F G\). In what ratio does the height \(BH\) divide segment \(FE\)?

Answer: \sqrt{2}: 1


When \(2x^2\) is added to a quadratic polynomial \(f(x)\), its maximum value increases by 10, and when \(5x^2\) is subtracted from it, its maximum value decreases by \(\frac{15}{2}\). By how much will the maximum value of \(f(x)\) change if \(3x^2\) is added to it?

Answer: \frac{45}{2}


Find the real solution \( x, y, z \) to the equations \( x + y + z = 5 \) and \( xy + yz + zx = 3 \) such that \( z \) is the largest possible value.

Answer: \frac{13}{3}


A circular sheet of iron with a radius of 6 has a sector removed, which is $\frac{1}{6}$ of the original area. The remaining part is rolled into the lateral surface of a cone. The volume of the cone is \_\_\_\_\_\_.

Answer: \frac{25\sqrt{11}}{3}\pi


Let $ABC$ be a triangle with orthocenter $H$ and let $P$ be the second intersection of the circumcircle of triangle $AHC$ with the internal bisector of the angle $\angle BAC$.  Let $X$ be the circumcenter of triangle $APB$ and $Y$ the orthocenter of triangle $APC$.  Prove that the length of segment $XY$ is equal to the circumradius of triangle $ABC$.

Answer: XY=R


Given the function $f(x)=x- \frac {1}{x}+2a\ln x$ $(a\in\mathbb{R})$.

$(1)$ Discuss the monotonicity of $f(x)$;

$(2)$ If $f(x)$ has two extreme values $x_{1}$ and $x_{2}$, where $x_{2}\in[e,+\infty)$, find the minimum value of $f(x_{1})-f(x_{2})$.

Answer: \frac {4}{e}


Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=2$, $|\overrightarrow{b}|=1$, and $\overrightarrow{a}\cdot \overrightarrow{b}=-\sqrt{2}$, calculate the angle between vector $\overrightarrow{a}$ and $\overrightarrow{b}$.

Answer: \dfrac{3\pi}{4}


Triangle $ABC$ has $AC = 450$ and $BC = 300$. Points $K$ and $L$ are located on $\overline{AC}$ and $\overline{AB}$ respectively so that $AK = CK$, and $\overline{CL}$ is the angle bisector of angle $C$. Let $P$ be the point of intersection of $\overline{BK}$ and $\overline{CL}$, and let $M$ be the point on line $BK$ for which $K$ is the midpoint of $\overline{PM}$. If $AM = 180$, find $LP$.
[asy] import markers; defaultpen(fontsize(8)); size(300); pair A=(0,0), B=(30*sqrt(331),0), C, K, L, M, P; C = intersectionpoints(Circle(A,450), Circle(B,300))[0]; K =  midpoint(A--C); L = (3*B+2*A)/5; P = extension(B,K,C,L); M = 2*K-P; draw(A--B--C--cycle); draw(C--L);draw(B--M--A); markangle(n=1,radius=15,A,C,L,marker(markinterval(stickframe(n=1),true))); markangle(n=1,radius=15,L,C,B,marker(markinterval(stickframe(n=1),true))); dot(A^^B^^C^^K^^L^^M^^P); label("$A$",A,(-1,-1));label("$B$",B,(1,-1));label("$C$",C,(1,1)); label("$K$",K,(0,2));label("$L$",L,(0,-2));label("$M$",M,(-1,1)); label("$P$",P,(1,1)); label("$180$",(A+M)/2,(-1,0));label("$180$",(P+C)/2,(-1,0));label("$225$",(A+K)/2,(0,2));label("$225$",(K+C)/2,(0,2)); label("$300$",(B+C)/2,(1,1)); [/asy]

Answer: 072


Find all triples $(x, y, z)$ of nonnegative integers such that
$$ x^5+x^4+1=3^y7^z $$

Answer: (0, 0, 0), (1, 1, 0), (2, 0, 2)


Nan Nan and Feng Feng are calculating the same addition problem. Nan Nan got 982, which is correct, but Feng Feng got 577, which is incorrect. The reason Feng Feng got it wrong is that he missed a 0 in the units place of one of the addends. What are the two addends in this addition problem?

Answer: 450,532


The real number sequence \(a_{0}, a_{1}, a_{2}, \cdots, a_{n}, \cdots\) satisfies the following equation: \(a_{0}=a\), where \(a\) is a real number,
\[
a_{n}=\frac{a_{n-1} \sqrt{3}+1}{\sqrt{3}-a_{n-1}}, \quad n \in \mathbf{N}
\]

Find \(a_{1994}=\) ______

Answer: \frac{\sqrt{3}}{1-\sqrt{3}}


Let $f(n)$ and $g(n)$ be functions satisfying
\[f(n) = \begin{cases}\sqrt{n} & \text{ if } \sqrt{n} \text{ is an integer}\\ 1 + f(n+1) & \text{ otherwise} \end{cases}\]and
\[g(n) = \begin{cases}\sqrt{n} & \text{ if } \sqrt{n} \text{ is an integer}\\ 2 + g(n+2) & \text{ otherwise} \end{cases}\]for positive integers $n$. Find the least positive integer $n$ such that $\tfrac{f(n)}{g(n)} = \tfrac{4}{7}$.

Answer: 258


The sum of the product and the sum of two integers is 95. The difference between the product and the sum of these integers is 59. Find the integers.

Answer: 11\\7


Given two circles $\Gamma_{1}$ and $\Gamma_{2}$ intersecting at points $A$ and $B$, let $PQ$ be a common tangent of the two circles where point $P$ lies on $\Gamma_{1}$ and point $Q$ lies on $\Gamma_{2}$. Let $X$ be an arbitrary point on $\Gamma_{1}$, and let the line $AX$ intersect $\Gamma_{2}$ at a second point $Y$. Let $Y'$ be a point on $\Gamma_{2}$ such that $QY = QY'$. The line $Y'B$ intersects $\Gamma_{1}$ at a second point $X'$. Prove that $PX = PX'$.

Answer: PX=PX'


Let $A(2,0)$ be a fixed point in the plane, and let $P\left(\sin \left(2 t-60^{\circ}\right), \cos \left(2 t-60^{\circ}\right)\right)$ be a moving point. Find the area swept by the line segment $AP$ as $t$ changes from $15^{\circ}$ to $45^{\circ}$.

Answer: \frac{\pi}{6}


Given a circle that is tangent to the side \(DC\) of a regular pentagon \(ABCDE\) at point \(D\) and tangent to the side \(AB\) at point \(A\), what is the degree measure of the minor arc \(AD\)?

Answer: 144


The height of a right-angled triangle, dropped to the hypotenuse, divides this triangle into two triangles. The distance between the centers of the inscribed circles of these triangles is 1. Find the radius of the inscribed circle of the original triangle.

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


At the clock tower in the railway station, there is an electronic clock reporting time. Along the boundary of the circular clock face, there is a small colored light at each minute mark. At 9:35:20 PM, how many small colored lights are within the acute angle formed by the minute hand and the hour hand?

Answer: 12


A parallelogram-shaped paper WXYZ with an area of 7.17 square centimeters is placed on another parallelogram-shaped paper EFGH, as shown in the diagram. The intersection points A, C, B, and D are formed, and AB // EF and CD // WX. What is the area of the paper EFGH in square centimeters? Explain the reasoning.

Answer: 7.17


Find the sum of all numbers $x$ satisfying $x+25/x = 10.$

Answer: 5


The function  $f: \mathbb{N}\to\mathbb{N}_{0}$  satisfies for all  $m,n\in\mathbb{N}$ : \[f(m+n)-f(m)-f(n)=0\text{ or }1, \; f(2)=0, \; f(3)>0, \; \text{ and }f(9999)=3333.\] Determine  $f(1982)$ .

