A frog is placed at the [origin](https://artofproblemsolving.com/wiki/index.php/Origin) on the [number line](https://artofproblemsolving.com/wiki/index.php/Number_line), and moves according to the following rule: in a given move, the frog advances to either the closest [point](https://artofproblemsolving.com/wiki/index.php/Point) with a greater [integer](https://artofproblemsolving.com/wiki/index.php/Integer) [coordinate](https://artofproblemsolving.com/wiki/index.php/Coordinate) that is a multiple of 3, or to the closest point with a greater integer coordinate that is a multiple of 13. A move sequence is a [sequence](https://artofproblemsolving.com/wiki/index.php/Sequence) of coordinates that correspond to valid moves, beginning with 0 and ending with 39. For example, $0,\ 3,\ 6,\ 13,\ 15,\ 26,\ 39$ is a move sequence. How many move sequences are possible for the frog?

Answer: 169


The sphere with radius 1 and center $(0,0,1)$ rests on the $xy$-plane.  A light source is at $P = (0,-1,2).$  Then the boundary of the shadow of the sphere can be expressed in the form $y = f(x),$ for some function $f(x).$  Find the function $f(x).$

Answer: \frac{x^2}{4} - 1


A sphere and a point inside it are given. Three mutually perpendicular planes pass through the point and intersect the sphere in three circles. Prove that the sum of the areas of these three circles is constant, and find this sum if the radius of the sphere is \( R \), and the distance from the intersection point of the planes to the center of the sphere is \( d \).

Answer: \pi(3R^2-^2)


Petya writes different three-digit natural numbers on the board, each of which is divisible by 3, and the first two digits differ by 2. What is the maximum number of such numbers he can write down if they end in 6 or 7?

Answer: 9


A square with sides of 10 inches is shown. If $P$ is a point such that the segments $\overline{PA}$, $\overline{PB}$, and $\overline{PC}$ are equal in length, and segment $\overline{PC}$ is perpendicular to segment $\overline{GD}$, what is the area, in square inches, of triangle $APB$? Here, $G$ is the midpoint of side $AD$.
[asy]
pair A, B, C, D, G, P;
A = (0,0); B= (10,0); C = (5,10); D = (10,10); G = (0,10); P = (5,5);
draw(A--B--D--G--cycle);
draw(C--P); draw(P--A); draw(P--B);
label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,N); label("$D$",D,NE); label("$G$",G,NW);label("$P$",P,NW);
label("$10''$",(5,0),S);
[/asy]

Answer: \frac{75}{4}


Point \( P \) lies inside equilateral triangle \( ABC \) so that \( \angle BPC = 120^\circ \) and \( AP \sqrt{2} = BP + CP \). The ratio \( \frac{AP}{AB} \) can be written as \( \frac{a \sqrt{b}}{c} \), where \( a, b, c \) are integers, \( c \) is positive, \( b \) is square-free, and \(\operatorname{gcd}(a, c) = 1 \). Find \( 100a + 10b + c \).

Answer: 255


Let
\[f(x) = (\arccos x)^3 + (\arcsin x)^3.\]Find the range of $f(x).$  All functions are in radians.

Answer: \left[ \frac{\pi^3}{32}, \frac{7 \pi^3}{8} \right]


Given the real numbers \(\alpha\) and \(\beta\) that satisfy the following system of equations:
\[
\begin{cases}
\alpha^{3} - 3\alpha^{2} + 5\alpha - 17 = 0 \\
\beta^{3} - 3\beta^{2} + 5\beta + 11 = 0
\end{cases}
\]
find the value of \(\alpha + \beta\).

Answer: 2


Triangle  $ABC$  is a triangle with side lengths  $13$ ,  $14$ , and  $15$ . A point  $Q$  is chosen uniformly at random in the interior of  $\triangle{ABC}$ . Choose a random ray (of uniformly random direction) with endpoint  $Q$  and let it intersect the perimeter of  $\triangle{ABC}$  at  $P$ . What is the expected value of  $QP^2$ ?

*2018 CCA Math Bonanza Tiebreaker Round #4*

Answer: \frac{84}{\pi}


A circle is circumscribed around a square with side length $a$, and a regular hexagon is circumscribed around the circle. Determine the area of the hexagon.

Answer: \sqrt{3}^2


Determine the smallest integer $k \geq 1$ such that the value of the expression $\sin(kx) \cdot \sin^k(x) + \cos(kx) \cdot \cos^k(x) - \cos^k(2x)$ does not depend on $x$.

Answer: 3


Let \( n \) be a given positive integer. Find the smallest positive integer \( u_n \) such that, for every positive integer \( d \), the number of numbers divisible by \( d \) in any \( u_n \) consecutive positive odd numbers is no less than the number of numbers divisible by \( d \) in the set of odd numbers \( 1, 3, 5, \ldots, 2n-1 \).

Answer: 2n-1


Let \( A_{1} A_{2} \ldots A_{6} \) be a regular hexagon with side length \( 11 \sqrt{3} \), and let \( B_{1} B_{2} \ldots B_{6} \) be another regular hexagon completely inside \( A_{1} A_{2} \ldots A_{6} \) such that for all \( i \in \{1,2, \ldots, 5\} \), \( A_{i} A_{i+1} \) is parallel to \( B_{i} B_{i+1} \). Suppose that the distance between lines \( A_{1} A_{2} \) and \( B_{1} B_{2} \) is 7, the distance between lines \( A_{2} A_{3} \) and \( B_{2} B_{3} \) is 3, and the distance between lines \( A_{3} A_{4} \) and \( B_{3} B_{4} \) is 8. Compute the side length of \( B_{1} B_{2} \ldots B_{6} \).

Answer: 3\sqrt{3}


The sequence \( \{a_n\} \) satisfies: \( a_1 = 1 \), and for each \( n \in \mathbf{N}^{*} \), \( a_n \) and \( a_{n+1} \) are the roots of the equation \( x^2 + 3n x + b_n = 0 \). Find the value of \( \sum_{k=1}^{20} b_k \).

Answer: 6385


Calculate the probability that four randomly chosen vertices of a cube form a tetrahedron (triangular pyramid).

Answer: \frac{29}{35}


Six distinguishable players are participating in a tennis tournament. Each player plays one match of tennis against every other player. The outcome of each tennis match is a win for one player and a loss for the other players; there are no ties. Suppose that whenever  $A$  and  $B$  are players in the tournament for which  $A$  won (strictly) more matches than  $B$  over the course of the tournament, it is also the case that  $A$  won the match against  $B$  during the tournament. In how many ways could the tournament have gone?

Answer: 720


In the fourth grade, there are 20 boys and 26 girls. The percentage of the number of boys to the number of girls is     %.

Answer: 76.9


$ M$  is a subset of  $ \{1, 2, 3, \ldots, 15\}$  such that the product of any three distinct elements of  $ M$  is not a square. Determine the maximum number of elements in  $ M.$

Answer: 10


Given an acute triangle \( \triangle ABC \), the isosceles triangles \( \triangle DAC \), \( \triangle EAB \), and \( \triangle FBC \) are constructed externally to \( \triangle ABC \), such that \( DA = DC \), \( EA = EB \), \( FB = FC \), and the angles \( \angle ADC = 2 \angle BAC \), \( \angle BEA = 2 \angle ABC \), and \( \angle CFB = 2 \angle ACB \). Let \( D' \) be the intersection of line \( DB \) and \( EF \), \( E' \) be the intersection of line \( EC \) and \( DF \), and \( F' \) be the intersection of line \( FA \) and \( DE \). Find the value of \( \frac{DB}{DD'} + \frac{EC}{EE'} + \frac{FA}{FF'} \).

Answer: 4


Determine all functions \( f: \mathbb{N}^{*} \rightarrow \mathbb{N}^{*} \) such that, for all \( m, n \in \mathbb{N}^{*} \), the following holds:

\[ f(m)^{2} + f(n) \mid (m^{2} + n)^{2} \]

Answer: f(n)=n


Given that \( a, b, c, d \) are certain real values, the equation \( x^{4}+a x^{3}+b x^{2}+c x+d=0 \) has 4 non-real roots. Among these roots, the product of 2 of the roots is \( 13 + \mathrm{i} \) and the sum of the other 2 roots is \( 3 + 4\mathrm{i} \). Here, \( \mathrm{i} \) represents the imaginary unit. Find \( b \).

Answer: 51


A finite set  $M$  of real numbers has the following properties:  $M$  has at least  $4$  elements, and there exists a bijective function  $f:M\to M$ , different from the identity, such that  $ab\leq f(a)f(b)$  for all  $a\neq b\in M.$  Prove that the sum of the elements of  $M$  is  $0.$

Answer: 0


How many ways can you mark 8 squares of an $8 \times 8$ chessboard so that no two marked squares are in the same row or column, and none of the four corner squares is marked? (Rotations and reflections are considered different.)

Answer: 21600


A massive vertical plate is fixed to a car moving at a speed of $5 \, \text{m/s}$. A ball is flying towards it at a speed of $6 \, \text{m/s}$ with respect to the ground. Determine the speed of the ball with respect to the ground after a perfectly elastic normal collision.

Answer: 16


Find the minimum value of
\[(15 - x)(8 - x)(15 + x)(8 + x).\]

Answer: -6480.25


Let \( n \geq 3 \) be a positive integer and let \( \left(a_{1}, a_{2}, \ldots, a_{n}\right) \) be a strictly increasing sequence of \( n \) positive real numbers with sum equal to 2. Let \( X \) be a subset of \( \{1,2, \ldots, n\} \) such that the value of
$$
\left|1-\sum_{i \in X} a_{i}\right|
$$
is minimized. Prove that there exists a strictly increasing sequence of \( n \) positive real numbers \( \left(b_{1}, b_{2}, \ldots, b_{n}\right) \) with sum equal to 2 such that
$$
\sum_{i \in X} b_{i}=1
$$

Answer: 1


A biased coin has a probability of $\frac{3}{4}$ of landing heads and $\frac{1}{4}$ of landing tails on each toss. The outcomes of the tosses are independent. The probability of winning Game C, where the player tosses the coin four times and wins if either all four outcomes are heads or all four are tails, can be compared to the probability of winning Game D, where the player tosses the coin five times and wins if the first two tosses are the same, the third toss is different from the first two, and the last two tosses are the same as the first two. Determine the difference in the probabilities of winning Game C and Game D.

Answer: \frac{61}{256}


180 grams of 920 purity gold was alloyed with 100 grams of 752 purity gold. What is the purity of the resulting alloy?

Answer: 860


In which of the expressions

$$
\left(1 + x^2 - x^3\right)^{1000} \quad \text{and} \quad \left(1 - x^2 + x^3\right)^{1000}
$$

will the coefficient for \( x^{20} \) be greater after expanding and combining like terms?

Answer: (1+x^{2}-x^{3})^{1000}


A rectangular piece of paper  $ABCD$  has sides of lengths  $AB = 1$ ,  $BC = 2$ . The rectangle is folded in half such that  $AD$  coincides with  $BC$  and  $EF$  is the folding line. Then fold the paper along a line  $BM$  such that the corner  $A$  falls on line  $EF$ . How large, in degrees, is  $\angle ABM$ ?
[asy]
size(180); pathpen = rgb(0,0,0.6)+linewidth(1); pointpen = black+linewidth(3); pointfontpen = fontsize(10); pen dd = rgb(0,0,0.6) + linewidth(0.7) + linetype("4 4"), dr = rgb(0.8,0,0), dg = rgb(0,0.6,0), db = rgb(0,0,0.6)+linewidth(1);
pair A=(0,1), B=(0,0), C=(2,0), D=(2,1), E=A/2, F=(2,.5), M=(1/3^.5,1), N=reflect(B,M)*A;
D(B--M--D("N",N,NE)--B--D("C",C,SE)--D("D",D,NE)--M); D(D("M",M,plain.N)--D("A",A,NW)--D("B",B,SW),dd); D(D("E",E,W)--D("F",F,plain.E),dd);
[/asy]

Answer: 30


The base of a triangle is  $80$ , and one side of the base angle is  $60^\circ$ .  The sum of the lengths of the other two sides is  $90$ .  The shortest side is:

Answer: 17


What is the result of adding 12.8 to a number that is three times more than 608?

Answer: 2444.8


A right triangle and a circle are drawn such that the circle is tangent to the legs of the right triangle. The circle cuts the hypotenuse into three segments of lengths 1, 24, and 3, and the segment of length 24 is a chord of the circle. Compute the area of the triangle.

Answer: 192


Kelvin the Frog lives in the 2-D plane.  Each day, he picks a uniformly random direction (i.e. a uniformly random bearing  $\theta\in [0,2\pi]$ ) and jumps a mile in that direction.  Let  $D$  be the number of miles Kelvin is away from his starting point after ten days.  Determine the expected value of  $D^4$ .

Answer: 200


There is a unique positive integer $n$ such that\[\log_2{(\log_{16}{n})} = \log_4{(\log_4{n})}.\]What is the sum of the digits of $n?$
$\textbf{(A) } 4 \qquad \textbf{(B) } 7 \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 11 \qquad \textbf{(E) } 13$

Answer: \textbf{(E)}13


Introduce a standard scalar product in  $\mathbb{R}^4.$  Let  $V$  be a partial vector space in  $\mathbb{R}^4$  produced by  $\left(
\begin{array}{c}
1
-1
-1
1
\end{array}
\right),\left(
 \begin{array}{c}
1  -1
1
-1
\end{array}
\right).$ Find a pair of base of orthogonal complement  $W$  for  $V$  in  $\mathbb{R}^4.$

Answer: {\begin{pmatrix}1\\1\\0\\0\end{pmatrix},\begin{pmatrix}0\\0\\1\\1\end{pmatrix}}


Find all $t$ such that $x-t$ is a factor of $4x^2 + 11x - 3$.

