The front tires of a car wear out after 25,000 km, and the rear tires wear out after 15,000 km. When should the tires be swapped so that they wear out at the same time?

Answer: 9375


The edges of tetrahedron $ABCD$ are given as $AB=c$, $BC=a$, $CA=b$, $DA=a_{1}$, $DB=b_{1}$, and $DC=c_{1}$. Let $h$ be the length of the median line from vertex $D$. Prove that

$$
h^{2}=\frac{1}{3}\left(a_{1}^{2}+b_{1}^{2}+c_{1}^{2}\right)-\frac{1}{9}\left(a^{2}+b^{2}+c^{2}\right)
$$

Answer: ^2=\frac{1}{3}(a_1^2+b_1^2+c_1^2)-\frac{1}{9}(^2+b^2+^2)


The largest three-digit number divided by an integer, with the quotient rounded to one decimal place being 2.5, will have the smallest divisor as:

Answer: 392


Mr. Lee V. Soon starts his morning commute at 7:00 AM to arrive at work by 8:00 AM. If he drives at an average speed of 30 miles per hour, he is late by 5 minutes, and if he drives at an average speed of 70 miles per hour, he is early by 4 minutes. Find the speed he needs to maintain to arrive exactly at 8:00 AM.

Answer: 32.5


Suppose convex hexagon  $ \text{HEXAGN}$  has  $ 120^\circ$ -rotational symmetry about a point  $ P$ —that is, if you rotate it  $ 120^\circ$  about  $ P$ , it doesn't change. If  $ PX\equal{}1$ , find the area of triangle  $ \triangle{GHX}$ .

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


A firecracker was thrown vertically upward with a speed of $20 \, \mathrm{m/s}$. One second after the flight began, it exploded into two unequal parts, with their mass ratio being $1:2$. Immediately after the explosion, the smaller fragment flew horizontally with a speed of $16 \, \mathrm{m/s}$. Find the speed (in m/s) of the second fragment immediately after the explosion. Assume the acceleration due to gravity is $10 \, \mathrm{m/s}^2$.

Answer: 17\,


In rectangle $PQRS$, $PQ=8$ and $QR=6$. Points $A$ and $B$ lie on $\overline{PQ}$, points $C$ and $D$ lie on $\overline{QR}$, points $E$ and $F$ lie on $\overline{RS}$, and points $G$ and $H$ lie on $\overline{SP}$ so that $AP=BQ<4$ and the convex octagon $ABCDEFGH$ is equilateral. The length of a side of this octagon can be expressed in the form $k+m\sqrt{n}$, where $k$, $m$, and $n$ are integers and $n$ is not divisible by the square of any prime. What is $k+m+n$?

Answer: 7


The integer 119 is a multiple of which number?

Answer: 7


Given the function \( f(x)=\sqrt{3} \sin 2x + 2 \cos^2 x + a \), if the minimum value of \( f(x) \) on the interval \(\left[ 0, \frac{\pi}{2} \right] \) is \(-1\), find the value of \( a \).

Answer: -1


Find all functions  $f : \mathbb{N} \to \mathbb{N}$  such that for any positive integers  $m, n$  the number  $$ (f(m))^2+ 2mf(n) + f(n^2) $$  is the square of an integer.

*Proposed by Fedir Yudin*

Answer: f(n)=n


Define the function \( f \) on positive integers such that

\[ f(1) = 1, \quad f(3) = 3 \]

and

\[
\begin{aligned}
f(2n) &= f(n), \\
f(4n+1) &= 2f(2n+1) - f(n), \\
f(4n+3) &= 3f(2n+1) - 2f(n)
\end{aligned}
\]

for every positive integer \( n \).

Determine the number of integers \( n \) satisfying \( 1 \leq n \leq 1988 \) for which \( f(n) = n \).

Answer: 92


A convex pentagon  $P=ABCDE$  is inscribed in a circle of radius  $1$ . Find the maximum area of  $P$  subject to the condition that the chords  $AC$  and  $BD$  are perpendicular.

Answer: 1 + \frac{3\sqrt{3}}{4}


A square with an integer side length was cut into 2020 smaller squares. It is known that the areas of 2019 of these squares are 1, while the area of the 2020th square is not equal to 1. Find all possible values that the area of the 2020th square can take. Provide the smallest of these possible area values in the answer.