Answer: 660


Given the function $y=\sin (2x+1)$, determine the direction and magnitude of the horizontal shift required to obtain this graph from the graph of the function $y=\sin 2x$.

Answer: \frac{1}{2}


The squares in a  $7\times7$  grid are colored one of two colors: green and purple. The coloring has the property that no green square is directly above or to the right of a purple square. Find the total number of ways this can be done.

Answer: 3432


Given the function $f(x)=x+\sin \pi x-3$, calculate the value of $f\left( \dfrac {1}{2015}\right)+f\left( \dfrac {2}{2015}\right)+f\left( \dfrac {3}{2015}\right)+\ldots+f\left( \dfrac {4029}{2015}\right)$.

Answer: -8058


For positive integers \( n \), let \( L(n) \) be the largest factor of \( n \) other than \( n \) itself. Determine the number of ordered pairs of composite positive integers \( (m, n) \) for which \( L(m) L(n) = 80 \).

Answer: 12


Given positive real numbers \( p \) and \( q \). A positive real number sequence \(\{a_{n}\}\) satisfies:
\[ a_{0} = 1, a_{n+2} = p a_{n} - q a_{n+1} \ (n \in \mathbf{N}). \]

Find all possible values of \( a_{1} \) in terms of \( p \) and \( q \).

Answer: \frac{-\sqrt{q^2+4p}}{2}


In an equilateral triangle \(ABC\) with a side length of \(8 \, \text{cm}\), point \(D\) is the midpoint of side \(BC\), and point \(E\) is the midpoint of side \(AC\). Point \(F\) lies on the segment \(BC\) such that the area of triangle \(ABF\) is equal to the area of quadrilateral \(ABDE\).

Calculate the length of segment \(BF\).

Answer: 6\,


Given that 6 rational numbers are placed sequentially on a given circle, now select any 3 adjacent numbers $a$, $b$, $c$ in a clockwise direction, satisfying $a = |b - c|$. It is also known that the total sum of all the placed numbers is 1. What are the values of these 6 numbers?

Answer: \frac{1}{4}


Central High School is competing against Northern High School in a backgammon match. Each school has three players, and the contest rules require that each player play two games against each of the other school's players. The match takes place in six rounds, with three games played simultaneously in each round. In how many different ways can the match be scheduled?

Answer: 900


Among the positive integers less than $10^{4}$, how many positive integers $n$ are there such that $2^{n} - n^{2}$ is divisible by 7?

Answer: 2857


Arnold has plates weighing  $5$ ,  $15$ ,  $25$ ,  $35$ , or  $45$  pounds. He lifts a barbell, which consists of a  $45$ -pound bar and any number of plates that he has. Vlad looks at Arnold's bar and is impressed to see him bench-press  $600$  pounds. Unfortunately, Vlad mistook each plate on Arnold's bar for the plate one size heavier, and Arnold was actually lifting  $470$  pounds. How many plates did Arnold have on the bar?

*2018 CCA Math Bonanza Team Round #2*

Answer: 13


Six pepperoni circles will exactly fit across the diameter of a $12$-inch pizza when placed. If a total of $24$ circles of pepperoni are placed on this pizza without overlap, what fraction of the pizza is covered by pepperoni?
$\textbf{(A)}\ \frac 12 \qquad\textbf{(B)}\ \frac 23 \qquad\textbf{(C)}\ \frac 34 \qquad\textbf{(D)}\ \frac 56 \qquad\textbf{(E)}\ \frac 78$

Answer: \textbf{(B)}\\frac{2}{3}


If \( n \in \mathbf{N}^{*} \), then \( \lim_{n \rightarrow \infty} \sin^{2}\left(\pi \sqrt{n^{2}+n}\right) = \) ?

Answer: 1


Triangle  $ABC$  has sidelengths  $AB=1$ ,  $BC=\sqrt{3}$ , and  $AC=2$ . Points  $D,E$ , and  $F$  are chosen on  $AB, BC$ , and  $AC$  respectively, such that  $\angle EDF = \angle DFA = 90^{\circ}$ . Given that the maximum possible value of  $[DEF]^2$  can be expressed as  $\frac{a}{b}$  for positive integers  $a, b$  with  $\gcd (a, b) = 1$ , find  $a + b$ . (Here  $[DEF]$  denotes the area of triangle  $DEF$ .)

*Proposed by Vismay Sharan*

Answer: 67


Consider sequences of positive real numbers of the form $x, 2000, y, \dots$ in which every term after the first is 1 less than the product of its two immediate neighbors. For how many different values of $x$ does the term 2001 appear somewhere in the sequence?

Answer: 4


Six circles with radii equal to 1 are placed on a plane such that the distance between the centers of any two of them is greater than \( d \). What is the smallest \( d \) for which there exists a line that does not intersect any of the circles, and on each side of which lie three circles?

Answer: 2


How many sequences of 0s and 1s are there of length 10 such that there are no three 0s or 1s consecutively anywhere in the sequence?

Answer: 178


For each positive integer $n$, let $f(n)$ be the sum of the digits in the base-four representation of $n$ and let $g(n)$ be the sum of the digits in the base-eight representation of $f(n)$. For example, $f(2020) = f(133210_{\text{4}}) = 10 = 12_{\text{8}}$, and $g(2020) = \text{the digit sum of }12_{\text{8}} = 3$. Let $N$ be the least value of $n$ such that the base-sixteen representation of $g(n)$ cannot be expressed using only the digits $0$ through $9$. Find the remainder when $N$ is divided by $1000$.

Answer: 151


Find all non-negative integers \( x, y, z \) such that \( x^{3} + 4y^{3} = 16z^{3} + 4xyz \).

Answer: (0,0,0)


Given a geometric sequence $\{a_n\}$ with the sum of its first n terms denoted as $S_n$, if $S_5$, $S_4$, and $S_6$ form an arithmetic sequence, determine the common ratio $q$ of the sequence $\{a_n\}$.

Answer: -2


Define the Fibonacci numbers by \( F_{0}=0, F_{1}=1, F_{n}=F_{n-1}+F_{n-2} \) for \( n \geq 2 \). For how many \( n, 0 \leq n \leq 100 \), is \( F_{n} \) a multiple of 13?

Answer: 15


For real numbers  $a$  and  $b$ , define  $$ f(a,b) = \sqrt{a^2+b^2+26a+86b+2018}. $$  Find the smallest possible value of the expression   $$ f(a, b) + f (a,-b) + f(-a, b) + f (-a, -b). $$

Answer: 4 \sqrt{2018}


For which values of \( n \) is the fraction

$$
\frac{3n^{2}+2n+4}{n+1}
$$

irreducible?

Answer: n\not\equiv4\pmod{5}


Triangle  $ABC$  has  $BC=1$  and  $AC=2$ . What is the maximum possible value of  $\hat{A}$ .

Answer: 30


If one of the 13 provinces or territories is chosen at random, what is the probability that it joined Canadian Confederation between 1890 and 1969?

Answer: $\frac{4}{13}$


Joey has 30 thin sticks, each stick has a length that is an integer from 1 cm to 30 cm. Joey first places three sticks on the table with lengths of 3 cm, 7 cm, and 15 cm, and then selects a fourth stick such that it, along with the first three sticks, forms a convex quadrilateral. How many different ways are there for Joey to make this selection?