Answer: -3


A triangle in the $x y$-plane is such that when projected onto the $x$-axis, $y$-axis, and the line $y=x$, the results are line segments whose endpoints are $(1,0)$ and $(5,0),(0,8)$ and $(0,13)$, and $(5,5)$ and $(7.5,7.5)$, respectively. What is the triangle's area?

Answer: \frac{17}{2}


Katya is passing time while her parents are at work. On a piece of paper, she absentmindedly drew Cheburashkas in two rows (at least one Cheburashka was drawn in each row).

Afterwards, she drew a Crocodile Gena between every two adjacent Cheburashkas in both rows. Then she drew an Old Lady Shapoklyak to the left of each Cheburashka. Finally, she drew a Krakozyabra between each pair of characters in the rows.

Upon closer inspection of the drawing, she realized that only the Krakozyabras turned out nicely, so she angrily erased all the other characters. In the end, her parents saw two rows of Krakozyabras: a total of 29.

How many Cheburashkas were erased?

Answer: 11


Find the minimum value of the function \( f(x, y) = 6\left(x^{2} + y^{2}\right)(x + y) - 4\left(x^{2} + xy + y^{2}\right) - 3(x + y) + 5 \) in the region \( D = \{(x, y) \mid x > 0, y > 0\} \).

Answer: 2


Find all functions  $f:\mathbb{N}\rightarrow \mathbb{N}$  such that the inequality   $$ f(x)+yf(f(x))\le x(1+f(y)) $$  holds for all positive integers  $x, y$ .

Proposed by Adrian Beker.

Answer: f(x)=x


G.H. Hardy once went to visit Srinivasa Ramanujan in the hospital, and he started the conversation with: "I came here in taxi-cab number 1729. That number seems dull to me, which I hope isn't a bad omen." "Nonsense," said Ramanujan. "The number isn't dull at all. It's quite interesting. It's the smallest number that can be expressed as the sum of two cubes in two different ways." Ramanujan had immediately seen that $1729 = 12^{3} + 1^{3} = 10^{3} + 9^{3}$. What is the smallest positive integer representable as the sum of the cubes of three positive integers in two different ways?

Answer: 251


The area of trapezoid \(ABCD\) is 30. Point \(P\) is the midpoint of the lateral side \(AB\). Point \(R\) on side \(CD\) is chosen such that \(2CD = 3RD\). Lines \(AR\) and \(PD\) intersect at point \(Q\). Find the area of triangle \(APQ\) if \(AD = 2BC\).

Answer: \frac{10}{3}


Find the smallest natural number \( n \) for which the number \( A = n^3 + 12n^2 + 15n + 180 \) is divisible by 23.

Answer: 10


On each of the lines \(x=2\) and \(x=15\), there are 400 points marked with ordinates \(1, 2, 3, \ldots, 400\). In how many ways can you select three points from the 800 marked points so that they form the vertices of a right triangle?

Answer: 320868


We define $|\begin{array}{l}{a}&{c}\\{b}&{d}\end{array}|=ad-bc$. For example, $|\begin{array}{l}{1}&{3}\\{2}&{4}\end{array}|=1\times 4-2\times 3=4-6=-2$. If $x$ and $y$ are integers, and satisfy $1 \lt |\begin{array}{l}{2}&{y}\\{x}&{3}\end{array}| \lt 3$, then the minimum value of $x+y$ is ____.

Answer: -5


On a $15 \times 15$ grid, a closed line that does not intersect itself is drawn by connecting the centers of adjacent cells (by a side). We assume that the line is symmetric with respect to one of the two main diagonals. Show that its length is less than or equal to 200.

Answer: 200


A math teacher randomly selects 3 questions for analysis from a test paper consisting of 12 multiple-choice questions, 4 fill-in-the-blank questions, and 6 open-ended questions. The number of different ways to select questions such that at least one multiple-choice question and at least one open-ended question are selected is __________.

Answer: 864


Determine the smallest number \( M \) such that the inequality
$$
\left|a b\left(a^{2}-b^{2}\right)+b c\left(b^{2}-c^{2}\right)+c a\left(c^{2}-a^{2}\right)\right| \leq M\left(a^{2}+b^{2}+c^{2}\right)^{2}
$$
holds for all real numbers \( a, b, c \).

Answer: \frac{9}{32}\sqrt{2}


The logarithm of $27\sqrt[4]{9}\sqrt[3]{9}$ to the base $3$ is:
$\textbf{(A)}\ 8\frac{1}{2} \qquad \textbf{(B)}\ 4\frac{1}{6} \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ \text{none of these}$

Answer: (B)\4\frac{1}{6}


Given that the sequence $\{a_n\}$ is an arithmetic sequence, and if $\frac{a_{12}}{a_{11}} < -1$, find the maximum value of $n$ for which the sum of its first $n$ terms, $s_n$, is greater than $0$.

Answer: 21


The number \( x \) is such that among the four numbers \( x - \sqrt{2} \), \( x - \frac{1}{x} \), \( x + \frac{1}{x} \), \( x^2 + 2 \sqrt{2} \), exactly one is not an integer. Find all such \( x \).

Answer: \sqrt{2} - 1


Olya drew $N$ different lines on a plane, any two of which intersect. It turned out that among any 15 lines, there must be two with an angle of $60^\circ$ between them. What is the largest possible $N$ for which this is possible?

Answer: 42


Given the equation $(2a-1) \sin x + (2-a) \sin 2x = \sin 3x$ in $x \in [0, 2\pi]$, where all solutions in this interval form an arithmetic sequence, find the range of the real number $a$.

Answer: (-\infty,-2]\cup{0}\cup[2,+\infty)


Let \( f(x) = x^2 + ax + b \) have two real roots in the interval \([0,1]\). Then, find the range of values of \( a^2 - 2b \).

Answer: [0,2]


The distance between two squares of an infinite chessboard is defined as the minimum number of moves a king needs to travel between these squares. Three squares are marked on the board, where each pair of distances between them equals 100. How many squares are there from which the distance to all three marked squares is 50?

Answer: 1


Given that $a, b, c$ are not all zero, find the maximum value of $\frac{a b + 2 b c}{a^{2} + b^{2} + c^{2}}$.

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


Let $S$ be the set of positive real numbers.  Let $g : S \to \mathbb{R}$ be a function such that
\[g(x) g(y) = g(xy) + 3003 \left( \frac{1}{x} + \frac{1}{y} + 3002 \right)\]for all $x,$ $y > 0.$

Let $m$ be the number of possible values of $g(2),$ and let $t$ be the sum of all possible values of $g(2).$  Find $m \times t.$

Answer: \frac{6007}{2}


Given \(\sin x + \sin y = 0.6\) and \(\cos x + \cos y = 0.8\), find \(\cos x \cdot \cos y\).

Answer: -\frac{11}{100}


For how many integers $n$ between 1 and 20 (inclusive) is $\frac{n}{18}$ a repeating decimal?

Answer: 18


What is the maximum number of rooks that can be placed in an \(8 \times 8 \times 8\) cube so that they do not attack each other?

Answer: 64


Brothers Lyosha and Sasha decided to get from home to the skate park. They left at the same time, but Lyosha walked with the skateboard in his hands, while Sasha rode the skateboard. It is known that Sasha rides the skateboard 3 times faster than Lyosha walks with the skateboard. After some time, they simultaneously changed their mode of transportation: Lyosha started riding the skateboard, and Sasha started walking. As a result, the speed of each of them changed by a factor of 2: Lyosha's speed increased, and Sasha's speed decreased. They both arrived at the skate park at the same time. How many meters did Sasha ride on the skateboard if the distance from home to the skate park is 3300 meters?

Answer: 1100


Given the sequence ${a_n}$ satisfying $a_1=1$, $a_2=2$, $a_3=3$, $a_{n+3}=a_n$ ($n\in\mathbb{N}^*$). If $a_n=A\sin(\omega n+\varphi)+c$ $(ω>0,|\varphi|<\frac{\pi}{2})$, find the real number $A$.

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


Determine to how many decimal places the following approximate equality holds:

\[ 495 \sqrt{2} - 388 \sqrt{3} \approx 28 \]

In other words, how many zeros can be written after the decimal point in 28 so that the resulting number is still a rounded value of the left-hand side.

Answer: 7


Given any positive integer $n$, calculate the sum $\sum_{k=0}^{\infty}\left\lfloor\frac{n+2^{k}}{2^{k+1}}\right\rfloor$.

Answer: n


Let $n$ be a positive integer. Determine, in terms of $n$, the largest integer $m$ with the following property: There exist real numbers $x_1,\dots,x_{2n}$ with $-1 < x_1 < x_2 < \cdots < x_{2n} < 1$ such that the sum of the lengths of the $n$ intervals \[ [x_1^{2k-1}, x_2^{2k-1}], [x_3^{2k-1},x_4^{2k-1}], \dots, [x_{2n-1}^{2k-1}, x_{2n}^{2k-1}] \] is equal to 1 for all integers $k$ with $1 \leq k \leq m$.

Answer: n


A circle is divided into seven arcs such that the sum of any two adjacent arcs does not exceed $103^\circ$. Determine the largest possible value of $A$ such that, in any such division, each of the seven arcs contains at least $A^\circ$.

Answer: 51


A $\textit{palindrome}$ is a number which reads the same forward as backward, for example 313 or 1001. Ignoring the colon, how many different palindromes are possible on a 12-hour digital clock displaying only the hours and minutes? (Notice a zero may not be inserted before a time with a single-digit hour value. Therefore, 01:10 may not be used.)

Answer: 57


Prove that \[ \lim_{n \to \infty} n \left( \frac{\pi}{4} - n \int_0^1 \frac{x^n}{1+x^{2n}} \, dx \right) = \int_0^1 f(x) \, dx ,  \] where  $f(x) = \frac{\arctan x}{x}$  if  $x \in \left( 0,1 \right]$  and  $f(0)=1$ .

*Dorin Andrica, Mihai Piticari*

Answer: \int_0^1\frac{\arctanx}{x}\,dx


In an opaque bag, there are four identical balls labeled with numbers $3$, $4$, $5$, and $6$ respectively. Outside the bag, there are two balls labeled with numbers $3$ and $6$. Determine the probability that a triangle with the drawn ball and the numbers on the two balls outside the bag forms an isosceles triangle.

Answer: \frac{1}{4}


Tom has twelve slips of paper which he wants to put into five cups labeled $A$, $B$, $C$, $D$, $E$. He wants the sum of the numbers on the slips in each cup to be an integer. Furthermore, he wants the five integers to be consecutive and increasing from $A$ to $E$. The numbers on the papers are $2, 2, 2, 2.5, 2.5, 3, 3, 3, 3, 3.5, 4,$ and $4.5$. If a slip with $2$ goes into cup $E$ and a slip with $3$ goes into cup $B$, then the slip with $3.5$ must go into what cup?
$\textbf{(A) } A \qquad \textbf{(B) } B \qquad \textbf{(C) } C \qquad \textbf{(D) } D \qquad \textbf{(E) } E$

Answer: \textbf{(D)}~D


All the roots of the polynomial $z^6-10z^5+Az^4+Bz^3+Cz^2+Dz+16$ are positive integers, possibly repeated. What is the value of $B$?
$\textbf{(A) }{-}88 \qquad \textbf{(B) }{-}80 \qquad \textbf{(C) }{-}64 \qquad \textbf{(D) }{-}41\qquad \textbf{(E) }{-}40$

Answer: \textbf{(A)}{-}88


In a round-robin tournament with 6 teams, each team plays one game against each other team, and each game results in one team winning and one team losing. At the end of the tournament, the teams are ranked by the number of games won. What is the maximum number of teams that could be tied for the most wins at the end of the tournament?
$\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6$

Answer: \textbf{(D)}\5


Mother decides to take Xiaohua to 10 cities for a vacation by car. Xiaohua checked the map and was surprised to find that any three of these 10 cities have either all highways between them or only two of the cities have no highway between them. How many highways are there at least between these 10 cities? (Note: There can be at most one highway between any two cities.)

Answer: 40


Find the last two digits of  $\tbinom{200}{100}$ . Express the answer as an integer between  $0$  and  $99$ . (e.g. if the last two digits are  $05$ , just write  $5$ .)

Answer: 20


Suppose that  $10$  points are given in the plane, such that among any five of them there are four lying on a circle. Find the minimum number of these points which must lie on a circle.

Answer: 9


Given the function $f(x) = \frac{x+3}{x^2+1}$, and $g(x) = x - \ln(x-p)$.
(I) Find the equation of the tangent line to the graph of $f(x)$ at the point $\left(\frac{1}{3}, f\left(\frac{1}{3}\right)\right)$;
(II) Determine the number of zeros of the function $g(x)$, and explain the reason;
(III) It is known that the sequence $\{a_n\}$ satisfies: $0 < a_n \leq 3$, $n \in \mathbb{N}^*$, and $3(a_1 + a_2 + \ldots + a_{2015}) = 2015$. If the inequality $f(a_1) + f(a_2) + \ldots + f(a_{2015}) \leq g(x)$ holds for $x \in (p, +\infty)$, find the minimum value of the real number $p$.