Answer: 112225


Find the smallest natural number \( n \) such that in a simple graph with 10 vertices and \( n \) edges that is 2-colored, there always exists a monochromatic triangle or a monochromatic quadrilateral.

Answer: 31


Given vectors $\overrightarrow{a}, \overrightarrow{b}$ satisfy $\overrightarrow{a}=(1, \sqrt {3}), |\overrightarrow{b}|=1, |\overrightarrow{a}+ \overrightarrow{b}|= \sqrt {3}$, then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is ______.

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


Given two non-zero vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ that satisfy $| \overrightarrow {a} + \overrightarrow {b} | = | \overrightarrow {a} - \overrightarrow {b} | = \sqrt {2} | \overrightarrow {a} |$, find the angle between vector $\overrightarrow {b}$ and $\overrightarrow {a} + \overrightarrow {b}$.

Answer: \frac{\pi}{4}


Determine all positive integers $n$ for which the equation $$x^{n}+(2+x)^{n}+(2-x)^{n}=0$$ has an integer as a solution.

Answer: n=1


Before the soccer match between the "North" and "South" teams, five predictions were made:

a) There will be no draw;

b) "South" will concede goals;

c) "North" will win;

d) "North" will not lose;

e) Exactly 3 goals will be scored in the match.

After the match, it was found that exactly three predictions were correct. What was the final score of the match?

Answer: 2-1


Let  $ n > 1, n \in \mathbb{Z}$  and  $ B \equal{}\{1,2,\ldots, 2^n\}.$  A subset  $ A$  of  $ B$  is called weird if it contains exactly one of the distinct elements  $ x,y \in B$  such that the sum of  $ x$  and  $ y$  is a power of two. How many weird subsets does  $ B$  have?

Answer: 2^{n+1}


Investigate the stability of the equilibrium point $x=0, y=0$ for the system

$$
\left\{\begin{array}{l}
\frac{d x}{d t}=y^{3}+x^{5} \\
\frac{d y}{d t}=x^{3}+y^{5}
\end{array}\right.
$$

Answer: Unstable


What is the value of $x + y$ if the sequence $3, ~9, ~15, \ldots, ~x, ~y, ~39$ is an arithmetic sequence?

Answer: 60


Consider the region $A^{}_{}$ in the complex plane that consists of all points $z^{}_{}$ such that both $\frac{z^{}_{}}{40}$ and $\frac{40^{}_{}}{\overline{z}}$ have real and imaginary parts between $0^{}_{}$ and $1^{}_{}$, inclusive. Find the area of $A.$

Answer: 1200 - 200 \pi


The diagram shows an octagon consisting of $10$ unit squares. The portion below $\overline{PQ}$ is a unit square and a triangle with base $5$. If $\overline{PQ}$ bisects the area of the octagon, what is the ratio $\dfrac{XQ}{QY}$?

Answer: \frac{2}{3}


Calculate the definite integral:

$$
\int_{0}^{\sqrt{2} / 2} \frac{x^{4} \cdot d x}{\sqrt{\left(1-x^{2}\right)^{3}}}
$$

Answer: \frac{5}{4} - \frac{3\pi}{8}


From the vertex of the right angle of triangle \(ABC\), the median \(CM\) is drawn. The circle inscribed in triangle \(CAM\) touches \(CM\) at its midpoint. Find the angle \(BAC\).

Answer: 60


Answer: \2


Let  $S$  be the set of  $2\times2$ -matrices over  $\mathbb{F}_{p}$  with trace  $1$  and determinant  $0$ . Determine  $|S|$ .

Answer: p(p+1)


Let  $n\ge 2$  be a positive integer. There are  $n$  real coefficient polynomials  $P_1(x),P_2(x),\cdots ,P_n(x)$  which is not all the same, and their leading coefficients are positive. Prove that  $$ \deg(P_1^n+P_2^n+\cdots +P_n^n-nP_1P_2\cdots P_n)\ge (n-2)\max_{1\le i\le n}(\deg P_i) $$  and find when the equality holds.

Answer: P_1^n+P_2^n+\cdots+P_n^n-nP_1P_2\cdotsP_n)\ge(n-2)\max_{1\lei\len}(\P_i)


Let  $p(x)$  be a polynomial of degree strictly less than  $100$  and such that it does not have  $(x^3-x)$  as a factor. If   $$ \frac{d^{100}}{dx^{100}}\bigg(\frac{p(x)}{x^3-x}\bigg)=\frac{f(x)}{g(x)} $$   for some polynomials  $f(x)$  and  $g(x)$  then find the smallest possible degree of  $f(x)$ .