Answer: 17


In the movie "The Wandering Earth 2," there are many UEG (United Earth Government) mechanical devices that are drivable, operable, and deformable, all of which are from the leading Chinese engineering machinery brand - XCMG. Many of the hardcore equipment in the movie are not special effects, but are actually designed and modified from cutting-edge domestic equipment. Many of the equipment can find prototypes in reality. A new device has been developed in a workshop of the group. The specific requirement of the group for the new device is: the product with a part inner diameter (unit: mm) within the range of $(199.82, 200.18)$ is qualified, otherwise it is defective; the part inner diameter $X$ follows a normal distribution $X \sim N(200, 0.0036)$.

$(1)$ If the workshop installed and debugged the new device and produced 5 parts for trial, with measured inner diameters (unit: mm) of $199.87$, $199.91$, $199.99$, $200.13$, $200.19$, if you are in charge of the workshop, try to determine whether this device needs further debugging based on the $3\sigma$ principle? Explain your reasoning.

$(2)$ If the device meets the production requirements of the group, and now 10,000 parts produced by this device are tracked.

① Approximately how many parts out of the 10,000 parts have an inner diameter exceeding $200.12$ mm?

② What is the most likely number of defective parts out of the 10,000 parts?

Reference data:

If a random variable $X \sim N(\mu, \sigma^2)$, then $P(\mu - \sigma < X < \mu + \sigma) \approx 0.683$, $P(\mu - 2\sigma < X < \mu + 2\sigma) \approx 0.955$, $P(\mu - 3\sigma < X < \mu + 3\sigma) \approx 0.997$, $0.997^4 \approx 0.988$, $0.997^5 \approx 0.985$.

Answer: 30


Define the sequence \( a_{n} \) by \[ \sum_{d \mid n} a_{d} = 2^{n}. \] Show that \( n \mid a_{n} \).

Answer: n\mida_n


In the plane rectangular coordinate system $xOy$, the parametric equations of the line $l$ are $\left\{{\begin{array}{l}{x=t}\\{y=-1+\sqrt{3}t}\end{array}}\right.$ (where $t$ is a parameter). Establish a polar coordinate system with the coordinate origin $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis. The polar coordinate equation of the curve $C$ is $\rho =2\sin \theta +2\cos \theta$.
$(1)$ Find the general equation of the line $l$ and the rectangular coordinate equation of the curve $C$;
$(2)$ Let point $P(0,-1)$. If the line $l$ intersects the curve $C$ at points $A$ and $B$, find the value of $\frac{1}{{|{PA}|}}+\frac{1}{{|{PB}|}}$.

Answer: \frac{2\sqrt{3}+1}{3}


In a plane Cartesian coordinate system, a point whose x and y coordinates are both integers is called a "lattice point." How many lattice points are there inside and on the boundaries of the triangle formed by the line $7x + 11y = 77$ and the coordinate axes?

Answer: 49


A non-zero digit is chosen in such a way that the probability of choosing digit $d$ is $\log_{10}{(d+1)}-\log_{10}{d}$. The probability that the digit $2$ is chosen is exactly $\frac{1}{2}$ the probability that the digit chosen is in the set

Answer: {4, 5, 6, 7, 8}


Find the smallest positive integer $a$ such that $x^4 + a^2$ is not prime for any integer $x.$

Answer: 8


Let  $n$  be a positive integer,  $k\in \mathbb{C}$  and  $A\in \mathcal{M}_n(\mathbb{C})$  such that  $\text{Tr } A\neq 0$  and   $$ \text{rank } A +\text{rank } ((\text{Tr } A) \cdot I_n - kA) =n. $$  Find  $\text{rank } A$ .

Answer: k


A quadrilateral \(ABCD\) with perpendicular diagonals is inscribed in a circle.

Perpendiculars dropped from vertices \(B\) and \(C\) to side \(AD\) intersect the diagonals \(AC\) and \(BD\) at points \(E\) and \(F\) respectively. Find \(EF\), given that \(BC = 1\).

Answer: 1


A palindrome is a string that does not change when its characters are written in reverse order. Let \( S \) be a 40-digit string consisting only of 0s and 1s, chosen uniformly at random out of all such strings. Let \( E \) be the expected number of nonempty contiguous substrings of \( S \) which are palindromes. Compute the value of \( \lfloor E \rfloor \).

Answer: 113


An $8 \times 8$ chessboard has one black piece placed in the top right corner. Each move allows placing one white piece on any empty square and changing the colors of any pieces in adjacent squares (white to black, black to white). Determine if it is possible for all squares on the chessboard to eventually have white pieces.

Answer: No


Luna has an infinite supply of red, blue, orange, and green socks. She wants to arrange 2012 socks in a line such that no red sock is adjacent to a blue sock and no orange sock is adjacent to a green sock. How many ways can she do this?

Answer: 4\cdot3^{2011}


How many prime numbers are there among the four-digit numbers whose digits are $1, 2, 3,$ and $4$ in any order?

Answer: 4


On a certain math exam, $10\%$ of the students got $70$ points, $25\%$ got $80$ points, $20\%$ got $85$ points, $15\%$ got $90$ points, and the rest got $95$ points.  What is the difference between the [mean](https://artofproblemsolving.com/wiki/index.php/Mean) and the [median](https://artofproblemsolving.com/wiki/index.php/Median) score on this exam?
$\textbf{(A) }\ 0 \qquad \textbf{(B) }\ 1 \qquad \textbf{(C) }\ 2 \qquad \textbf{(D) }\ 4 \qquad \textbf{(E) }\ 5$

Answer: \textbf{(B)}\1


On a straight road running from north to south, there is a cart driven by Lebed. Exactly at midnight, Rak and Pike choose natural numbers \( m > n \). Every \( n \) minutes (i.e., every \( n, 2n, 3n, \ldots \) minutes after midnight) Pike commands "South!", and every \( m \) minutes (i.e., every \( m, 2m, 3m, \ldots \) minutes after midnight) Rak commands "North!". Upon hearing any command, Lebed immediately starts (or continues) pulling the cart in the specified direction at a speed of 1 m/min. Until the first command, the cart remained stationary. After \( mn \) minutes past midnight, Rak and Pike simultaneously gave Lebed two different commands for the first time, and the exhausted Lebed stopped the cart. At what distance from the starting point did it end up at that moment?

Answer: (-n)n


In $\triangle ABC$, let $a$, $b$, and $c$ be the lengths of the sides opposite angles $A$, $B$, and $C$ respectively, and the three interior angles $A$, $B$, $C$ satisfy $A+C=2B$.
$\text{(1)}$ If $b=2$, find the maximum value of the area of $\triangle ABC$ and determine the shape of the triangle when the maximum area is achieved;
$\text{(2)}$ If $\dfrac {1}{\cos A} + \dfrac {1}{\cos C} = -\dfrac {\sqrt {2}}{\cos B}$, find the value of $\cos \dfrac {A-C}{2}$.

Answer: \dfrac{\sqrt{2}}{2}


Alex and Bobette are playing on a $20 \times 20$ grid where the cells are square and have a side length of 1. The distance between two cells is the distance between their centers. They take turns playing as follows: Alex places a red stone on a cell, ensuring that the distance between any two cells with red stones is never $\sqrt{5}$, then Bobette places a blue stone on the grid without any restriction. The game ends when one of them cannot place a stone anymore. Find the largest $K$ such that Alex can always place at least $K$ stones, no matter how Bobette responds.

Answer: 100


The team wants to win $\frac{3}{4}$ of all their matches in the season. However, in the first third, they won only $55 \%$ of the matches.

a) What percentage of the remaining matches must the team win to achieve their intended goal?

b) If the team won all the remaining matches, what percentage of their matches would they win in the entire season?