Answer: 6044


Given the sequence $\{a_n\}$ such that the sum of the first $n$ terms is $S_n$, $S_1=6$, $S_2=4$, $S_n>0$ and $S_{2n}$, $S_{2n-1}$, $S_{2n+2}$ form a geometric progression, while $S_{2n-1}$, $S_{2n+2}$, $S_{2n+1}$ form an arithmetic progression. Determine the value of $a_{2016}$.
Options:
A) $-1009$
B) $-1008$
C) $-1007$
D) $-1006$

Answer: -1009


Find all naturals  $k$  such that  $3^k+5^k$  is the power of a natural number with the exponent  $\ge 2$ .

Answer: 1


There are 5 ping pong balls, among which 3 are new and 2 are old (i.e., used at least once). Each time a match occurs, 2 balls are selected and used, and then all balls are returned. Let $\xi$ be the number of new balls picked in the second match. What is the expected value $E(\xi)$?

Answer: \frac{18}{25}


Two people are flipping a coin: one flipped it 10 times, and the other flipped it 11 times. Find the probability that the second person got heads more times than the first person.

Answer: \frac{1}{2}


Given a finite arithmetic sequence \(\left\{a_{n}\right\}\) with the first term equal to 1 and the last term \(a_{n} = 1997\) (where \(n > 3\)), and the common difference being a natural number, find the sum of all possible values of \(n\).

Answer: 3501


Independently flip a coin \( n \) times (where \( n \) is a positive integer), each flip resulting in either heads or tails. Let \( a(n) \) denote the number of ways for the coin to come up heads exactly a multiple of 3 times in \( n \) coin flips, and let \( b(n) \) denote the number of ways for the coin to come up heads exactly a multiple of 6 times in \( n \) coin flips.
1. Find \( a(2016) \) and \( b(2016) \).
2. For \( n \leqslant 2016 \), find the number of positive integers \( n \) such that \( 2b(n) - a(n) > 0 \).

Answer: 840


I had $\$30$ in allowance money and spent it as indicated in the pie graph shown. How many dollars did I spend on burgers?

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O=(0,0);
A=(.707,.707);
B=(-.966,.259);
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D=(.342,-.940);
draw(Circle(O, 1));
draw(O--A);
draw(O--B);
draw(O--C);
draw(O--D);
W=(-.1,.5);
label("Movies", W, N);
label("$\frac{1}{3}$", W, S);
X=(-.55, 0);
label("Burgers", X, S);
Y=(-.17,-.7);
label("Ice Cream", Y, N);
label("$\frac{1}{5}$", Y, S);
Z=(.5, -.15);
label("Music", Z, N);
label("$\frac{3}{10}$", Z, S);
[/asy]

Answer: 5


Let \(a\) and \(b\) be integers. Show that the equation

$$(x-a)(x-b)(x-3) + 1 = 0$$

has at most one integer solution.

Answer: 1


In an arithmetic sequence \(\{a_{n}\}\), if \(\frac{a_{11}}{a_{10}} < -1\) and the sum of the first \(n\) terms \(S_{n}\) has a maximum value, then the value of \(n\) when \(S_{n}\) attains its smallest positive value is \(\qquad\).

Answer: 19


Let the domain of the function $f(x)$ be $R$. $f(x+1)$ is an odd function, and $f(x+2)$ is an even function. When $x\in [1,2]$, $f(x)=ax^{2}+b$. If $f(0)+f(3)=6$, calculate $f(\frac{9}{2})$.

Answer: \frac{5}{2}


Prove that for the sequence of Fibonacci numbers \( a_1, a_2, \ldots \) defined by the relations

\[
a_{1} = a_{2} = 1, \quad a_{n+2} = a_{n+1} + a_{n} \quad \text{for} \quad n \in \mathbb{N},
\]

there exists a unique triple of numbers \( a, b, c \in \mathbb{N} \) satisfying the conditions \( b < a, c < a \) and for any \( n \in \mathbb{N} \) the number \( a_n - nbc^n \) is divisible by \( a \).

Answer: (5,2,3)


The distance from platform $A$ to platform $B$ was covered by an electric train in $X$ minutes ($0<X<60$). Find $X$ if it is known that at both the moment of departure from $A$ and the moment of arrival at $B$, the angle between the hour and minute hands of the clock was $X$ degrees.

Answer: 48


In triangle $ABC$, $AB = 10$, $BC = 14$, and $CA = 16$. Let $D$ be a point in the interior of $\overline{BC}$. Let points $I_B$ and $I_C$ denote the incenters of triangles $ABD$ and $ACD$, respectively. The circumcircles of triangles $BI_BD$ and $CI_CD$ meet at distinct points $P$ and $D$. The maximum possible area of $\triangle BPC$ can be expressed in the form $a - b\sqrt {c}$, where $a$, $b$, and $c$ are positive integers and $c$ is not divisible by the square of any prime. Find $a + b + c$.
Diagram
[asy] defaultpen(fontsize(11)+0.8); size(300); pair A,B,C,D,Ic,Ib,P; A=MP("A",origin,down+left); B=MP("B",8*right,down+right); C=MP("C",IP(CR(A,5), CR(B,7)),2*up); real t=0.505; D=MP("",B+t*(C-B),SW); draw(A--B--C--A--D); path c1=incircle(A,D,C); path c2=incircle(A,D,B); draw(c1, gray+0.25); draw(c2, gray+0.25); Ic=MP("I_C",incenter(A,D,C),down+left); Ib=MP("I_B",incenter(A,D,B),left); path c3=circumcircle(Ic,D,C); path c4=circumcircle(Ib,D,B); draw(c3, fuchsia+0.2); draw(c4, fuchsia+0.2); P=MP("P",OP(c3,c4),up); draw(arc(circumcenter(B,C,P),B,C), royalblue+0.5+dashed); draw(C--Ic--D--P--C^^P--Ic, black+0.3); draw(B--Ib--D--P--B^^P--Ib, black+0.3); label("10",A--B,down); label("16",A--C,left); [/asy]

Answer: 150


For a triangle \( T = ABC \) we take the point \( X \) on the side \( AB \) such that \( \frac{AX}{XB} = \frac{4}{5} \), the point \( Y \) on the segment \( CX \) such that \( CY = 2YX \), and, if possible, the point \( Z \) on the ray \( CA \) such that \( \angle CXZ = 180^{\circ} - \angle ABC \). We denote by \( \Sigma \) the set of all triangles \( T \) for which \( \angle XYZ = 45^{\circ} \).
Prove that all the triangles from \( \Sigma \) are similar and find the measure of their smallest angle.

Answer: 45


In a store, there are 50 light bulbs in stock, 60% of which are produced by Factory A and 40% by Factory B. The first-class rate of the light bulbs produced by Factory A is 90%, and the first-class rate of the light bulbs produced by Factory B is 80%.

(1) If one light bulb is randomly selected from these 50 light bulbs (each light bulb has an equal chance of being selected), what is the probability that it is a first-class product produced by Factory A?

(2) If two light bulbs are randomly selected from these 50 light bulbs (each light bulb has an equal chance of being selected), and the number of first-class products produced by Factory A among these two light bulbs is denoted as $\xi$, find the value of $E(\xi)$.

Answer: 1.08


Evaluate the following expression:   $$ 0 - 1 -2 + 3 - 4 + 5 + 6 + 7 - 8 + ... + 2000 $$   The terms with minus signs are exactly the powers of two.

Answer: 1996906


Determine all positive integers  $n$  such that   $$ n\cdot 2^{n-1}+1 $$   is a perfect square.

Answer: 5


Let  $ABC$  be an isosceles right-angled triangle. Point  $D$  is chosen on the prolongation of the hypothenuse  $AB$  beyond point  $A$  so that  $AB = 2AD$ . Points  $M$  and  $N$  on side  $AC$  satisfy the relation  $AM = NC$ . Point  $K$  is chosen on the prolongation of  $CB$  beyond point  $B$  so that  $CN = BK$ . Determine the angle between lines  $NK$  and  $DM$ .

(M.Kungozhin)

Answer: 45


In a triangle  $ABC, L$  and  $K$  are the points of intersections of the angle bisectors of  $\angle ABC$  and  $\angle BAC$  with the segments  $AC$  and  $BC$ , respectively. The segment  $KL$  is angle bisector of  $\angle AKC$ , determine  $\angle BAC$ .

Answer: 120


Two circles of different radii touch each other externally. Find the angle determined by the chords connecting the point of tangency of the circles with the points of tangency of their common external tangent.

Answer: 90


Given that $21^{-1} \equiv 15 \pmod{61}$, find $40^{-1} \pmod{61}$, as a residue modulo 61.  (Provide a number between 0 and 60, inclusive.)

Answer: 46


Determine the value of

$$
z=a \sqrt{a} \sqrt[4]{a} \sqrt[8]{a} \ldots \sqrt[2^{n}]{a} \ldots
$$

if \( n \) is infinitely large.

Answer: ^2


On the side \(AB\) of quadrilateral \(ABCD\), points \(A_1\) and \(B_1\) are taken, and on the side \(CD\), points \(C_1\) and \(D_1\) are taken, such that \(AA_1 = BB_1 = p \cdot AB\) and \(CC_1 = DD_1 = p \cdot CD\), where \(p < 0.5\). Prove that \(\frac{S_{A_1 B_1 C_1 D_1}}{S_{ABCD}} = 1 - 2p\).

Answer: 1-2p


Several peasants collectively own 128 sheep. If one peasant ends up with at least half of all the sheep, the others conspire to expropriate that peasant: each of the others takes as many sheep as they already have. If two peasants each have 64 sheep, they expropriate one of them. It is known that 7 expropriations occurred. Prove that all the sheep ended up with one peasant.

Answer: 128


Express the sum

$$
1 \cdot 2^{2} \cdot 3 + 2 \cdot 3^{2} \cdot 4 + \ldots + (n-2)(n-1)^{2} n
$$

in closed form, where $n$ is a natural number greater than three.

Answer: \frac{(n-2)(n-1)n(n+1)(2n-1)}{10}


Answer: 200


Answer: 1975


Two players play the following game. Each player alternately strikes out 9 numbers (of their choice) from the sequence \(1,2,3, \ldots, 100,101\). After eleven such rounds, two numbers will remain. The second player then awards the first player points equal to the difference between these remaining numbers. Prove that the first player can always score at least 55 points, regardless of how the second player plays.

Answer: 55


Given: $\mathrm{H}_{\mathrm{n}}=\sum_{\mathrm{j}=1}^{\mathrm{n}} \frac{1}{\mathrm{j}}$, prove: $\sum_{\mathrm{i}=1}^{2012} \frac{1}{\mathrm{iH}_{\mathrm{i}}{ }^{2}}<2$.

Answer: 2


The polynomial \( P(x) \) of degree \( n \) satisfies the equations \( P(k) = \frac{k}{k+1} \) for \( k = 0, 1, \ldots, n \). Find \( P(n+1) \).

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


Let the following system of equations hold for positive numbers \(x, y, z\):

\[ \left\{\begin{array}{l}
x^{2}+x y+y^{2}=48 \\
y^{2}+y z+z^{2}=25 \\
z^{2}+x z+x^{2}=73
\end{array}\right. \]

Find the value of the expression \(x y + y z + x z\).

Answer: 40


A street has 20 houses on each side, for a total of 40 houses. The addresses on the south side of the street form an arithmetic sequence, as do the addresses on the north side of the street. On the south side, the addresses are 4, 10, 16, etc., and on the north side they are 3, 9, 15, etc. A sign painter paints house numbers on a house for $\$1$ per digit. If he paints the appropriate house number once on each of these 40 houses, how many dollars does he collect?

Answer: 84


Using one each of the coins and bills of 1 jiao, 2 jiao, 5 jiao, 1 yuan, 2 yuan, and 5 yuan, how many different monetary values can be formed?

Answer: 63


Let  $\theta$  be a constant number such that  $0\leq \theta \leq \pi.$
Evaluate

\[ \int_0^{2\pi} \sin 8x|\sin (x-\theta)|\ dx. \]

Answer: -\frac{4}{63}\sin8\theta


Given the cubic polynomial \( p(x) = x^3 - 3x^2 + 5x \), if \( h \) is a real root of \( p(x) = 1 \) and \( k \) is a real root of \( p(x) = 5 \), find \( h + k \).

Answer: 2


If \(a, b, c\) are positive numbers such that \(a + b + c = 3\), prove that \(ab + bc + ca \leq \sqrt{a} + \sqrt{b} + \sqrt{c} \leq 3\).

Answer: ++ca\le\sqrt{}+\sqrt{b}+\sqrt{}\le3


Let $g(n)$ be the product of the proper positive integer divisors of $n$. (Recall that a proper divisor of $n$ is a divisor other than $n$.) For how many values of $n$ does $n$ not divide $g(n)$, given that $2 \le n \le 50$?

Answer: 19


Answer: \sqrt{3}


If point \( P \) is the circumcenter of \(\triangle ABC\) and \(\overrightarrow{PA} + \overrightarrow{PB} + \lambda \overrightarrow{PC} = \mathbf{0}\), where \(\angle C = 120^\circ\), then find the value of the real number \(\lambda\).