Answer: 200


Using the six digits 0, 1, 2, 3, 4, 5 to form a six-digit number without repeating any digit, how many such numbers are there where the unit digit is less than the ten's digit?

Answer: 300


What is the minimum number of cells that need to be marked on a chessboard so that

1) none of the marked cells are adjacent (they do not share a side or a corner),
2) adding any one cell to these marked cells would violate condition 1?

Answer: 9


Using each of the nine digits exactly once, form prime numbers (numbers that are divisible only by 1 and themselves) such that their sum is minimized.

Answer: 207


A ray of light passing through the point $A = (-3,9,11),$ reflects off the plane $x + y + z = 12$ at $B,$ and then passes through the point $C = (3,5,9).$  Find the point $B.$

[asy]
import three;

size(180);
currentprojection = perspective(6,3,2);

triple A, B, C;

A = (0,-0.5,0.5*1.5);
B = (0,0,0);
C = (0,0.8,0.8*1.5);

draw(surface((-1,-1,0)--(-1,1,0)--(1,1,0)--(1,-1,0)--cycle),paleyellow,nolight);
draw((-1,-1,0)--(-1,1,0)--(1,1,0)--(1,-1,0)--cycle);
draw(A--B--C,Arrow3(6));

label("$A$", A, NW);
label("$B$", B, S);
label("$C$", C, NE);
[/asy]

Answer: \left( -\frac{5}{3}, \frac{16}{3}, \frac{25}{3} \right)


In the plane rectangular coordinate system $xOy$, the equation of the hyperbola $C$ is $x^{2}-y^{2}=1$. Find all real numbers $a$ greater than 1 that satisfy the following requirement: Through the point $(a, 0)$, draw any two mutually perpendicular lines $l_{1}$ and $l_{2}$. If $l_{1}$ intersects the hyperbola $C$ at points $P$ and $Q$, and $l_{2}$ intersects $C$ at points $R$ and $S$, then $|PQ| = |RS|$ always holds.

Answer: \sqrt{2}


Given six cards with the digits $1, 2, 4, 5, 8$ and a comma. Using each card exactly once, various numbers are formed (the comma cannot be at the beginning or at the end of the number). What is the arithmetic mean of all such numbers?

(M. V. Karlukova)

Answer: 1234.4321


Suppose that  $P(x)$  is a monic cubic polynomial with integer roots, and suppose that  $\frac{P(a)}{a}$  is an integer for exactly  $6$  integer values of  $a$ . Suppose furthermore that exactly one of the distinct numbers  $\frac{P(1) + P(-1)}{2}$  and  $\frac{P(1) - P(-1)}{2}$  is a perfect square. Given that  $P(0) > 0$ , find the second-smallest possible value of  $P(0).$ *Proposed by Andrew Wu*

Answer: 25


Given a sphere O with a radius of 2, a cone is inscribed in the sphere O. When the volume of the cone is maximized, find the radius of the sphere inscribed in the cone.

Answer: \frac{4(\sqrt{3} - 1)}{3}


Let \( f(x) \) be an odd function defined on \( \mathbf{R} \), and for \( x \geq 0 \), \( f(x) = x^2 \). If the inequality \( f(x+a) \geq 2 f(x) \) holds for any \( x \in [a, a+2] \), determine the range of the real number \( a \).

Answer: [\sqrt{2},+\infty)


An infinite sequence of positive numbers \(a_{1}, a_{2}, a_{3}, \ldots\) is defined by the rule: \(a_{1}=1\), and \(a_{n+1}^{2}=a_{n}^{2}+\frac{1}{a_{n}}\) for \(n=1, 2, 3, \ldots\). Prove that the sequence \(b_{1}, b_{2}, b_{3}, \ldots\), where \(b_{n}=a_{n+1}-a_{n}\), is decreasing, i.e., \(b_{1} > b_{2} > b_{3} > \ldots\).