Answer: 85


Let \( O = (0,0) \), \( Q = (13,4) \), \( A = (a, a) \), \( B = (b, 0) \), where \(a\) and \(b\) are positive real numbers with \(b \geq a\). The point \( Q \) is on the line segment \( A B \).

(a) Determine the values of \( a \) and \( b \) for which \( Q \) is the midpoint of \( A B \).

(b) Determine all values of \( a \) and \( b \) for which \( Q \) is on the line segment \( A B \) and the triangle \( O A B \) is isosceles and right-angled.

(c) There are infinitely many line segments \( A B \) that contain the point \( Q \). For how many of these line segments are \( a \) and \( b \) both integers?

Answer: 6


A sequence of 11 positive real numbers, $a_{1}, a_{2}, a_{3}, \ldots, a_{11}$, satisfies $a_{1}=4$ and $a_{11}=1024$ and $a_{n}+a_{n-1}=\frac{5}{2} \sqrt{a_{n} \cdot a_{n-1}}$ for every integer $n$ with $2 \leq n \leq 11$. For example when $n=7, a_{7}+a_{6}=\frac{5}{2} \sqrt{a_{7} \cdot a_{6}}$. There are $S$ such sequences. What are the rightmost two digits of $S$?

Answer: 20


Given a hyperbola \( x^{2} - y^{2} = t \) (where \( t > 0 \)), the right focus is \( F \). Any line passing through \( F \) intersects the right branch of the hyperbola at points \( M \) and \( N \). The perpendicular bisector of \( M N \) intersects the \( x \)-axis at point \( P \). When \( t \) is a positive real number (not equal to zero), find the value of \( \frac{|F P|}{|M N|} \).

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


If an item is sold for $x$ dollars, there is a loss of $15\%$ based on the cost. If, however, the same item is sold for $y$ dollars, there is a profit of $15\%$ based on the cost. The ratio of $y:x$ is:

Answer: 23:17


A top hat contains 3 red chips and 2 green chips. Chips are drawn randomly, one at a time without replacement, until all 3 of the reds are drawn or until both green chips are drawn. What is the probability that the 3 reds are drawn?
$\textbf{(A) }\dfrac{3}{10}\qquad\textbf{(B) }\dfrac{2}{5}\qquad\textbf{(C) }\dfrac{1}{2}\qquad\textbf{(D) }\dfrac{3}{5}\qquad \textbf{(E) }\dfrac{7}{10}$

Answer: \textbf{(B)}\frac{2}{5}


Find the area of the axial cross-section of a cylinder inscribed in a unit cube such that the axis of the cylinder lies on the diagonal of the cube and each base touches three faces of the cube at their centers.

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


Find all real numbers $x$ such that
\[\frac{8^x + 27^x}{12^x + 18^x} = \frac{7}{6}.\]Enter all the solutions, separated by commas.

Answer: -1,1


Let  $p>3$  be a prime and let  $a_1,a_2,...,a_{\frac{p-1}{2}}$  be a permutation of  $1,2,...,\frac{p-1}{2}$ . For which  $p$  is it always possible to determine the sequence  $a_1,a_2,...,a_{\frac{p-1}{2}}$  if it for all  $i,j\in\{1,2,...,\frac{p-1}{2}\}$  with  $i\not=j$  the residue of  $a_ia_j$  modulo  $p$  is known?

Answer: p>5


Among the numbers $1^{2}, 2^{2}, 3^{2}, \cdots, 95^{2}$, how many of these 95 numbers have an odd tens digit?

Answer: 19


Let \( f(x) \) and \( g(x) \) be functions defined on \(\mathbf{R}\), with \( f(x) \) being an odd function and \( g(x) \) being an even function, such that \( f(x) + g(x) = 2^x \). If the inequality \( a f(x) + g(2x) \geqslant 0 \) holds for all \( x \in [1, 2] \), determine the range of the real number \( a \).

Answer: [-\frac{17}{6},+\infty)


Find the number of positive integer divisors of \(12!\) that leave a remainder of 1 when divided by 3.

Answer: 66


Jana had to calculate the product of two six-digit numbers for her homework. While copying from the board, she missed one digit in one of the numbers, and instead of a six-digit number, she wrote 85522. When she got home, she realized her mistake but remembered that the number she copied incorrectly was divisible by three. She decided to try to determine what the original number could have been.

Determine how many such six-digit numbers exist.

Answer: 13


During Teacher's Day at a certain school, there are 200 teachers who each send a greeting message to exactly one other teacher. Determine the minimum value of \( k \) such that it is possible to select at least \( k \) representatives to attend a recognition ceremony, where no one in this group has sent a message to anyone else in the group.

Answer: 67


There are boys and girls, $n$ of them in total, sitting around a circle $(n \geq 3)$. If there are $a$ groups of 3 consecutive people having exactly one boy, and $b$ groups of 3 consecutive people having exactly one girl, prove that $3 \mid a - b$.

Answer: 3\mid(b)


Trapezoid $ABCD$ has $\overline{AB}\parallel\overline{CD},BC=CD=43$, and $\overline{AD}\perp\overline{BD}$. Let $O$ be the intersection of the diagonals $\overline{AC}$ and $\overline{BD}$, and let $P$ be the midpoint of $\overline{BD}$. Given that $OP=11$, the length of $AD$ can be written in the form $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. What is $m+n$?
$\textbf{(A) }65 \qquad \textbf{(B) }132 \qquad \textbf{(C) }157 \qquad \textbf{(D) }194\qquad \textbf{(E) }215$

Answer: \textbf{(D)}194


Arrange the 5 numbers \(2, 0, 1, 9, 2019\) in any order to form an 8-digit number (the first digit cannot be 0). Determine the number of different 8-digit numbers that can be produced.

Answer: 95


Given a rectangular grid of dots with 5 rows and 6 columns, determine how many different squares can be formed using these dots as vertices.

Answer: 40


Given that \( x \) and \( y \) are greater than 0, and \( x^{2}+y \geq x^{3}+y^{2} \), find the maximum value of \( x^{2}+y^{2} \).

Answer: 2


Suppose
\[\frac{1}{x^3-7x^2+11x+15} = \frac{A}{x-5} + \frac{B}{x+3} + \frac{C}{(x+3)^2}\]
where $A$, $B$, and $C$ are real constants. What is $A$?

Answer: \frac{1}{64}


From point $P$ on the line $l: x - 2y - 20 = 0$, tangents $PM$ and $PN$ are drawn to the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$ with points of tangency $M$ and $N$, respectively. Connect $M$ and $N$.
(I) Prove that when point $P$ moves on the line $l$, the line $MN$ always passes through a fixed point $Q$ and find the coordinates of point $Q$.
(II) When $MN \parallel l$, prove that the fixed point $Q$ bisects the segment $MN$.

Answer: (\frac{4}{5},-\frac{9}{10})


In $\triangle PQR$, the sum of $\angle P$ and $\angle Q$ is $60^\circ$. The measure of $\angle R$ is:
(A) $60^\circ$
(B) $300^\circ$
(C) $120^\circ$
(D) $30^\circ$
(E) $40^\circ$

Answer: 120


The expression $a^3-a^{-3}$ equals:
$\textbf{(A) \ }\left(a-\frac{1}{a}\right)\left(a^2+1+\frac{1}{a^2}\right) \qquad \textbf{(B) \ }\left(\frac{1}{a}-a\right)\left(a^2-1+\frac{1}{a^2}\right) \qquad \textbf{(C) \ }\left(a-\frac{1}{a}\right)\left(a^2-2+\frac{1}{a^2}\right) \qquad$
$\textbf{(D) \ }\left(\frac{1}{a}-a\right)\left(\frac{1}{a^2}+1+a^2\right) \qquad \textbf{(E) \ }\text{none of these}$

Answer: \textbf{(A)}\(-\frac{1}{})(^2+1+\frac{1}{^2})


At the mountain hut, the coach said, "If we continue at this comfortable pace of $4 \mathrm{~km}$ per hour, we will arrive at the station 45 minutes after the departure of our train."