Answer: -1


Prove: If \(\alpha, \beta, \gamma\) are the interior angles of a right triangle, then
$$
\begin{array}{l}
\sin \alpha \sin \beta \sin (\alpha-\beta)+\sin \beta \sin \gamma \sin (\beta-\gamma)+\sin \gamma \sin \alpha \sin (\gamma-\alpha)+ \\
\sin (\alpha-\beta) \sin (\beta-\gamma) \sin (\gamma-\alpha)=0 .
\end{array}
$$

Answer: 0


Let $S$ be the sum of all integers $b$ for which the polynomial $x^2+bx+2008b$ can be factored over the integers. Compute $|S|$.

Answer: 88352


Find all natural numbers whose own divisors can be paired such that the numbers in each pair differ by 545. An own divisor of a natural number is a natural divisor different from one and the number itself.

Answer: 1094


In the acute triangle \(ABC\), the altitudes \(AA_1\) and \(BB_1\) are drawn. The bisector of the exterior angle at vertex \(C\) intersects lines \(AB\) and \(A_1B_1\) at points \(L\) and \(K\) respectively. It turns out that \(CL = 2CK\). Find the angle \(C\).

Answer: 60


The triangle ABC has AB and AC unequal. The angle bisector of A meets the perpendicular bisector of BC at X. The line joining the feet of the perpendiculars from X to AB and AC meets BC at D. Find BD/DC.

Answer: 1


Given a sequence $\{a_{n}\}$ with all positive terms and the sum of the first $n$ terms $S_{n}$ satisfying $6 S_{n} = a_{n}^{2} + 3a_{n} + 2$. If $a_{2}$, $a_{4}$, and $a_{9}$ form a geometric sequence, find the general term formula of this sequence.

Answer: a_n=3n-2


Given that $| \overrightarrow{a}|=12$, $| \overrightarrow{b}|=9$, and $\overrightarrow{a} \cdot \overrightarrow{b}=-54 \sqrt {2}$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$.

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


Let $z$ be a non-real complex number with $z^{23}=1$. Compute $$ \sum_{k=0}^{22} \frac{1}{1+z^{k}+z^{2 k}} $$

Answer: 46 / 3


Point \( D \) lies on side \( AC \) of triangle \( ABC \). A circle with diameter \( BD \) intersects sides \( AB \) and \( BC \) at points \( P \) and \( T \) respectively. Points \( M \) and \( N \) are the midpoints of segments \( AD \) and \( CD \) respectively. It is known that \( PM \parallel TN \).

a) Find the angle \( ABC \).

b) Additionally, it is given that \( MP = \frac{1}{2} \), \( NT = 1 \), and \( BD = \frac{4}{3} \). Find the area of triangle \( ABC \).

Answer: \frac{\sqrt{35}}{3}


Betty Lou and Peggy Sue take turns flipping switches on a  $100 \times 100$  grid. Initially, all switches are "off". Betty Lou always flips a horizontal row of switches on her turn; Peggy Sue always flips a vertical column of switches. When they finish, there is an odd number of switches turned "on'' in each row and column. Find the maximum number of switches that can be on, in total, when they finish.

Answer: 9802


Answer: 7


Given the sequence $\left\{a_{n}\right\}$ defined as: $a_{1}=1599$, and for $k^{2}<n \leqslant(k+1)^{2}$, $a_{n}=\left[\frac{k+1}{2}\right]+\left[\frac{k}{2}\right]+1$ where $k \in \mathbf{N}^{*}$, is there a positive integer $n$ such that the equation $\sum_{i=1}^{n} a_{i}=2006$ holds? If yes, find all such numbers $n$; if no, explain why.

Answer: 68


Prove that if the product of positive numbers $a, b, c$ is 1, then

$$
\frac{1}{a^{2}(b+c)}+\frac{1}{b^{2}(c+a)}+\frac{1}{c^{2}(a+b)} \geq \frac{3}{2}
$$

Answer: \frac{3}{2}


Find all the integers $x, y$ and $z$ greater than or equal to $0$ such that $2^x + 9 \cdot  7^y = z^3$

Answer: (0, 1, 4)


Vasya, Petya, and Kolya are in the same class. Vasya always lies in response to any question, Petya alternates between lying and telling the truth, and Kolya lies in response to every third question but tells the truth otherwise. One day, each of them was asked six consecutive times how many students are in their class. The responses were "Twenty-five" five times, "Twenty-six" six times, and "Twenty-seven" seven times. Can we determine the actual number of students in their class based on their answers?

Answer: 27


Suppose \( a, b, c, d \) are real numbers such that
\[ |a-b| + |c-d| = 99 \]
\[ |a-c| + |b-d| = 1 \]

Determine all possible values of \( |a-d| + |b-c| \).

Answer: 99


Is it possible to place 19 bishops on a $4 \times 16$ chessboard so that no two bishops attack each other?

Answer: No


Evaluate the expression $-20 + 15 \times (4^{\div -1} \times 2)$.

Answer: -12.5


The number  $201212200619$  has a factor  $m$  such that  $6 \cdot 10^9 <m <6.5 \cdot 10^9$ . Find  $m$ .

Answer: 6490716149


The figure below shows line $\ell$ with a regular, infinite, recurring pattern of squares and line segments.

How many of the following four kinds of rigid motion transformations of the plane in which this figure is drawn, other than the identity transformation, will transform this figure into itself?

some rotation around a point of line $\ell$
some translation in the direction parallel to line $\ell$
the reflection across line $\ell$
some reflection across a line perpendicular to line $\ell$

Answer: 2


Six IMO competitions are hosted sequentially by two Asian countries, two European countries, and two African countries, where each country hosts once but no continent can host consecutively. How many such arrangements are possible?

Answer: 240


Anna thinks of an integer that is not a multiple of three, not a perfect square, and the sum of its digits is a prime number. What could the integer be?

Answer: 14


If \( x^{4} + ax^{2} + bx + c = 0 \) has roots 1, 2, and 3 (one root is repeated), find \( a + c \).

(17th Annual American High School Mathematics Examination, 1966)

Answer: -61


Given an ellipse $(C)$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$, the minor axis is $2 \sqrt{3}$, and the eccentricity $e= \frac{1}{2}$,
(1) Find the standard equation of ellipse $(C)$;
(2) If $F_{1}$ and $F_{2}$ are the left and right foci of ellipse $(C)$, respectively, a line $(l)$ passes through $F_{2}$ and intersects with ellipse $(C)$ at two distinct points $A$ and $B$, find the maximum value of the inradius of $\triangle F_{1}AB$.

Answer: \frac{3}{4}


Having graduated from a technical university, Oleg started his own business producing water heaters. This year, Oleg plans to sell 5000 units of water heaters. The variable costs of producing and selling one water heater are 800 rubles, and the total fixed costs are 1000 thousand rubles. Oleg wants his revenues to exceed expenses by 1500 thousand rubles. At what price should Oleg sell the water heaters? (The answer should be an integer, without spaces or units of measure.)

Answer: 1300


ABCD is a rectangle, D is the center of the circle, and B is on the circle.  If AD=4 and CD=3, then the area of the shaded region is between

Answer: 7 and 8


Point \(M\) is the midpoint of chord \(AB\). Chord \(CD\) intersects \(AB\) at point \(M\). A semicircle is constructed on segment \(CD\) as its diameter. Point \(E\) lies on this semicircle, and \(ME\) is perpendicular to \(CD\). Find the angle \(AEB\).

Answer: 90


Two cubic equations. Let \( a, b, c \) be the roots of the equation \( x^3 + qx + r = 0 \). Write the equation whose roots will be the numbers

\[ \frac{b+c}{a^2}, \quad \frac{c+a}{b^2}, \quad \frac{a+b}{c^2} \]

Answer: rx^3-^2-1=0


In the Cartesian coordinate system $xOy$, the parametric equation of line $l$ is $\begin{cases} x=- \dfrac { \sqrt {3}}{2}t \\ y=1+ \dfrac {1}{2}t\end{cases}$ (where $t$ is the parameter), with the origin as the pole and the positive half-axis of $x$ as the polar axis, the polar coordinate system is established. The equation of curve $C_{1}$ is $\rho= \dfrac {36}{4 \sqrt {3}\sin \theta-12\cos \theta-\rho}$. Let $M(6,0)$ be a fixed point, and $N$ be a moving point on curve $C_{1}$, with $Q$ being the midpoint of $MN$.
$(1)$ Find the Cartesian coordinate equation of the trajectory $C_{2}$ of point $Q$;
$(2)$ Given that the intersection of line $l$ with the $x$-axis is point $P$, and its intersections with curve $C_{2}$ are points $A$ and $B$. If the midpoint of $AB$ is $D$, find the length of $|PD|$.

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


How many integers between 100 and 10,000 contain exactly 3 identical digits in their representation?

Answer: 324


Let $a$, $b$, and $c$ be positive integers with $a \ge b \ge c$ such that
$a^2-b^2-c^2+ab=2011$ and
$a^2+3b^2+3c^2-3ab-2ac-2bc=-1997$.
What is $a$?

Answer: 253


Cars A and B travel the same distance. Car A travels half that distance at $u$ miles per hour and half at $v$ miles per hour. Car B travels half the time at $u$ miles per hour and half at $v$ miles per hour. The average speed of Car A is $x$ miles per hour and that of Car B is $y$ miles per hour. Then we always have

Answer: $x \leq y$


How many subsets \( S \) of the set \(\{1,2, \ldots, 10\}\) satisfy the property that, for all \( i \in[1,9] \), either \( i \) or \( i+1 \) (or both) is in \( S \)?

Answer: 144


Fourteen foxes met ninety-two Koloboks, felt pity, and did not eat them. Prove that there are at least two foxes that met the same number of Koloboks. (Assume that a fox could have met no Koloboks at all.)

Answer: 2


In the triangle \(ABC\), \(\angle A = 20^{\circ}\), \(\angle C = 90^{\circ}\). \(O\) is a point on \(AB\) and \(D\) is the midpoint of \(OB\). Suppose the circle centered at \(O\) with radius \(OD\) touches the side \(AC\) at \(T\). Determine the size of \(\angle BCD\) in degrees.

Answer: 35


Is it possible for the sum of the lengths of the three medians of a triangle to be less than $3 / 4$ of its perimeter?

Answer: No


A bullet with a mass of \( m = 10 \) g, flying horizontally with a speed of \( v_{1} = 400 \) m/s, passes through a massive board and emerges from it with a speed of \( v_{2} = 100 \) m/s. Find the amount of work done on the bullet by the resistive force of the board.

Answer: 750


According to the notice from the Ministry of Industry and Information Technology on the comprehensive promotion of China's characteristic enterprise new apprenticeship system and the strengthening of skills training, our region clearly promotes the new apprenticeship system training for all types of enterprises, deepens the integration of production and education, school-enterprise cooperation, and the apprenticeship training goal is to cultivate intermediate and senior technical workers that meet the needs of business positions. In the year 2020, a certain enterprise needs to train 200 apprentices. After the training, an assessment is conducted, and the statistics of obtaining corresponding job certificates are as follows:

| Job Certificate | Junior Worker | Intermediate Worker | Senior Worker | Technician | Senior Technician |
|-----------------|---------------|---------------------|--------------|-----------|------------------|
| Number of People | 20 | 60 | 60 | 40 | 20 |

$(1)$ Now, using stratified sampling, 10 people are selected from these 200 people to form a group for exchanging skills and experiences. Find the number of people in the exchange group who have obtained job certificates in the technician category (including technicians and senior technicians).

$(2)$ From the 10 people selected in (1) for the exchange group, 3 people are randomly chosen as representatives to speak. Let the number of technicians among these 3 people be $X$. Find the probability distribution and the mathematical expectation of the random variable $X$.

Answer: \frac{9}{10}


$ABCD$ is a regular tetrahedron. Let $N$ be the midpoint of $\overline{AB}$, and let $G$ be the centroid of triangle $ACD$. What is $\cos \angle BNG$?

Answer: \frac{1}{3}


There are \( n \geq 2 \) coins, each with a different positive integer value. Call an integer \( m \) sticky if some subset of these \( n \) coins have a total value \( m \). We call the entire set of coins a stick if all the sticky numbers form a consecutive range of integers. Compute the minimum total value of a stick across all sticks containing a coin of value 100.

Answer: 199


Adams plans a profit of $10$ % on the selling price of an article and his expenses are $15$ % of sales. The rate of markup on an article that sells for $ $5.00$ is:

Answer: 33\frac {1}{3}\%


In the rectangular coordinate system on the plane, establish a polar coordinate system with the coordinate origin as the pole and the positive semi-axis of the $x$-axis as the polar axis. The polar coordinates of point $A$ are $\left( 4\sqrt{2}, \frac{\pi}{4} \right)$, and the polar equation of line $l$ is $\rho \cos \left( \theta - \frac{\pi}{4} \right) = a$, which passes through point $A$. The parametric equations of curve $C_1$ are given by $\begin{cases} x = 2 \cos \theta \\ y = \sqrt{3} \sin \theta \end{cases}$ ($\theta$ is the parameter).

(1) Find the maximum and minimum distances from points on curve $C_1$ to line $l$.
(2) Line $l_1$, which is parallel to line $l$ and passes through point $B(-2, 2)$, intersects curve $C_1$ at points $M$ and $N$. Compute $|BM| \cdot |BN|$.

Answer: \frac{32}{7}


$k$ points are arranged on a plane such that any triangle with vertices at these points has an area of no more than 1. Prove that all these points can be placed within a triangle of area 4.