Answer: b_1>b_2>b_3>\ldots


There are three boxes of stones. Each hour, Sisyphus moves a stone from one box to another. For each transfer of a stone, he receives from Zeus a number of coins equal to the number of stones in the box from which the stone is drawn minus the number of stones in the recipient box, with the stone Sisyphus just carried not counted. If this number is negative, Sisyphus pays the corresponding amount (and can pay later if he is broke).

After 1000 years, all the stones lie in their initial boxes. What is the greatest possible earning of Sisyphus at that moment?

Answer: 0


Are there five consecutive natural numbers such that if they are designated by the letters \(a, b, c, d, e\) in some order, the equality
\[
(a+b)(b+c)(c+d)(d+e)(e+a) = (a+c)(c+e)(e+b)(b+d)(d+a)
\]
holds?

Answer: No


An electronic clock always displays the date as an eight-digit number. For example, January 1, 2011, is displayed as 20110101. What is the last day of 2011 that can be evenly divided by 101? The date is displayed as $\overline{2011 \mathrm{ABCD}}$. What is $\overline{\mathrm{ABCD}}$?

Answer: 1221


A square pyramid with a base edge of $26 \mathrm{~cm}$ has adjacent triangular faces that form a $120^{\circ}$ angle with each other. How tall is the pyramid?

Answer: 13\,


The solid shown has a square base of side length $s$. The upper edge is parallel to the base and has length $2s$. All other edges have length $s$. Given that $s=6\sqrt{2}$, what is the volume of the solid?
[asy] size(180); import three; pathpen = black+linewidth(0.65); pointpen = black; currentprojection = perspective(30,-20,10); real s = 6 * 2^.5; triple A=(0,0,0),B=(s,0,0),C=(s,s,0),D=(0,s,0),E=(-s/2,s/2,6),F=(3*s/2,s/2,6); draw(A--B--C--D--A--E--D); draw(B--F--C); draw(E--F); label("A",A,W); label("B",B,S); label("C",C,SE); label("D",D,NE); label("E",E,N); label("F",F,N); [/asy]

Answer: 288


Given an ellipse $C$: $\dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1$ passes through points $M(2,0)$ and $N(0,1)$.

$(1)$ Find the equation of ellipse $C$ and its eccentricity;

$(2)$ A line $y=kx (k \in \mathbb{R}, k \neq 0)$ intersects ellipse $C$ at points $A$ and $B$, point $D$ is a moving point on ellipse $C$, and $|AD| = |BD|$. Does the area of $\triangle ABD$ have a minimum value? If it exists, find the equation of line $AB$; if not, explain why.

Answer: \dfrac{8}{5}


A car departed from point A to point B, and with some delay, a second car followed. When the first car had traveled half the distance, the second car had traveled $26 \frac{1}{4}$ km. When the second car had traveled half the distance, the first car had traveled $31 \frac{1}{5}$ km. After overtaking the first car, the second car reached point B, immediately turned back, and after driving 2 km, met the first car. Find the distance between points A and B. Give your answer as a number without units.

Answer: 58


Two spheres touch the plane of triangle \(ABC\) at points \(A\) and \(B\) and are located on opposite sides of this plane. The sum of the radii of these spheres is 9, and the distance between their centers is \(\sqrt{305}\). The center of a third sphere with a radius of 7 is at point \(C\), and it externally touches each of the first two spheres. Find the radius of the circumcircle of triangle \(ABC\).

Answer: 2\sqrt{14}


In the subtraction shown, $K, L, M$, and $N$ are digits. What is the value of $K+L+M+N$?\n$$\begin{array}{r}6 K 0 L \\ -\quad M 9 N 4 \\ \hline 2011\end{array}$$

Answer: 17


A rectangular box has a total surface area of 94 square inches. The sum of the lengths of all its edges is 48 inches. What is the sum of the lengths in inches of all of its interior diagonals?
$\textbf{(A)}\ 8\sqrt{3}\qquad\textbf{(B)}\ 10\sqrt{2}\qquad\textbf{(C)}\ 16\sqrt{3}\qquad\textbf{(D)}\ 20\sqrt{2}\qquad\textbf{(E)}\ 40\sqrt{2}$

Answer: \textbf{(D)}\20\sqrt{2}


Determine the largest natural number  $m$  such that for each non negative real numbers  $a_1 \ge a_2 \ge ... \ge a_{2014} \ge 0$  , it is true that   $$ \frac{a_1+a_2+...+a_m}{m}\ge \sqrt{\frac{a_1^2+a_2^2+...+a_{2014}^2}{2014}} $$