Then he pointed to a group that had just passed us: "They are using poles, and therefore achieve an average speed of $6 \mathrm{~km}$ per hour. They will arrive at the station half an hour before the departure of our train."

How far was the station from the mountain hut?

Answer: 15


Show that for any positive real number \( x \), the following inequality holds: \( 1 + x^{2018} \geq \frac{(2x)^{2017}}{(1 + x)^{2016}} \).

Answer: 1+x^{2018}\ge\frac{(2x)^{2017}}{(1+x)^{2016}}


Find all functions $f: \mathbf{N} \rightarrow \mathbf{N}$ such that for all $m, n \in \mathbf{N}$, the equation $f(m^{2}+n^{2}) = f^{2}(m) + f^{2}(n)$ is satisfied and $f(1) > 0$.

Answer: f(n)=n


The diagram shows a solid with six triangular faces and five vertices. Andrew wants to write an integer at each of the vertices so that the sum of the numbers at the three vertices of each face is the same. He has already written the numbers 1 and 5 as shown. What is the sum of the other three numbers he will write?

Answer: 11


Solve the system of equations $\left\{\begin{array}{l}x+\sqrt{x+2 y}-2 y=\frac{7}{2}, \\ x^{2}+x+2 y-4 y^{2}=\frac{27}{2}\end{array}\right.$.

Answer: (\frac{19}{4},\frac{17}{8})


In the subtraction shown, $K, L, M$, and $N$ are digits. What is the value of $K+L+M+N$?

Answer: 20


A student rolls two dice simultaneously, and the scores obtained are a and b respectively. The eccentricity e of the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1$ (a > b > 0) is greater than $\frac{\sqrt{3}}{2}$. What is the probability of this happening?

Answer: \frac{1}{6}


Let  $a,b$  be real numbers such that  $|a| \neq |b|$  and  $\frac{a+b}{a-b}+\frac{a-b}{a+b}=6.$  Find the value of the expression  $\frac{a^3+b^3}{a^3-b^3}+\frac{a^3-b^3}{a^3+b^3}.$

Answer: 18/7


Let $M$ be the number of ways to write $3050$ in the form $3050 = b_3 \cdot 10^3 + b_2 \cdot 10^2 + b_1 \cdot 10 + b_0$, where the $b_i$'s are integers, and $0 \le b_i \le 99$. Find $M$.

Answer: 306


Two triangles  $ ABC$ and  $ A'B'C'$  are positioned in the space such that the length of every side of  $ \triangle ABC$  is not less than  $ a$ , and the length of every side of  $ \triangle A'B'C'$  is not less than  $ a'$ . Prove that one can select a vertex of  $ \triangle ABC$  and a vertex of  $ \triangle A'B'C'$  so that the distance between the two selected vertices is not less than  $ \sqrt {\frac {a^2 \plus{} a'^2}{3}}$ .

Answer: \sqrt{\frac{^2+'^2}{3}}


A four-digit palindrome is defined as any four-digit natural number that has the same digit in the units place as in the thousands place, and the same digit in the tens place as in the hundreds place. How many pairs of four-digit palindromes exist whose difference is 3674?

Answer: 35


Given that the ceiling is 3 meters above the floor, the light bulb is 15 centimeters below the ceiling, Alice is 1.6 meters tall and can reach 50 centimeters above the top of her head, and a 5 centimeter thick book is placed on top of a stool to reach the light bulb, find the height of the stool in centimeters.

Answer: 70


Determine all triples  $(x, y, z)$  of positive real numbers which satisfies the following system of equations
\[2x^3=2y(x^2+1)-(z^2+1), \] \[ 2y^4=3z(y^2+1)-2(x^2+1), \] \[ 2z^5=4x(z^2+1)-3(y^2+1).\]

Answer: (1,1,1)


Given complex numbers $\mathrm{z}_{\mathrm{i}} (\mathrm{i} = 1, 2, 3, 4, 5)$ satisfying:
$$\left\{\begin{array}{c}
\left|z_{1}\right| \leq 1, \quad \left|z_{2}\right| \leq 1 \\
\left|2 z_{3}-(\mathrm{z}_{1}+\mathrm{z}_{2})\right| \leq \left|\mathrm{z}_{1}-\mathrm{z}_{2}\right| \\
\left|2 \mathrm{z}_{4}-(\mathrm{z}_{1}+\mathrm{z}_{2})\right| \leq \left|\mathrm{z}_{1}-\mathrm{z}_{2}\right| \\
\left|2 z_{5}-(\mathrm{z}_{3}+\mathrm{z}_{4})\right| \leq \left|\mathrm{z}_{3}-\mathrm{z}_{4}\right|
\end{array} \right.$$
Find the maximum value of $\left|\mathrm{z}_{5}\right|$.

Answer: \sqrt{3}


In a circular circus arena with a radius of 10 meters, a lion is running. Moving along a broken line, it has traveled 30 kilometers. Prove that the sum of all the angles the lion turned is not less than 2998 radians.

Answer: 2998


Call an \( 2n \)-digit number special if we can split its digits into two sets of size \( n \) such that the sum of the numbers in the two sets is the same. Let \( p_{n} \) be the probability that a randomly-chosen \( 2n \)-digit number is special (we will allow leading zeros in the number). The sequence \( p_{n} \) converges to a constant \( c \). Find \( c \).

Answer: \frac{1}{2}


What is the sum of $x + y$ if the sequence $3, ~8, ~13, \ldots, ~x, ~y, ~33$ forms an arithmetic sequence?

Answer: 51


Given $f(α) = \frac{\sin(2π-α)\cos(π+α)\cos\left(\frac{π}{2}+α\right)\cos\left(\frac{11π}{2}-α\right)}{2\sin(3π + α)\sin(-π - α)\sin\left(\frac{9π}{2} + α\right)}$.
(1) Simplify $f(α)$;
(2) If $α = -\frac{25}{4}π$, find the value of $f(α)$.

Answer: -\frac{\sqrt{2}}{4}


In a group, every pair of people is either friendly or hostile. Suppose this group has $n$ people and $q$ friendly pairs, and in any group of three people, at least one pair is hostile. Prove that there is at least one member of this group for whom the set of their enemies contains no more than $q\left(1-4q/n^{2}\right)$ friendly pairs.

Answer: q(1-\frac{4q}{n^2})


In $\triangle ABC$, $\angle ABC=45^\circ$. Point $D$ is on $\overline{BC}$ so that $2\cdot BD=CD$ and $\angle DAB=15^\circ$. Find $\angle ACB.$
$\text{(A) }54^\circ \qquad \text{(B) }60^\circ \qquad \text{(C) }72^\circ \qquad \text{(D) }75^\circ \qquad \text{(E) }90^\circ$

Answer: \textbf{(D)}75


Lines parallel to the sides of a square form a small square whose center coincides with the center of the original square. It is known that the area of the cross, formed by the small square, is 17 times larger than the area of the small square. By how many times is the area of the original square larger than the area of the small square?

Answer: 81


Two positive rational numbers $x$ and $y$, when written in lowest terms, have the property that the sum of their numerators is 9 and the sum of their denominators is 10 . What is the largest possible value of $x+y$ ?