Answer: 4


In a certain competition, the rules are as follows: among the 5 questions preset by the organizer, if a contestant can answer two consecutive questions correctly, they will stop answering and advance to the next round. Assuming the probability of a contestant correctly answering each question is 0.8, and the outcomes of answering each question are independent of each other, then the probability that the contestant will exactly answer 4 questions before advancing to the next round is

Answer: 0.128


A doctor told Mikael to take a pill every 75 minutes. He took his first pill at 11:05. At what time did he take his fourth pill?

Answer: 14:50


Given two points  $A$  and  $B$  on a circle  $\varGamma$ . Let the tangents to this circle  $\varGamma$  at the points  $A$  and  $B$  meet at a point  $X$ . Let  $C$  and  $D$  be two points on the circle  $\varGamma$  such that the points  $C$ ,  $D$ ,  $X$  are collinear in this order and such that the lines  $CA$  and  $BD$  are perpendicular.

Let the line  $CA$  intersect the line  $BD$  at a point  $F$ .
Let the line  $CD$  intersect the line  $AB$  at a point  $G$ .
Let  $H$  be the point of intersection of the segment  $BD$  and the perpendicular bisector of the segment  $GX$ .

Prove that the four points  $X$ ,  $F$ ,  $G$ ,  $H$  lie on one circle.

Answer: X,F,G,H


Let \( n \) be the answer to this problem. Hexagon \( ABCDEF \) is inscribed in a circle of radius 90. The area of \( ABCDEF \) is \( 8n \), \( AB = BC = DE = EF \), and \( CD = FA \). Find the area of triangle \( ABC \).

Answer: 2592


A corner of a tiled floor is shown. If the entire floor is tiled in this way and each of the four corners looks like this one, then what fraction of the tiled floor is made of darker tiles?

Answer: \frac{4}{9}


The sum of ten numbers is zero. The sum of all their pairwise products is also zero. Prove that the sum of the cubes of these numbers is also zero.

Answer: 0


What is the maximum number of rays in space that form obtuse angles pairwise?

Answer: 4


Given vectors $m=(\sqrt{3}\cos x,-1)$, $n=(\sin x,\cos ^{2}x)$.

$(1)$ When $x=\frac{\pi}{3}$, find the value of $m\cdot n$;

$(2)$ If $x\in\left[ 0,\frac{\pi}{4} \right]$, and $m\cdot n=\frac{\sqrt{3}}{3}-\frac{1}{2}$, find the value of $\cos 2x$.

Answer: \frac{3 \sqrt{2}- \sqrt{3}}{6}


For several identical books, 104 rubles were paid. The price of one book is expressed as a natural number. How much does one book cost if more than 10 but fewer than 60 books were bought?

Answer: 2


There are 111 balls in a box, each being red, green, blue, or white. It is known that if 100 balls are drawn, it ensures getting balls of all four colors. Find the smallest integer $N$ such that if $N$ balls are drawn, it can ensure getting balls of at least three different colors.

Answer: 88


In 2016, the number of books in a school library's collection increased by $0.4\%$, and in 2017 it increased by $0.8\%$, remaining less than 50,000. By how many books did the school library's collection increase in 2017?

Answer: 251


We define five-digit numbers like 31024 and 98567 as "Shenma numbers", where the middle digit is the smallest, the digits increase as they move away from the middle, and all the digits are different. How many such five-digit numbers are there?

Answer: 1512


A store sells jellybeans at a fixed price per gram. The price for \(250 \mathrm{~g}\) of jellybeans is \(\$7.50\). What mass of jellybeans sells for \(\$1.80\)?
(A) \(6 \mathrm{~g}\)
(B) \(54 \mathrm{~g}\)
(C) \(60 \mathrm{~g}\)
(D) \(120 \mathrm{~g}\)
(E) \(190 \mathrm{~g}\)

Answer: 60


Suppose  $n$  is an integer  $\geq 2$ . Determine the first digit after the decimal point in the decimal expansion of the number \[\sqrt[3]{n^{3}+2n^{2}+n}\]

Answer: 6


In the number \(2016 * * * * 02 * *\), each of the 6 asterisks needs to be replaced with any of the digits \(0, 2, 4, 5, 7, 9\) (digits can repeat) so that the resulting 12-digit number is divisible by 15. How many ways can this be done?

Answer: 5184


Let \( R \) be the rectangle in the Cartesian plane with vertices at \((0,0)\), \((2,0)\), \((2,1)\), and \((0,1)\). \( R \) can be divided into two unit squares. Pro selects a point \( P \) uniformly at random in the interior of \( R \). Find the probability that the line through \( P \) with slope \(\frac{1}{2}\) will pass through both unit squares.

Answer: 3/4


Given the hyperbola $C:\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1(a > 0,b > 0)$, the line $l$ passing through point $P(3,6)$ intersects $C$ at points $A$ and $B$, and the midpoint of $AB$ is $N(12,15)$. Determine the eccentricity of the hyperbola $C$.

Answer: \frac{3}{2}


Members of the Rockham Soccer League buy socks and T-shirts. Socks cost $4 per pair and each T-shirt costs $5 more than a pair of socks. Each member needs one pair of socks and a shirt for home games and another pair of socks and a shirt for away games. If the total cost is $2366, how many members are in the League?
$\mathrm{(A) \ } 77\qquad \mathrm{(B) \ } 91\qquad \mathrm{(C) \ } 143\qquad \mathrm{(D) \ } 182\qquad \mathrm{(E) \ } 286$

Answer: \mathrm{(B)}\91


Sangho uploaded a video to a website where viewers can vote that they like or dislike a video. Each video begins with a score of $0$, and the score increases by $1$ for each like vote and decreases by $1$ for each dislike vote. At one point Sangho saw that his video had a score of $90$, and that $65\%$ of the votes cast on his video were like votes. How many votes had been cast on Sangho's video at that point?
$\textbf{(A) }   200   \qquad        \textbf{(B) }   300   \qquad    \textbf{(C) }   400   \qquad   \textbf{(D) }  500  \qquad  \textbf{(E) }   600$

Answer: \textbf{(B)}300


How many ways can one color the squares of a $6 \times 6$ grid red and blue such that the number of red squares in each row and column is exactly 2?

Answer: 67950


Point \( M \) lies on the edge \( AB \) of cube \( ABCD A_1 B_1 C_1 D_1 \). Rectangle \( MNLK \) is inscribed in square \( ABCD \) in such a way that one of its vertices is at point \( M \), and the other three vertices are located on different sides of the base square. Rectangle \( M_1N_1L_1K_1 \) is the orthogonal projection of rectangle \( MNLK \) onto the plane of the upper face \( A_1B_1C_1D_1 \). The ratio of the side lengths \( MK_1 \) and \( MN \) of quadrilateral \( MK_1L_1N \) is \( \sqrt{54}:8 \). Find the ratio \( AM:MB \).

Answer: 1:4


Compute the number of functions $f:\{1,2, \ldots, 9\} \rightarrow\{1,2, \ldots, 9\}$ which satisfy $f(f(f(f(f(x)))))=$ $x$ for each $x \in\{1,2, \ldots, 9\}$.

Answer: 3025


In each cell of a 200 x 200 square grid, a real number not exceeding 1 in absolute value was written. It turned out that the sum of all the numbers is zero. What is the smallest value of \( S \) for which it can be asserted that the sum of the numbers in some row or some column is necessarily not exceeding \( S \) in absolute value?

Answer: 100


There is a multiple-choice test paper with six questions. The scoring criteria are: 8 points for a correct answer, 0 points for a wrong answer, and 2 points for no answer. A student scored a total of 20 points. Determine which of the following statements is true:
(A) The student answered at most one question correctly.
(B) The student answered at least three questions correctly.
(C) The student left at least three questions unanswered.
(D) The student answered two questions incorrectly.

Answer: 2


Find the modulus of the complex number \( z^{3}+z^{5} \), where \( z=\cos \alpha+i \sin \alpha \) and \( \alpha \in\left(\pi, \frac{3 \pi}{2}\right) \).

Answer: 2|\cos\alpha|


Two players, A and B, are competing in a table tennis match. The winner of each game scores 1 point, and the loser scores 0 points. The match ends when one player has 2 points more than the other or after 6 games, whichever comes first. The probability of A winning a game is $\frac{2}{3}$, and the probability of B winning a game is $\frac{1}{3}$. The outcomes of the games are independent. Find the expected number of games $\xi$ played when the match ends, $\mathrm{E} \xi$.

Answer: \frac{266}{81}


A natural number $\mathrm{K}$ increases by 54 or 77 every second. Prove that eventually, the number will have the last two digits the same.

Answer: 2


Consider seven-digit natural numbers in which each of the digits \(1, 2, 3, 4, 5, 6, 7\) appears exactly once.

(a) How many of these numbers have the digits from the first to the sixth in ascending order and the digits from the sixth to the seventh in descending order?

(b) How many of these numbers have the digits from the first to the fifth in ascending order and the digits from the fifth to the seventh in descending order?

Answer: 15


In the obtuse triangle $ABC$ with $\angle C>90^\circ$, $AM=MB$, $MD\perp BC$, and $EC\perp BC$ ($D$ is on $BC$, $E$ is on $AB$, and $M$ is on $EB$). If the area of $\triangle ABC$ is $24$, then the area of $\triangle BED$ is

Answer: 12


What is the area enclosed by the graph of the equation $(x - 1)^2 + (y - 1)^2 = |x - 1| + |y - 1|$?
A) $\frac{\pi}{4}$
B) $\frac{\pi}{2}$
C) $\frac{\pi}{3}$
D) $\pi$

Answer: \frac{\pi}{2}


The "Hua Luogeng" Golden Cup Junior Math Invitational Contest was first held in 1986, the second in 1988, and the third in 1991, and has subsequently been held every 2 years. The sum of the digits of the year of the first "Hua Cup" is: \( A_1 = 1 + 9 + 8 + 6 = 24 \). The sum of the digits of the years of the first two "Hua Cup" contests is: \( A_2 = 1 + 9 + 8 + 6 + 1 + 9 + 8 + 8 = 50 \). Find the sum of the digits of the years of the first 50 "Hua Cup" contests, \( A_{50} \).

Answer: 629


A regular tetrahedron has a square shadow of area 16 when projected onto a flat surface with light shone perpendicular to the plane. Compute the side length of the regular tetrahedron.

Answer: 4\sqrt{2}


Find the minimum value of
\[
\sqrt{x^2 + (2 - x)^2} + \sqrt{(2 - x)^2 + (2 + x)^2}
\]
over all real numbers $x$.

Answer: 2\sqrt{5}


Let \(a\) and \(b\) be strictly positive integers such that \(a! + b!\) divides \(a! b!\). Show that \(3a \geq 2b + 2\).

Answer: 3a\ge2b+2


A magic square is an array of numbers in which the sum of the numbers in each row, in each column, and along the two main diagonals are equal. The numbers in the magic square shown are not written in base 10. For what base will this be a magic square?

[asy]
unitsize(0.75cm);
for (int i=0; i<4; ++i) {
draw((0,i)--(3,i),linewidth(0.7));
draw((i,0)--(i,3),linewidth(0.7));
}
label("1",(1.5,2),N);
label("2",(2.5,0),N);
label("3",(0.5,1),N);
label("4",(0.5,0),N);
label("10",(1.5,1),N);
label("11",(2.5,2),N);
label("12",(2.5,1),N);
label("13",(0.5,2),N);
label("14",(1.5,0),N);
[/asy]

Answer: 5


Triangle $ABC$ has an area 1.  Points $E,F,G$ lie, respectively, on sides $BC$, $CA$, $AB$ such that $AE$ bisects $BF$ at point $R$, $BF$ bisects $CG$ at point $S$, and $CG$ bisects $AE$ at point $T$. Find the area of the triangle $RST$.

Answer: \frac{7 - 3 \sqrt{5}}{4}


If a number is selected from the whole numbers 1 to \( m \), and if each number has an equal chance of being selected, the probability that the number is a factor of \( m \) is \(\frac{p}{39}\). Find \( p \).

Answer: 4


We have a calculator with two buttons that displays an integer $x$. Pressing the first button replaces $x$ by $\left\lfloor\frac{x}{2}\right\rfloor$, and pressing the second button replaces $x$ by $4 x+1$. Initially, the calculator displays 0. How many integers less than or equal to 2014 can be achieved through a sequence of arbitrary button presses? (It is permitted for the number displayed to exceed 2014 during the sequence. Here, $\lfloor y\rfloor$ denotes the greatest integer less than or equal to the real number $y$.)

Answer: 233


Find all functions \( f: \mathbb{Q} \rightarrow \mathbb{Q} \) satisfying

$$
\forall x, y \in \mathbb{Q}, \quad f(x+y)=f(x)+f(y)
$$

Answer: f(x)=ax


The minimum sum of the distances from a point in space to the vertices of a regular tetrahedron with side length 1 is:

Answer: $\sqrt{6}$


Define mutually externally tangent circles  $\omega_1$ ,  $\omega_2$ , and  $\omega_3$ . Let  $\omega_1$  and  $\omega_2$  be tangent at  $P$ . The common external tangents of  $\omega_1$  and  $\omega_2$  meet at  $Q$ . Let  $O$  be the center of  $\omega_3$ . If  $QP = 420$  and  $QO = 427$ , find the radius of  $\omega_3$ .

*Proposed by Tanishq Pauskar and Mahith Gottipati*

Answer: 77


In triangle $ABC$, $AB = 8$, $BC = 8$, and $CA = 6$.

Point $P$ is randomly selected inside triangle $ABC$. What is the probability that $P$ is closer to vertex $C$ than it is to either vertex $A$ or $B$?