Answer: 73 / 9


\(ABCD - A_{1}B_{1}C_{1}D_{1}\) is a cube with side length 1. Points \(D, E, F\), and \(G\) are the midpoints of \(BA\), \(BC\), \(BB_{1}\), and \(A_{1}D_{1}\) respectively. What is the volume of the tetrahedron \(G-DEF\)?

Answer: \frac{1}{12}


At exactly noon, Anna Kuzminichna looked out the window and saw Klava, the village shop clerk, going on a break. Two minutes past twelve, Anna Kuzminichna looked out the window again, and no one was at the closed store. Klava was absent for exactly 10 minutes, and when she returned, she found that Ivan and Foma were waiting at the door, with Foma evidently arriving after Ivan. Find the probability that Foma had to wait no more than 4 minutes for the store to open.

Answer: 1/2


In an isosceles triangle \( ABC \), the bisectors \( AD, BE, CF \) are drawn.

Find \( BC \), given that \( AB = AC = 1 \), and the vertex \( A \) lies on the circle passing through the points \( D, E, \) and \( F \).

Answer: \frac{\sqrt{17} - 1}{2}


How many nine-digit integers of the form 'pqrpqrpqr' are multiples of 24? (Note that p, q, and r need not be different.)

Answer: 112


Four students participate in a competition where each chooses one question from two options, A and B. The rules result in the following point system: 21 points for correct A, -21 points for incorrect A, 7 points for correct B, and -7 points for incorrect B. If the total score of the four students is 0, calculate the number of different scoring situations.

Answer: 44


The surface of a 3 x 3 x 3 Rubik's Cube consists of 54 cells. What is the maximum number of cells you can mark such that the marked cells do not share any vertices?

Answer: 14


Let $z_1,$ $z_2,$ $\dots,$ $z_{20}$ be the twenty (complex) roots of the equation
\[z^{20} - 4z^{19} + 9z^{18} - 16z^{17} + \dots + 441 = 0.\]Calculate $\cot \left( \sum_{k = 1}^{20} \operatorname{arccot} z_k \right).$  Note that the addition formula for cotangent is still valid when working with complex numbers.

Answer: \frac{241}{220}


Given a prime number \( p \) and a positive integer \( n \) (\( p \geq n \geq 3 \)), the set \( A \) consists of different sequences of length \( n \) formed from the elements of the set \(\{1, 2, \cdots, p\}\) (such that not all elements in a sequence are the same). If, for any two sequences \((x_{1}, x_{2}, \cdots, x_{n})\) and \((y_{1}, y_{2}, \cdots, y_{n})\) in the set \( A \), there exist three different positive integers \( k, l, m \) such that \( x_{k} \neq y_{k}, x_{l} \neq y_{l}, x_{m} \neq y_{m} \), find the maximum number of elements in the set \( A \).

Answer: p^{n-2}


The minimum positive period of the function $f(x) = \sin \omega x + \sqrt{3}\cos \omega x + 1$ ($\omega > 0$) is $\pi$. When $x \in [m, n]$, $f(x)$ has at least 5 zeros. The minimum value of $n-m$ is \_\_\_\_\_\_.

Answer: 2\pi


Xiaoming takes 100 RMB to the store to buy stationery. After returning, he counts the money he received in change and finds he has 4 banknotes of different denominations and 4 coins of different denominations. The banknotes have denominations greater than 1 yuan, and the coins have denominations less than 1 yuan. Furthermore, the total value of the banknotes in units of "yuan" must be divisible by 3, and the total value of the coins in units of "fen" must be divisible by 7. What is the maximum amount of money Xiaoming could have spent?

(Note: The store gives change in denominations of 100 yuan, 50 yuan, 20 yuan, 10 yuan, 5 yuan, and 1 yuan banknotes, and coins with values of 5 jiao, 1 jiao, 5 fen, 2 fen, and 1 fen.)

Answer: 63.37


In the country of Anchuria, a day can either be sunny, with sunshine all day, or rainy, with rain all day. If today's weather is different from yesterday's, the Anchurians say that the weather has changed. Scientists have established that January 1st is always sunny, and each subsequent day in January will be sunny only if the weather changed exactly one year ago on that day. In 2015, January in Anchuria featured a variety of sunny and rainy days. In which year will the weather in January first change in exactly the same pattern as it did in January 2015?

Answer: 2047


White paint is randomly thrown on the floor of a $2 \mathrm{~m} \times 2 \mathrm{~m}$ room. Show that there exist two points of the same color whose distance is exactly $1 \mathrm{~m}$.

Answer: 2


The ratio of the sides of a triangle, which is inscribed in a circle of radius \(2 \sqrt{3}\), is \(3: 5: 7\). Find the area of the triangle.

Answer: \frac{135}{49}\sqrt{3}


Find all pairs \((u, v)\) of natural numbers such that

\[ \frac{uv^3}{u^2 + v^2} \]

is a prime power.

Answer: (2^k,2^k)


The number of sets $A$ that satisfy $\{1, 2\} \subset A \subseteq \{1, 2, 3, 4, 5, 6\}$ must be determined.

Answer: 15


There are 207 different cards with numbers $1, 2, 3, 2^{2}, 3^{2}, \ldots, 2^{103}, 3^{103}$ (each card has one number, and each number appears exactly once). How many ways can you choose 3 cards such that the product of the numbers on the selected cards is a square of an integer that is divisible by 6?

Answer: 267903


In a kingdom, there are counts, dukes, and marquises. One day, each count dueled with three dukes and several marquises. Each duke dueled with two counts and six marquises. Each marquis dueled with three dukes and two counts. It is known that all counts dueled with an equal number of marquises. With how many marquises did each count duel?

Answer: 6


There are 6 locked suitcases and 6 keys for them. However, it is unknown which key opens which suitcase. What is the minimum number of attempts needed to ensure that all suitcases are opened? How many attempts are needed if there are 10 suitcases and 10 keys?

Answer: 45


In triangle  $ABC$ ,  $\angle ABC$  is obtuse.  Point  $D$  lies on side  $AC$  such that  $\angle ABD$  is right, and point  $E$  lies on side  $AC$  between  $A$  and  $D$  such that  $BD$  bisects  $\angle EBC$ .  Find  $CE$  given that  $AC=35$ ,  $BC=7$ , and  $BE=5$ .

Answer: 10


$\triangle PQR$ is similar to $\triangle XYZ$. What is the number of centimeters in the length of $\overline{YZ}$? Express your answer as a decimal to the nearest tenth.

[asy]
draw((0,0)--(10,-2)--(8,6)--cycle);
label("10cm",(6,3),NW);
label("7cm",(10.2,2.5),NE);
draw((15,0)--(23,-1.8)--(22,4.5)--cycle);
label("$P$",(10,-2),E);
label("4cm",(21.2,1.3),NE);
label("$Q$",(8,6),N);
label("$R$",(0,0),SW);
label("$X$",(23,-1.8),E);
label("$Y$",(22,4.5),NW);
label("$Z$",(15,0),SW);
[/asy]

Answer: 5.7


Given an isosceles triangle \( ABC \) (\( AB = BC \)). Point \( E \) is marked on ray \( BA \) beyond point \( A \), and point \( D \) is marked on side \( BC \). It is known that
$$ \angle ADC = \angle AEC = 60^\circ, \quad AD = CE = 13. $$
Find the length of segment \( AE \) if \( DC = 9 \).

Answer: 4


Famous skater Tony Hawk rides a skateboard (segment $AB$) in a ramp, which is a semicircle with a diameter $PQ$. Point $M$ is the midpoint of the skateboard, and $C$ is the foot of the perpendicular dropped from point $A$ to the diameter $PQ$. What values can the angle $\angle ACM$ take if the angular measure of arc $AB$ is $24^{\circ}$?

Answer: 12


For how many natural numbers  $n$  between  $1$  and  $2014$  (both inclusive) is  $\frac{8n}{9999-n}$  an integer?

Answer: 1


Given the curve \(\sqrt{1-(y-1)^{2}}=|x|-1\) and the line \(k x-y=2\) intersect at two different points, find the range of values for the real number \(k\).

Answer: [-2,-\frac{4}{3})\cup(\frac{4}{3},2]


Given an ellipse with the equation \\(\\dfrac{x^{2}}{a^{2}}+\\dfrac{y^{2}}{b^{2}}=1(a > b > 0)\\) and an eccentricity of \\(\\dfrac{\\sqrt{3}}{2}\\). A line $l$ is drawn through one of the foci of the ellipse, perpendicular to the $x$-axis, and intersects the ellipse at points $M$ and $N$, with $|MN|=1$. Point $P$ is located at $(-b,0)$. Point $A$ is any point on the circle $O:x^{2}+y^{2}=b^{2}$ that is different from point $P$. A line is drawn through point $P$ perpendicular to $PA$ and intersects the circle $x^{2}+y^{2}=a^{2}$ at points $B$ and $C$.

(1) Find the standard equation of the ellipse;

(2) Determine whether $|BC|^{2}+|CA|^{2}+|AB|^{2}$ is a constant value. If it is, find that value; if not, explain why.

Answer: 26


Given that each of the smaller rectangles has a shorter side of 7 feet and a longer side with a length three times that of the shorter side, and that two smaller rectangles are placed adjacent to each other by their longer sides and the third rectangle is placed adjacent by its long side to one of the other two rectangles, calculate the area of the larger rectangle EFGH in square feet.

Answer: 588


Find the maximum value of  $M =\frac{x}{2x + y} +\frac{y}{2y + z}+\frac{z}{2z + x}$  ,  $x,y, z > 0$

Answer: 1


Solve the equation \( 2 \sqrt{2} \sin ^{3}\left(\frac{\pi x}{4}\right) = \sin \left(\frac{\pi}{4}(1+x)\right) \).

How many solutions of this equation satisfy the condition: \( 2000 \leq x \leq 3000 \)?

Answer: 250


Isabella's house has $3$ bedrooms. Each bedroom is $12$ feet long, $10$ feet wide, and $8$ feet high. Isabella must paint the walls of all the bedrooms. Doorways and windows, which will not be painted, occupy $60$ square feet in each bedroom. How many square feet of walls must be painted?
$\mathrm{(A)}\ 678 \qquad \mathrm{(B)}\ 768 \qquad \mathrm{(C)}\ 786 \qquad \mathrm{(D)}\ 867 \qquad \mathrm{(E)}\ 876$

Answer: \textbf{(E)}876


On floor 0 of a weird-looking building, you enter an elevator that only has one button. You press the button twice and end up on floor 1. Thereafter, every time you press the button, you go up by one floor with probability $\frac{X}{Y}$, where $X$ is your current floor, and $Y$ is the total number of times you have pressed the button thus far (not including the current one); otherwise, the elevator does nothing. Between the third and the $100^{\text {th }}$ press inclusive, what is the expected number of pairs of consecutive presses that both take you up a floor?

Answer: 97


A six-digit square number is divided into two-digit parts. It is found that the last two-digit part is equal to the middle two-digit part, while the first two-digit part, when added to the middle two-digit part, sums to 100. What is this square number?

Answer: 316969


Two long cylindrical tubes of the same length but different diameters lie parallel to each other on a flat surface. The larger tube has radius $72$ and rolls along the surface toward the smaller tube, which has radius $24$. It rolls over the smaller tube and continues rolling along the flat surface until it comes to rest on the same point of its circumference as it started, having made one complete revolution. If the smaller tube never moves, and the rolling occurs with no slipping, the larger tube ends up a distance $x$ from where it starts. The distance $x$ can be expressed in the form $a\pi+b\sqrt{c},$ where $a,$ $b,$ and $c$ are integers and $c$ is not divisible by the square of any prime. Find $a+b+c.$

Answer: 179


On a plane, a regular hexagon with side length \( a \) is drawn. For any \( n \in \mathbf{N} \), greater than 1, construct a segment of length \( a / n \) using only a straightedge.

Answer: \frac{}{n}


What number is formed from five consecutive digits (not necessarily in order) such that the number formed by the first two digits, when multiplied by the middle digit, gives the number formed by the last two digits? (For example, if we take the number 12896, 12 multiplied by 8 gives 96. However, since 1, 2, 6, 8, 9 are not consecutive digits, this example is not suitable as a solution.)

Answer: 13452


$ABCD$ is a rectangular sheet of paper that has been folded so that corner $B$ is matched with point $B'$ on edge $AD.$ The crease is $EF,$ where $E$ is on $AB$ and $F$ is on $CD.$ The dimensions $AE=8, BE=17,$ and $CF=3$ are given. The perimeter of rectangle $ABCD$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
[asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair A=origin, B=(25,0), C=(25,70/3), D=(0,70/3), E=(8,0), F=(22,70/3), Bp=reflect(E,F)*B, Cp=reflect(E,F)*C; draw(F--D--A--E); draw(E--B--C--F, linetype("4 4")); filldraw(E--F--Cp--Bp--cycle, white, black); pair point=( 12.5, 35/3 ); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$E$", E, dir(point--E)); label("$F$", F, dir(point--F)); label("$B^\prime$", Bp, dir(point--Bp)); label("$C^\prime$", Cp, dir(point--Cp));[/asy]

Answer: 293


In space, two segments $A C$ and $B D$ of length 1 are given. Prove that:

a) At least one of the distances $A B, B C, C D$, and $D A$ is not less than $\frac{\sqrt{2}}{2}$.

b) If the segments $A C$ and $B D$ intersect, then at least one of the distances $A B, B C, C D$, and $D A$ is not greater than $\frac{\sqrt{2}}{2}$.

c) If the segments $A C$ and $B D$ lie in the same plane but do not intersect, then at least one of the distances $A B, B C$, $C D$, and $D A$ is greater than 1.

Answer: 1


In space, there are \(n\) planes. Any 2 planes intersect in a line, any 3 planes intersect at a point, and no 4 planes intersect at a single point. How many non-overlapping regions do these \(n\) planes divide the space into?

Answer: \frac{n^3+5n+6}{6}


Answer: 0


Let  $ABCD$  be a quadrilateral with an inscribed circle, centre  $O$ . Let
\[AO = 5, BO =6, CO = 7, DO = 8.\]
If  $M$  and  $N$  are the midpoints of the diagonals  $AC$  and  $BD$ , determine  $\frac{OM}{ON}$  .

Answer: 35/48


On side \( AC \) of triangle \( ABC \), a circle with center \( O \) is constructed as a diameter, intersecting side \( BC \) of the triangle at point \( D \). Find the radius of the inscribed circle of triangle \( ABC \) if \( OD = 4 \), \( DC = 3 \), and the center of the circumscribed circle of triangle \( ABC \) lies on \( BO \).

Answer: \frac{4\sqrt{55}}{11}


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

Answer: 1


In the bottom-left corner of a $7 \times 7$ chessboard, there is a king. In one move, it can move one square to the right, one square up, or one square diagonally (right and up). How many different paths can the king take to the top-right corner of the board if it is forbidden to visit the central cell?

Answer: 5020


From four non-zero digits, two four-digit numbers were formed: the largest possible and the smallest possible. The sum of the resulting numbers turned out to be 11990. What numbers could have been formed?