Answer: \frac{1}{4}


Given that \([x]\) represents the largest integer not exceeding \( x \), if \([x+0.1] + [x+0.2] + \ldots + [x+0.9] = 104\), what is the minimum value of \( x \)?

Answer: 11.5


Given \( xy + yz + zx = 1 \), where \( x, y, z \) are all positive numbers, determine the integer part of \( \sqrt{3xy + 1} + \sqrt{3yz + 1} + \sqrt{3zx + 1} \).

Answer: 4


Select some numbers from \(1, 2, 3, \cdots, 9, 10\) such that each number from \(1, 2, 3, \cdots, 19, 20\) is equal to the sum of one or two of the selected numbers (they can be the same). Find the minimum number of selected numbers needed.

Answer: 6


Each of the symbols $\diamond$ and $\circ$ represents an operation in the set $\{+,-,\times,\div\}$, and $\frac{15 \diamond 3}{8 \circ 2} = 3$. What is the value of $\frac{9 \diamond 4}{14 \circ 7}$? Express your answer as a common fraction.

Answer: \frac{13}{7}


Given four parallel planes $\alpha_{1}, \alpha_{2}, \alpha_{3}, \alpha_{4}$ arranged in sequence, where the distance between each consecutive pair of planes is 1. If a regular tetrahedron $A_{1} A_{2} A_{3} A_{4}$ has its four vertices such that $A_{i} \in \alpha_{i}$ for $i=1, 2, 3, 4$, find the volume of this regular tetrahedron.

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


Ten test papers are to be prepared for the National Olympiad. Each paper has 4 problems, and no two papers have more than 1 problem in common. At least how many problems are needed?

Answer: 13


A line of recruit soldiers was standing facing the sergeant. On the command "left turn," some turned to the left while the rest turned to the right. It turned out that six times as many soldiers were looking at the back of their neighbor's head as were looking at a face. Then, on the command "about turn," everyone turned 180 degrees. Now, seven times as many soldiers were looking at the back of their neighbor's head as were looking at a face. How many soldiers are in the line?

Answer: 98


Let \( n \) be a positive integer, and let \( x \) and \( y \) be positive real numbers such that \( x^{n} + y^{n} = 1 \). Prove that
$$
\left(\sum_{k=1}^{n} \frac{1+x^{2k}}{1+x^{4k}}\right)\left(\sum_{k=1}^{n} \frac{1+y^{2k}}{1+y^{4k}}\right) < \frac{1}{(1-x)(1-y)}.
$$

Answer: (\sum_{k=1}^{n}\frac{1+x^{2k}}{1+x^{4k}})(\sum_{k=1}^{n}\frac{1+y^{2k}}{1+y^{4k}})<\frac{1}{(1-x)(1-y)}


Given that \( f(x) \) is an odd function defined on \( (-\infty, 0) \cup (0, +\infty) \) and that for \( x > 0 \), \( x f'(x) < 0 \) and \( f(1) = 0 \), determine the solution set of the inequality \( x f(x) < 0 \).

Answer: (-\infty,-1)\cup(1,+\infty)


The internal and external angle bisectors of angle \( C \) of triangle \( ABC \) are equal. Prove that the following relationship holds between the sides of the triangle: \(\left(b^{2}-a^{2}\right)^{2}=c^{2}\left(a^{2}+b^{2}\right)\). Verify the truth of the converse theorem.

Answer: (b^2-^2)^2=^2(^2+b^2)


Let \( n \) be a positive integer. A sequence of \( n \) positive integers (not necessarily distinct) is called full if it satisfies the following condition: For each positive integer \( k \geq 2 \), if the number \( k \) appears in the sequence, then so does the number \( k-1 \), and moreover, the first occurrence of \( k-1 \) comes before the last occurrence of \( k \). For each \( n \), how many full sequences are there?

Answer: n!


A  $4\times4\times4$  cube is composed of  $64$  unit cubes. The faces of  $16$  unit cubes are to be coloured red. A colouring is called interesting if there is exactly  $1$  red unit cube in every  $1\times1\times 4$  rectangular box composed of  $4$  unit cubes. Determine the number of interesting colourings.

Answer: 576


Find the number of ordered triples of integers  $(a,b,c)$  such that   $$ a^2 + b^2 + c^2  - ab - bc - ca - 1 \le 4042b - 2021a - 2021c - 2021^2 $$   and  $|a|, |b|, |c| \le 2021.$

Answer: 14152


A circle $K$ is drawn through the vertices $A$ and $C$ of triangle $ABC$ such that its center lies on the circumcircle of triangle $ABC$. The circle $K$ intersects the extension of side $BA$ beyond point $A$ at point $M$.

Find angle $C$, given that $MA:AB = 2:5$ and $\angle B = \arcsin \frac{3}{5}$.

Answer: 45


In a regular tetrahedron \(ABCD\), \(MN\) is the line segment joining the midpoint of \(AD\) and the centroid of \(\triangle BCD\). \(PQ\) is the line segment joining the midpoint of \(CD\) and the centroid of \(\triangle ABC\). Find the angle between the lines \(MN\) and \(PQ\).

Answer: \arccos(\frac{1}{18})


There are 2008 red cards and 2008 white cards. 2008  players sit down in circular toward the inside of the circle in situation that 2 red cards and 2 white cards from each card are delivered to each person. Each person conducts the following procedure in one turn as follows.

 $ (*)$  If you have more than one red card, then you will pass one red card to the left-neighbouring player.

If you have no red card, then you will pass one white card to the left -neighbouring player.
Find the maximum value of the number of turn required for the state such that all person will have one red card and one white card first.

Answer: 1004


Given the equations $\left(625\right)^{0.24}$ and $\left(625\right)^{0.06}$, find the value of their product.

Answer: 5^{6/5}


Assuming that the clock hands move without jumps, determine how many minutes after the clock shows 8:00 will the minute hand catch up with the hour hand.

Answer: 43 \frac{7}{11}


Find minimum of  $x+y+z$  where  $x$ ,  $y$  and  $z$  are real numbers such that  $x \geq 4$ ,  $y \geq 5$ ,  $z \geq 6$  and  $x^2+y^2+z^2 \geq 90$

Answer: 16


The journey from Petya's home to school takes him 20 minutes. One day, on his way to school, Petya remembered that he had forgotten a pen at home. If he continues his journey at the same speed, he will arrive at school 3 minutes before the bell rings. However, if he returns home for the pen and then goes to school at the same speed, he will be 7 minutes late for the start of the lesson. What fraction of the way to school had he covered when he remembered about the pen?

Answer: \frac{1}{4}


Let $N$ be a positive integer. Brothers Michael and Kylo each select a positive integer less than or equal to $N$, independently and uniformly at random. Let $p_{N}$ denote the probability that the product of these two integers has a units digit of 0. The maximum possible value of $p_{N}$ over all possible choices of $N$ can be written as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100 a+b$.

Answer: 2800


For certain ordered pairs $(a,b)\,$ of real numbers, the system of equations
\[\begin{aligned} ax+by&=1 \\ x^2 + y^2 &= 50  \end{aligned}\]has at least one solution, and each solution is an ordered pair $(x,y)\,$ of integers. How many such ordered pairs $(a,b)\,$ are there?

Answer: 72


In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is given that $\sqrt{3}b\cos A=a\cos B$.

$(1)$ Find the angle $A$;

$(2)$ If $a= \sqrt{2}$ and $\frac{c}{a}= \frac{\sin A}{\sin B}$, find the perimeter of $\triangle ABC$.

Answer: 3 \sqrt{2}


Jia and Yi are playing a table tennis singles match using a best-of-five format (i.e., the first to win three games wins the championship). For each game, Jia's probability of winning is $\frac{2}{3}$ and Yi's probability of winning is $\frac{1}{3}$. Calculate the probability that Yi wins the championship.

Answer: \frac{17}{81}


In the triangle \( ABC \) the points \( M \) and \( N \) lie on the side \( AB \) such that \( AN = AC \) and \( BM = BC \). We know that \(\angle MCN = 43^\circ\). Find the size in degrees of \(\angle ACB\).

Answer: 94


The stem and leaf plot shows the heights, in inches, of the players on the Westvale High School boys' basketball team. Calculate the mean height of the players on the team. (Note: $6|2$ represents 62 inches.)

Height of the Players on the Basketball Team (inches)

$5|7$

$6|2\;4\;4\;5\;7\;8$

$7|0\;1\;2\;2\;3\;4\;5\;5$

Answer: 68.6


A circular cone has vertex \( I \), a base with radius 1, and a slant height of 4. Point \( A \) is on the circumference of the base and point \( R \) is on the line segment \( IA \) with \( IR = 3 \). Shahid draws the shortest possible path starting at \( R \), traveling once around the cone, and ending at \( A \). If \( P \) is the point on this path that is closest to \( I \), what is the length \( IP \)?

Answer: \frac{12}{5}


For a positive integer  $n$ , let  $\omega(n)$  denote the number of positive prime divisors of  $n$ . Find the smallest positive tinteger  $k$  such that  $2^{\omega(n)}\leq k\sqrt[4]{n}\forall n\in\mathbb{N}$ .

Answer: 5


$ABCDEF GH$  is a regular octagon with  $10$  units side . The circle with center  $A$  and radius  $AC$  intersects  the circle  with center  $D$  and radius  $CD$  at point  $ I$ , different from  $C$ . What is the length of the segment  $IF$ ?

Answer: 10


Chess piece called *skew knight*, if placed on the black square, attacks all the gray squares.

![Image](https://i.ibb.co/HdTDNjN/Kyiv-MO-2021-Round-1-11-2.png)

What is the largest number of such knights that can be placed on the  $8\times 8$  chessboard without them attacking each other?

*Proposed by Arsenii Nikolaiev*

Answer: 32


On a piece of paper, the number 686 is written. How can you obtain a number that is 303 greater than this one without writing anything down or performing any calculations?

Answer: 989


Let $n$ be an integer greater than or equal to $1$. Find, as a function of $n$, the smallest integer $k\ge 2$ such that, among any $k$ real numbers, there are necessarily two of which the difference, in absolute value, is either strictly less than $1 / n$, either  strictly greater than $n$.

Answer: n^2 + 2


Squares $ABCD$ and $EFGH$ are congruent, $AB=10$, and $G$ is the center of square $ABCD$. The area of the region in the plane covered by these squares is

Answer: 175


Given a number \( a \in (0,1) \). Positive numbers \( x_{0}, x_{1}, \ldots, x_{n} \) satisfy the conditions \( x_{0} + x_{1} + \ldots + x_{n} = n + a \) and \( \frac{1}{x_{0}} + \frac{1}{x_{1}} + \ldots + \frac{1}{x_{n}} = n + \frac{1}{a} \). Find the minimum value of the expression \( x_{0}^{2} + x_{1}^{2} + \ldots + x_{n}^{2} \).

Answer: n+^2


Suppose that positive integers  $m,n,k$  satisfy the equations   $$ m^2+1=2n^2, 2m^2+1=11k^2. $$   Find the residue when  $n$  is divided by  $17$ .

Answer: 5


Let  $\mathcal{F}$  be the set of all functions  $f : (0,\infty)\to (0,\infty)$  such that  $f(3x) \geq f( f(2x) )+x$  for all  $x$ . Find the largest  $A$  such that  $f(x) \geq A x$  for all  $f\in\mathcal{F}$  and all  $x$ .

Answer: 1/2


Let \( R \) be the reals and \( C \) the set of all functions \( f : [0, 1] \to R \) with a continuous derivative and satisfying \( f(0) = 0 \) and \( f(1) = 1 \). Find \( \inf_{f \in C} \int_{0}^{1} | f'(x) - f(x) | \, dx \).

Answer: \frac{1}{e}


Suppose \(AB = 1\), and the slanted segments form an angle of \(45^\circ\) with \(AB\). There are \(n\) vertices above \(AB\).

What is the length of the broken line?

Answer: \sqrt{2}


In the right trapezoid \( ABCD \), it is known that \(\angle A = \angle D = 90^\circ\), \(DE \perp AC\) at point \(E\), \(\angle ACD = \angle EBC = 30^\circ\), and \(AD = \sqrt{3}\). Find \(BC\).

Answer: 3


The last two digits of the integer part of $\left[\frac{10^{93}}{10^{31}+3}\right]$ are $\qquad$ (write the tens place digit first, followed by the units place digit; where $[x]$ denotes the greatest integer less than or equal to $x$).

Answer: 8


The diagonals of a trapezoid are 12 and 6, and the sum of the bases is 14. Find the area of the trapezoid.

Answer: 16\sqrt{5}


In a kindergarten, each child was given three cards, each of which has either "MA" or "NY" written on it. It turned out that 20 children can arrange their cards to spell the word "MAMA", 30 children can arrange their cards to spell the word "NYANYA", and 40 children can arrange their cards to spell the word "MANYA". How many children have all three cards that are the same?

Answer: 10


A farmer claims that with four rods, he can fence a square plot of land sufficient for one sheep. If this is true, what is the minimum number of rods needed to fence an area for ten sheep? The answer depends on the shape of your fence.

How many rods are required for 10 sheep?

Answer: 12


In a shooting contest, 8 targets are arranged in two columns with 3 targets and one column with 2 targets. The rules are:

- The shooter can freely choose which column to shoot at.
- He must attempt the lowest target not yet hit.

a) If the shooter ignores the second rule, in how many ways can he choose only 3 positions out of the 8 distinct targets to shoot?

b) If the rules are followed, in how many ways can the 8 targets be hit?

Answer: 560


With all angles measured in degrees, the product $\prod_{k=1}^{22} \sec^2(4k)^\circ=p^q$, where $p$ and $q$ are integers greater than 1. Find the value of $p+q$.

Answer: 46


A sequence of positive integers is defined by $a_{0}=1$ and $a_{n+1}=a_{n}^{2}+1$ for each $n \geq 0$. Find $\operatorname{gcd}(a_{999}, a_{2004})$.

Answer: 677


Determine all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) for which

\[
f(2x + f(y)) = x + y + f(x)
\]

holds for all \( x, y \in \mathbb{R} \).

Answer: f(x)=x


$p$  is a polynomial with integer coefficients and for every natural  $n$  we have  $p(n)>n$ .  $x_k $  is a sequence that:  $x_1=1, x_{i+1}=p(x_i)$  for every  $N$  one of  $x_i$  is divisible by  $N.$  Prove that  $p(x)=x+1$

Answer: p(x)=x+1


Let the set \( S_{n} = \{1, 2, \cdots, n\} \). If \( X \) is a subset of \( S_{n} \), the sum of all elements in \( X \) is called the "capacity" of \( X \) (stipulating that the capacity of the empty set is 0). If the capacity of \( X \) is an odd (even) number, then \( X \) is called an odd (even) subset of \( S_{n} \).

1. Prove that the number of odd subsets and even subsets of \( S_{n} \) are equal.

2. Prove that for \( n \geqslant 3 \), the sum of the capacities of all odd subsets of \( S_{n} \) is equal to the sum of the capacities of all even subsets of \( S_{n} \).

3. For \( n \geqslant 3 \), find the sum of the capacities of all odd subsets of \( S_{n} \).

Answer: 2^{n-3}n(n+1)


In a given right triangle  $ABC$ , the hypotenuse  $BC$ , of length  $a$ , is divided into  $n$  equal parts ( $n$  and odd integer). Let  $\alpha$  be the acute angel subtending, from  $A$ , that segment which contains the mdipoint of the hypotenuse. Let  $h$  be the length of the altitude to the hypotenuse fo the triangle. Prove that: \[ \tan{\alpha}=\dfrac{4nh}{(n^2-1)a}.  \]

Answer: \tan{\alpha}=\frac{4nh}{(n^2-1)}


Given two lines $l_1: y = m$ and $l_2: y = \frac{8}{2m+1}$ ($m > 0$), line $l_1$ intersects the graph of the function $y = |\log_2 x|$ from left to right at points $A$ and $B$, and line $l_2$ intersects the graph of the function $y = |\log_2 x|$ from left to right at points $C$ and $D$. The lengths of the projections of segments $AC$ and $BD$ on the $x$-axis are denoted as $a$ and $b$, respectively. When $m$ varies, the minimum value of $\frac{b}{a}$ is __________.

Answer: 8\sqrt{2}


Replace the symbols $\$ last\$ with the same numbers so that the equation becomes true: $\$ \$ |\frac{20}{x} - \frac{x}{15} = \frac{20}{15} \$ \$$

Answer: 10


For real numbers \( x \), \( \lfloor x \rfloor \) denotes the greatest integer less than or equal to \( x \). Find the largest positive integer \( n \) such that the following equation holds:
\[
\lfloor \log_2 1 \rfloor + \lfloor \log_2 2 \rfloor + \lfloor \log_2 3 \rfloor + \cdots + \lfloor \log_2 n \rfloor = 1994
\]
(12th Annual American Invitational Mathematics Examination, 1994)

Answer: 312


Find the maximum positive integer \( n \) such that
\[
n^{2} \leq 160 \times 170 \times 180 \times 190
\]

Answer: 30499


There are 456 natives on an island, each of whom is either a knight who always tells the truth or a liar who always lies. All residents have different heights. Once, each native said, "All other residents are shorter than me!" What is the maximum number of natives who could have then said one minute later, "All other residents are taller than me?"

Answer: 454


Find the total number of positive four-digit integers \( N \) satisfying both of the following properties:
(i) \( N \) is divisible by 7, and
(ii) when the first and last digits of \( N \) are interchanged, the resulting positive integer is also divisible by 7. (Note that the resulting integer need not be a four-digit number.)

Answer: 210


In regular hexagon $ABCDEF$, points $W$, $X$, $Y$, and $Z$ are chosen on sides $\overline{BC}$, $\overline{CD}$, $\overline{EF}$, and $\overline{FA}$ respectively, so lines $AB$, $ZW$, $YX$, and $ED$ are parallel and equally spaced. What is the ratio of the area of hexagon $WCXYFZ$ to the area of hexagon $ABCDEF$?
$\textbf{(A)}\ \frac{1}{3}\qquad\textbf{(B)}\ \frac{10}{27}\qquad\textbf{(C)}\ \frac{11}{27}\qquad\textbf{(D)}\ \frac{4}{9}\qquad\textbf{(E)}\ \frac{13}{27}$

Answer: \frac{11}{27}\implies\textbf{(C)}


Points $A(3,5)$ and $B(7,10)$ are the endpoints of a diameter of a circle graphed in a coordinate plane. How many square units are in the area of the circle? Express your answer in terms of $\pi$.

Answer: \frac{41\pi}{4}


Let \( \triangle ABC \) have centroid \( S \), midpoint of segment \( AS \) be \( H \), and midpoint of side \( AB \) be \( Q \). Let the line parallel to \( BC \) through \( H \) intersect \( AB \) at \( P \) and line \( CQ \) at \( R \). What is the ratio of the areas of triangles \( PQR \) and \( APH \)?

Answer: 1


(The full score for this question is 8 points) Arrange 3 male students and 2 female students in a row,
　　(1) The number of all different arrangements;
(2) The number of arrangements where exactly two male students are adjacent;
(3) The number of arrangements where male students are of different heights and are arranged from tallest to shortest from left to right.
[Answer all questions with numbers]

Answer: 20


Each side of the triangle $ABC$ is divided into $p$ equal parts, where $p$ is a prime number. Then, from each side, one division point is connected to the opposite vertex of the triangle such that these three lines intersect at a single point. Determine the possible values of $p$.

Answer: 2


Given that \( a \) and \( b \) are real numbers and the sets \( A = \{a, a^{2}, ab\} \) and \( B = \{1, a, b\} \), if \( A = B \), find the value of \( a^{2004} + b^{2004} \).

Answer: 1


Given \(0 < a < b\), two fixed points \(A(a, 0)\) and \(B(b, 0)\) are used to draw lines \(l\) and \(m\), respectively, which intersect the parabola \(y^2 = x\) at four different points. When these four points are concyclic, find the locus of the intersection point \(P\) of these lines \(l\) and \(m\).

Answer: \frac{b}{2}


A triangle \(ABC\) is inscribed with a circle centered at \(O\). The median \(AD\) intersects the circle at points \(X\) and \(Y\). Find the angle \( \angle XOY \) given that \( AC = AB + AD \).

Answer: 120


A number like 45132 is called a "wave number," which means the tens and thousands digits are both larger than their respective neighboring digits. What is the probability of forming a non-repeating five-digit "wave number" using the digits 1, 2, 3, 4, 5?

Answer: \frac{1}{15}


Answer: 67.35\%


Let $A B C$ be a triangle with $A B=13, B C=14, C A=15$. Let $O$ be the circumcenter of $A B C$. Find the distance between the circumcenters of triangles $A O B$ and $A O C$.

Answer: \frac{91}{6}


In 2021, a first-class postage stamp cost 85 pence and a second-class postage stamp cost 66 pence. In order to spend an exact number of pounds and to buy at least one of each type, what is the smallest total number of stamps that should be purchased?
A) 10
B) 8
C) 7
D) 5
E) 2

Answer: 7


A section of a book fell out. The first page of this section is numbered 143, and the number of the last page consists of the same digits but in a different order. How many pages fell out of the book?

Answer: 172


The product of two positive integers plus their sum is 119. The integers are relatively prime and each is less than 30. What is the sum of the two integers?

Answer: 20


Given a point $P$ on a triangular piece of paper $ABC,\,$ consider the creases that are formed in the paper when $A, B,\,$ and $C\,$ are folded onto $P.\,$ Let us call $P$ a fold point of $\triangle ABC\,$ if these creases, which number three unless $P$ is one of the vertices, do not intersect. Suppose that $AB=36, AC=72,\,$ and $\angle B=90^\circ.\,$ Then the area of the set of all fold points of $\triangle ABC\,$ can be written in the form $q\pi-r\sqrt{s},\,$ where $q, r,\,$ and $s\,$ are positive integers and $s\,$ is not divisible by the square of any prime. What is $q+r+s\,$?

Answer: 597


We have a spherical onion with a diameter of $4 \mathrm{~cm}$, which is cut into slices each $2 \mathrm{~mm}$ thick. How many times larger is the combined surface area of the 20 slices compared to the original surface area of the onion?

Answer: 7.65


If a number is a multiple of 4 or contains the digit 4, we say this number is a "4-inclusive number", such as 20, 34. Arrange all "4-inclusive numbers" in the range \[0, 100\] in ascending order to form a sequence. What is the sum of all items in this sequence?

Answer: 1883


Let  $A$  and  $B$  be distinct positive integers such that each has the same number of positive divisors that 2013 has.  Compute the least possible value of  $\left| A - B \right|$ .

Answer: 1


Ten 6-sided dice are rolled. What is the probability that exactly three of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth.

Answer: .155


Prove that if the parameter \( t \) is eliminated from the system of equations

$$
\begin{gathered}
x = 1 + 2t - 2t^{2}, \\
y = 2(1 + t) \sqrt{1 - t^{2}},
\end{gathered}
$$

then the following equation is obtained:

$$
y^{4} + 2 y^{2} \left( x^{2} - 12x + 9 \right) + x^{4} + 8x^{3} + 18x^{2} - 27 = 0.
$$

Answer: y^4+2y^2(x^2-12x+9)+x^4+8x^3+18x^2-27=0


How many circles of radius 1 are needed to cover a square with a side length of 2?

Answer: 4


In triangle $ABC$, let $a$, $b$, and $c$ be the lengths of the sides opposite to angles $A$, $B$, and $C$ respectively, with $b=4$ and $\frac{\cos B}{\cos C} = \frac{4}{2a - c}$.
(1) Find the measure of angle $B$;
(2) Find the maximum area of $\triangle ABC$.

Answer: 4\sqrt{3}


What is the value of \[2^{\left(0^{\left(1^9\right)}\right)}+\left(\left(2^0\right)^1\right)^9?\]
$\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 2 \qquad\textbf{(D) } 3 \qquad\textbf{(E) } 4$

Answer: \textbf{(C)}


Define $n!!$ to be $n(n-2)(n-4)\cdots 3\cdot 1$ for $n$ odd and $n(n-2)(n-4)\cdots 4\cdot 2$ for $n$ even. When $\sum_{i=1}^{2009} \frac{(2i-1)!!}{(2i)!!}$ is expressed as a fraction in lowest terms, its denominator is $2^ab$ with $b$ odd. Find $\dfrac{ab}{10}$.

Answer: 401


When \( n \) is a positive integer, the function \( f \) satisfies \( f(n+3)=\frac{f(n)-1}{f(n)+1} \), with \( f(1) \neq 0 \) and \( f(1) \neq \pm 1 \). Find the value of \( f(8) \cdot f(2018) \).

Answer: -1


Three boys \( B_{1}, B_{2}, B_{3} \) and three girls \( G_{1}, G_{2}, G_{3} \) are to be seated in a row according to the following rules:
1) A boy will not sit next to another boy and a girl will not sit next to another girl,
2) Boy \( B_{1} \) must sit next to girl \( G_{1} \).
If \( s \) is the number of different such seating arrangements, find the value of \( s \).

Answer: 40


Calculate $1010101_2 + 111000_2$ and write your answer in base $10$.

Answer: 141


Let  $ a_i,b_i$   $ (i\equal{}1,2,...,n)$  be real numbers such that the  $ a_i$  are distinct, and suppose that there is a real number  $ \alpha$  such that the product  $ (a_i\plus{}b_1)(a_i\plus{}b_2)...(a_i\plus{}b_n)$  is equal to  $ \alpha$  for each  $ i$ . Prove that there is a real number  $ \beta$  such that  $ (a_1\plus{}b_j)(a_2\plus{}b_j)...(a_n\plus{}b_j)$  is equal to  $ \beta$  for each  $ j$ .

Answer: \beta=(-1)^{n+1}\alpha


Let ellipse $\Gamma: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a > b > 0)$ have an eccentricity of $\frac{\sqrt{3}}{2}$. A line with slope $k (k > 0)$ passes through the left focus $F$ and intersects the ellipse $\Gamma$ at points $A$ and $B$. If $\overrightarrow{A F}=3 \overrightarrow{F B}$, find $k$.

Answer: \sqrt{2}


For real numbers \( x \) and \( y \), define the operation \( \star \) as follows: \( x \star y = xy + 4y - 3x \).

Compute the value of the expression

$$
((\ldots)(((2022 \star 2021) \star 2020) \star 2019) \star \ldots) \star 2) \star 1
$$

Answer: 12


Find the number of integers  $n$  for which  $\sqrt{\frac{(2020 - n)^2}{2020 - n^2}}$  is a real number.