Answer: 1999\\9991


The smallest positive integer greater than 1 that leaves a remainder of 1 when divided by 4, 5, and 6 lies between which of the following pairs of numbers?
$\textbf{(A) }2\text{ and }19\qquad\textbf{(B) }20\text{ and }39\qquad\textbf{(C) }40\text{ and }59\qquad\textbf{(D) }60\text{ and }79\qquad\textbf{(E) }80\text{ and }124$

Answer: \textbf{(D)}\


A systematic sampling method is used to select 5 representatives from 752 students, after removing 2 students randomly. Calculate the probability that student A is selected.

Answer: \frac{5}{752}


Let  $AB$  be a segment of unit length and let  $C, D$  be variable points of this segment. Find the maximum value of the product of the lengths of the six distinct segments with endpoints in the set  $\{A,B,C,D\}.$

Answer: \frac{\sqrt{5}}{125}


The Young Pioneers plan to make some lucky stars for the kindergarten children. If each person makes 10 stars, they will still be 6 stars short of completing the plan; if 4 of them each make 8 stars and the rest each make 12 stars, they will just complete the plan. How many lucky stars are planned to be made in total?

Answer: 116


Answer: -5


Paul starts with the number 19. In one step, he can add 1 to his number, divide his number by 2, or divide his number by 3. What is the minimum number of steps Paul needs to get to 1?

Answer: 6


Given a triangle \( \triangle ABC \) with interior angles \( A, B, C \) and opposite sides \( a, b, c \) respectively, consider the vectors \( \boldsymbol{m} = (\sin A, b+c) \) and \( \boldsymbol{n} = (\sin C - \sin B, a - b) \). Suppose there exists a real number \( \lambda \) such that \( \boldsymbol{m} = \lambda \boldsymbol{n} \).
(1) Find the measure of angle \( C \);
(2) If \( a + b = kc \), find the range of the real number \( k \).

Answer: (1,2]


The sum of the digits of the positive integer \(N\) is three times the sum of the digits of \(N+1\). What is the smallest possible sum of the digits of \(N\)?

Answer: 12


\(A_{1}, A_{2}, \ldots, A_{n}\) are distinct points in a plane. Color red the midpoints of all segments determined by these points. What is the minimum number of red points that can be produced?

Answer: 2n-3


In a Go championship participated by three players: A, B, and C, the matches are conducted according to the following rules: the first match is between A and B; the second match is between the winner of the first match and C; the third match is between the winner of the second match and the loser of the first match; the fourth match is between the winner of the third match and the loser of the second match. Based on past records, the probability of A winning over B is 0.4, B winning over C is 0.5, and C winning over A is 0.6.
(1) Calculate the probability of B winning four consecutive matches to end the competition;
(2) Calculate the probability of C winning three consecutive matches to end the competition.

Answer: 0.162


We define an operation * on the set of real numbers with the following properties:

a) \((x * y) * z = x * (y * z)\),

b) \((x * y) * z = (y * z) * x\),

c) if \(x \neq y\), then there exists a \(z\) such that \(z * x \neq z * y\).

Prove that \(x * y = y * x\).

Answer: x*y=y*x


Find the degree measure of the angle

$$
\delta = \arccos \left(\left(\sin 2537^{\circ} + \sin 2538^{\circ} + \cdots + \sin 6137^{\circ}\right)^{\cos 2520^{\circ}} + \cos 2521^{\circ} + \cdots + \cos 6120^{\circ}\right)
$$

Answer: 73


There are 12 stages in a racing tournament and $n$ participants. After each stage, all participants receive points $a_{k}$ based on their position $k$ ($a_{k}$ are natural numbers, and $a_{1}>a_{2}>\ldots>a_{n}$). For what is the smallest $n$ such that the tournament organizer can choose numbers $a_{1}, \ldots, a_{n}$ so that after the penultimate stage, at least two participants will still have a chance to take first place, regardless of the possible distribution of positions?

Answer: 13


How many 4-digit positive multiples of 4 can be formed from the digits 0, 1, 2, 3, 4, 5, 6 such that each digit appears without repetition?

Answer: 176


How many two digit numbers have exactly  $4$  positive factors?  $($ Here  $1$  and  the number  $n$  are also considered as factors of  $n. )$

Answer: 31


Let \( a, b, c \) be positive integers such that \( a \geq b \geq c \), and \(\frac{a-c}{2}\) is a prime number. Prove that if
\[ a^2 + b^2 + c^2 - 2(ab + bc + ca) = b, \]
then \( b \) is either a prime number or a perfect square.

Answer: b


In how many derangements of the 8 letters A, B, C, D, E, F, G, H do the four letters A, C, E, G not appear in their original positions?

Answer: 24024


Team A and Team B each have 7 players who will compete in a fixed order in a Go tournament. The first player of each team plays against each other, the loser is eliminated, and the winner faces the next player from the losing team, and so on, until all players from one team are eliminated. At this point, the other team wins. What is the total number of possible match sequences? (1988 National High School Competition Problem)

Answer: 3432


Solve the system of equations:
\[
\begin{cases}
x^2 + y - 2z = -3 \\
3x + y + z^2 = 14 \\
7x - y^2 + 4z = 25
\end{cases}
\]

Answer: 2,-1,3


Let \( ABC \) be an isosceles triangle with \( AB = AC \) and \(\widehat{A} = 30^\circ\). Let \( D \) be the midpoint of the base \( BC \). On \( AD \) and \( AB \), take two points \( P \) and \( Q \), respectively, such that \( PB = PQ \). Determine the measure of the angle \( \widehat{PQC} \).

Answer: 15


Every cell of a  $3\times 3$  board is coloured either by red or blue. Find the number of all colorings in which there are no  $2\times 2$  squares in which all cells are red.

Answer: 417


Given \(0 \leq x_0 < 1\), let
\[
x_n = \left\{ \begin{array}{ll}
2x_{n-1} & \text{if } 2x_{n-1} < 1 \\
2x_{n-1} - 1 & \text{if } 2x_{n-1} \geq 1
\end{array} \right.
\]
for all integers \(n > 0\). Determine the number of initial values of \(x_0\) that satisfy \(x_0 = x_6\).

Answer: 64


On the board, a certain two-digit number is written. Neznaika claimed that it is divisible by $3, 4, 5, 9, 10, 15, 18, 30$. Znayka, upon hearing this, told Neznaika that he was mistaken exactly 4 times. What number could have been written on the board? List all possible options.

Answer: 36,45,72


Pizzas are sized by diameter. What percent increase in area results if Lorrie’s pizza increases from a 16-inch pizza to an 18-inch pizza?

Answer: 26.5625\%


How many three-eighths are there in $8\frac{5}{3} - 3$?

Answer: 17\frac{7}{9}


Let  $ f : (0,\infty) \to \mathbb R$  be a continous function such that the sequences  $ \{f(nx)\}_{n\geq 1}$  are nondecreasing for any real number  $ x$ . Prove that  $ f$  is nondecreasing.

Answer: f


The area in square units of the region enclosed by parallelogram $ABCD$ is

Answer: 8


Using the six digits $0$, $1$, $2$, $3$, $4$, $5$, form integers without repeating any digit. Determine how many such integers satisfy the following conditions:

$(1)$ How many four-digit even numbers can be formed?

$(2)$ How many five-digit numbers that are multiples of $5$ and have no repeated digits can be formed?

$(3)$ How many four-digit numbers greater than $1325$ and with no repeated digits can be formed?

Answer: 270


Compute the unique positive integer \( n \) such that
\[
3 \cdot 2^3 + 4 \cdot 2^4 + 5 \cdot 2^5 + \dots + n \cdot 2^n = 2^{n + 11}.
\]

Answer: 1025