Answer: 90


Suppose \( A, B, C \), and \( D \) are four circles of radius \( r > 0 \) centered at the points \((0, r)\), \((r, 0)\), \((0, -r)\), and \((-r, 0)\) in the plane. Let \( O \) be a circle centered at \((0, 0)\) with radius \( 2r \). In terms of \( r \), what is the area of the union of circles \( A, B, C \), and \( D \) subtracted by the area of circle \( O \) that is not contained in the union of \( A, B, C \), and \( D \)?

Answer: 8r^2


There are  $100$  white points on a circle. Asya and Borya play the following game: they alternate, starting with Asya, coloring a white point in green or blue. Asya wants to obtain as much as possible pairs of adjacent points of distinct colors, while Borya wants these pairs to be as less as possible. What is the maximal number of such pairs Asya can guarantee to obtain, no matter how Borya plays.

Answer: 50


(Gabriel Carroll) Let $n$ be a positive integer. Denote by $S_n$ the set of points $(x, y)$ with integer coordinates such that
\[\left|x\right| + \left|y + \frac {1}{2}\right| < n\]
A path is a sequence of distinct points $(x_1 , y_1 ), (x_2 , y_2 ), \ldots , (x_\ell, y_\ell)$ in $S_n$ such that, for $i = 2, \ldots , \ell$, the distance between $(x_i , y_i )$ and $(x_{i - 1} , y_{i - 1} )$ is $1$ (in other words, the points $(x_i , y_i )$ and $(x_{i - 1} , y_{i - 1} )$ are neighbors in the lattice of points with integer coordinates). Prove that the points in $S_n$ cannot be partitioned into fewer than $n$ paths (a partition of $S_n$ into $m$ paths is a set $\mathcal{P}$ of $m$ nonempty paths such that each point in $S_n$ appears in exactly one of the $m$ paths in $\mathcal{P}$).

Answer: n


In the number $56439.2071$, the value of the place occupied by the digit 6 is how many times as great as the value of the place occupied by the digit 2?

Answer: 10,000


Let  $R$  and  $S$  be the numbers defined by
\[R = \dfrac{1}{2} \times \dfrac{3}{4} \times \dfrac{5}{6} \times \cdots \times \dfrac{223}{224} \text{ and } S = \dfrac{2}{3} \times \dfrac{4}{5} \times \dfrac{6}{7} \times \cdots \times \dfrac{224}{225}.\]Prove that  $R < \dfrac{1}{15} < S$ .

Answer: R<\frac{1}{15}<S


Prove the inequality for the elements of an arbitrary triangle:

$$
\frac{1}{(p-a)^{2}}+\frac{1}{(p-b)^{2}}+\frac{1}{(p-c)^{2}} \geq \frac{1}{r^{2}}
$$

Answer: \frac{1}{(p-)^{2}}+\frac{1}{(p-b)^{2}}+\frac{1}{(p-)^{2}}\ge\frac{1}{r^{2}}


Each segment with endpoints at the vertices of a regular 100-gon is painted red if there is an even number of vertices between its endpoints, and blue otherwise (in particular, all sides of the 100-gon are red). Numbers are placed at the vertices such that the sum of their squares equals 1, and the products of the numbers at the endpoints are placed on the segments. Then the sum of the numbers on the red segments is subtracted from the sum of the numbers on the blue segments. What is the largest possible result?

Answer: 1/2


Let \( [x] \) denote the greatest integer not exceeding \( x \). Determine the smallest natural number \( n \) that satisfies
$$
\left[\frac{1}{15}\right]+\left[\frac{2}{15}\right]+\left[\frac{3}{15}\right]+\cdots+\left[\frac{n-1}{15}\right]+\left[\frac{n}{15}\right]>2000.
$$

Answer: 252


Into how many regions can a circle be cut by 10 parabolas?

Answer: 201


Find the least natural number  $ n$  such that, if the set  $ \{1,2, \ldots, n\}$  is arbitrarily divided into two non-intersecting subsets, then one of the subsets contains 3 distinct numbers such that the product of two of them equals the third.

Answer: 96


The number $r$ can be expressed as a four-place decimal $0.abcd,$ where $a, b, c,$ and $d$ represent digits, any of which could be zero. It is desired to approximate $r$ by a fraction whose numerator is 1 or 2 and whose denominator is an integer. The closest such fraction to $r$ is $\frac 27.$ What is the number of possible values for $r$?

Answer: 417


A diagonal from a vertex of a polygon forms a triangle with the two adjacent sides, so the number of triangles formed by the polygon is at least the number of sides of the polygon. However, the number of triangles cannot exceed the number of ways to choose 2 sides from the polygon, which is given by the combination formula $\binom{n}{2}=\frac{n(n-1)}{2}$. If the number of triangles is at most 2021, determine the number of sides of the polygon.

Answer: 2023


Find all integers $n$ satisfying $n \geq 2$ and $\dfrac{\sigma(n)}{p(n)-1} = n$, in which $\sigma(n)$ denotes the sum of all positive divisors of $n$, and $p(n)$ denotes the largest prime divisor of $n$.

Answer: 6


A rational number written in base eight is $\underline{ab} . \underline{cd}$, where all digits are nonzero. The same number in base twelve is $\underline{bb} . \underline{ba}$. Find the base-ten number $\underline{abc}$.

Answer: 321


Let \( n \) be a positive integer, and denote by \( f(n) \) the smallest positive integer \( k \) for which \( \left\lfloor \frac{n^2}{k} \right\rfloor = \left\lfloor \frac{n^2}{k+1} \right\rfloor \). Determine the value of \( \left\lfloor \frac{n^2}{f(n)} \right\rfloor + f(n) \).

Answer: 2n


Given vectors $\overrightarrow{a} =(\cos x,\sin x)$, $\overrightarrow{b} =(3,-\sqrt{3} )$, with $x\in[0,\pi]$.
$(1)$ If $\overrightarrow{a} \parallel \overrightarrow{b}$, find the value of $x$;
$(2)$ Let $f(x)=\overrightarrow{a} \cdot \overrightarrow{b}$, find the maximum and minimum values of $f(x)$ and the corresponding values of $x$.

Answer: -2 \sqrt {3}


A secant line splits a circle into two segments. Inside those segments, one draws two squares such that both squares has two corners on a secant line and two on the circumference. The ratio of the square's side lengths is  $5:9$ . Compute the ratio of the secant line versus circle radius.

Answer: \frac{3\sqrt{10}}{5}


On the sides $\mathrm{AB}$ and $\mathrm{AC}$ of a right triangle $\mathrm{ABC}\left(\angle B C A=90^{\circ}\right)$, right triangles $\mathrm{ABT}$ and $\mathrm{ACK}$ are constructed externally such that $\angle A T B=\angle A K C=90^{\circ}$ and $\angle A B T=\angle A C K=60^{\circ}$. A point $\mathrm{M}$ is chosen on the side $\mathrm{BC}$ such that $\mathrm{BM}=\mathrm{MC}$. Determine the measure of angle $\angle KMT$ in degrees.

Answer: 120


A group of schoolchildren, heading to a school camp, was planned to be seated in buses so that there would be an equal number of passengers in each bus. Initially, 22 people were seated in each bus, but it turned out that three schoolchildren could not be seated. However, when one bus left empty, all the remaining schoolchildren seated equally in the other buses. How many schoolchildren were in the group, given that no more than 18 buses were provided for transporting the schoolchildren, and each bus can hold no more than 36 people? Give the answer as a number without indicating the units.

Answer: 135


Find the smallest natural number \( m \) such that the expression \( 148^{n} + m \cdot 141^{n} \) is divisible by 2023 for any odd natural \( n \).

Answer: 1735


Let $ABC$ be an acute-angled triangle, $CC_1$ its angle bisector, and $O$ the circumcenter. The intersection point of the line $OC_1$ with the perpendicular dropped from vertex $C$ onto side $AB$ lies on the circumcircle $\Omega$ of triangle $AOB$. Find angle $C$.

Answer: 60


Determine the maximum possible real value of the number  $ k$ , such that
\[ (a \plus{} b \plus{} c)\left (\frac {1}{a \plus{} b} \plus{} \frac {1}{c \plus{} b} \plus{} \frac {1}{a \plus{} c} \minus{} k \right )\ge k\]
for all real numbers  $ a,b,c\ge 0$  with  $ a \plus{} b \plus{} c \equal{} ab \plus{} bc \plus{} ca$ .

Answer: 1


In the quadrilateral pyramid \(PABCD\), the lateral face \(P A D\) is an equilateral triangle and is perpendicular to the base \(ABCD\). The base \(ABCD\) is a rhombus with side length 2, and \(\angle B A D = 60^\circ\). Let \(N\) be the midpoint of \(P B\), and the cross-section \(DAN\) intersects \(P C\) at \(M\). Find the dihedral angle between the planes \(P-AD-M\).

Answer: 45


The last two digits of the decimal representation of the square of a natural number are the same and are not zero. What are these digits? Find all solutions.

Answer: 44


In the convex quadrilateral  $ABCD$  angle  $\angle{BAD}=90$ , $\angle{BAC}=2\cdot\angle{BDC}$  and  $\angle{DBA}+\angle{DCB}=180$ . Then find the angle  $\angle{DBA}$

Answer: 45


Once in a restaurant ***Dr. Strange*** found out that there were 12 types of food items from 1 to 12 on the menu. He decided to visit the restaurant 12 days in a row and try a different food everyday. 1st day, he tries one of the items from the first two. On the 2nd day, he eats either item 3 or the item he didn’t tried on the 1st day. Similarly, on the 3rd day, he eats either item 4 or the item he didn’t tried on the 2nd day. If someday he's not able to choose items that way, he eats the item that remained uneaten from the menu. In how many ways can he eat the items for 12 days?

Answer: 2048


Answer: \frac{221}{2}


Three people, A, B, and C, arrive at the shore of a reservoir. A swims to the opposite shore at a speed of \( v \) km/h. Meanwhile, B and C take a motorboat moving at a speed of \( 10v \) km/h. After some time, C decides to swim the remaining distance at the same speed as A. B immediately turns the boat around to pick up A, who quickly gets into the boat and continues the journey with B. All three arrive at the opposite shore simultaneously. Determine the time it takes to cross if it is known that the width of the reservoir is \( b \) km and the speed of the current is zero.

Answer: \frac{31b}{130v}


The non-zero numbers \( a, b, \) and \( c \) are such that the doubled roots of the quadratic polynomial \( x^{2}+a x+b \) are the roots of the polynomial \( x^{2}+b x+c \). What can the ratio \( a / c \) equal?

Answer: \frac{1}{8}


Let $f(x)=c x(x-1)$, where $c$ is a positive real number. We use $f^{n}(x)$ to denote the polynomial obtained by composing $f$ with itself $n$ times. For every positive integer $n$, all the roots of $f^{n}(x)$ are real. What is the smallest possible value of $c$?

Answer: 2


Piglet has balloons of five different colors. He managed to arrange them in a row in such a way that for any two different colors in the row, there are always two adjacent balloons of these colors. What is the minimum number of balloons Piglet could have?

Answer: 11


Let \( \triangle ABC \) be a triangle with circumcenter \( O \), incenter \( I \), \(\angle B = 45^{\circ}\), and \( OI \parallel BC \). Find \( \cos \angle C \).

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


Three students proposed different methods to tackle the problem: "For the inequality $x^{2}+25+\left|x^{3}-5 x^{2}\right| \geqslant a x$ to hold for all $x$ in the interval $[1,12]$, find the range of the real number $a$."

Each student suggested a different approach:
- Student A: "We only need the minimum value on the left-hand side of the inequality to be no smaller than the maximum value on the right-hand side."
- Student B: "Transform the inequality into a function involving variable $x$ on the left side and a constant on the right side, then find the extremum of the function."
- Student C: "Consider both sides of the inequality as functions of $x$ and plot their graphs."

Based on their proposed methods, determine the correct conclusion for the range of $a$.

Answer: (-\infty,10]


Points \( K \) and \( M \) are located on sides \( AB \) and \( BC \) of triangle \( ABC \), with \( BK : KA = 1 : 4 \) and \( BM : MC = 3 : 2 \). Lines \( MK \) and \( AC \) intersect at point \( N \).

Find the ratio \( AC : CN \).

Answer: 5:1


Given the six whole numbers 10-15, compute the largest possible value for the sum, S, of the three numbers on each side of the triangle.

Answer: 39


At a chemistry conference, there were $k$ scientists, both chemists and alchemists, with more chemists than alchemists present. It is known that chemists always tell the truth, while alchemists sometimes tell the truth and sometimes lie. A mathematician attending the conference wants to determine whether each scientist is a chemist or an alchemist. To this end, the mathematician can ask any scientist the following question: "Is this person a chemist or an alchemist?" (In particular, they can ask about the identity of the scientist themselves). Prove that the mathematician can determine this with:

a) $4k$ questions
b) $2k-2$ questions.

Answer: 2k-2


Calculate the limit of the function:

$$\lim _{x \rightarrow 0}\left(\frac{\arcsin ^{2} x}{\arcsin ^{2} 4 x}\right)^{2 x+1}$$

Answer: 1/16


Given that \( m \) and \( n \) are integers such that \( m, n \in \{1, 2, \dots, 1981\} \), and \( (n^2 - mn - m^2)^2 = 1 \), determine the maximum value of \( m^2 + n^2 \).

Answer: 3524578


Compute the smallest positive integer \( k \) such that 49 divides \( \binom{2k}{k} \).

Answer: 25