Answer: 44


What is the smallest positive angle \( x \) for which

\[
2^{\sin^2 x} \cdot 4^{\cos^2 x} \cdot 2^{\tan x} = 8
\]

Answer: 60


Prove that there exists an unique sequence  $ \left( c_n \right)_{n\ge 1} $  of real numbers from the interval  $ (0,1) $  such that  $$  \int_0^1 \frac{dx}{1+x^m} =\frac{1}{1+c_m^m } ,  $$   for all natural numbers  $ m, $  and calculate  $ \lim_{k\to\infty } kc_k^k. $ *Radu Pop*

Answer: \ln(2)


A cube with a side length of 1 meter was cut into smaller cubes with a side length of 1 centimeter and arranged in a straight line. What is the length of the resulting line?

Answer: 10000


Each of the numbers \(1, 2, 3, 4, 5, 6\) is to be placed in the cells of a \(2 \times 3\) table, with one number in each cell. In how many ways can this be done so that in each row and in each column the sum of the numbers is divisible by 3?

Answer: 48


In a right-angled triangle, let \( s_{a} \) and \( s_{b} \) be the medians to the legs, and \( s_{c} \) be the median to the hypotenuse. Determine the maximum value of the expression \( \frac{s_{a} + s_{b}}{s_{c}} \).

Answer: \sqrt{10}


Find the maximum of the expression
$$
|| \ldots|| x_{1}-x_{2}\left|-x_{3}\right|-\ldots\left|-x_{2023}\right|,
$$
where \( x_{1}, x_{2}, \ldots, x_{2023} \) are distinct natural numbers between 1 and 2023.

Answer: 2022


On a particular street in Waterloo, there are exactly 14 houses, each numbered with an integer between 500 and 599, inclusive. The 14 house numbers form an arithmetic sequence in which 7 terms are even and 7 terms are odd. One of the houses is numbered 555 and none of the remaining 13 numbers has two equal digits. What is the smallest of the 14 house numbers?

(An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding a constant. For example, 3, 5, 7, 9 is an arithmetic sequence with four terms.)

Answer: 506


The arithmetic sequence \( a, a+d, a+2d, a+3d, \ldots, a+(n-1)d \) has the following properties:

- When the first, third, fifth, and so on terms are added, up to and including the last term, the sum is 320.
- When the first, fourth, seventh, and so on, terms are added, up to and including the last term, the sum is 224.

What is the sum of the whole sequence?

Answer: 608


Given the function $f(x) = x^2 - 2\cos{\theta}x + 1$, where $x \in \left[-\frac{\sqrt{3}}{2}, \frac{1}{2}\right]$.

(1) When $\theta = \frac{\pi}{3}$, find the maximum and minimum values of $f(x)$.

(2) If $f(x)$ is a monotonous function on $x \in \left[-\frac{\sqrt{3}}{2}, \frac{1}{2}\right]$ and $\theta \in [0, 2\pi)$, find the range of $\theta$.

(3) If $\sin{\alpha}$ and $\cos{\alpha}$ are the two real roots of the equation $f(x) = \frac{1}{4} + \cos{\theta}$, find the value of $\frac{\tan^2{\alpha} + 1}{\tan{\alpha}}$.

Answer: \frac{16 + 4\sqrt{11}}{5}


Is it possible to select 1000 points in a plane so that at least 6000 distances between two of them are equal?

Answer: Yes


For positive integers $n,$ let $\tau (n)$ denote the number of positive integer divisors of $n,$ including 1 and $n.$ For example, $\tau (1)=1$ and $\tau(6) =4.$ Define $S(n)$ by $S(n)=\tau(1)+ \tau(2) + \cdots + \tau(n).$ Let $a$ denote the number of positive integers $n \leq 2005$ with $S(n)$ odd, and let $b$ denote the number of positive integers $n \leq 2005$ with $S(n)$ even. Find $|a-b|.$

Answer: 25


Given a cube \( ABCDA_1B_1C_1D_1 \) with edge length \( a \), a point \( M \) is taken on the line \( AA_1 \), and a point \( N \) is taken on the line \( BC \) such that the line \( MN \) intersects the edge \( C_1D_1 \). Find the minimum value of the length \(|MN|\).

Answer: 3a


Determine the largest value of \( x \) for which
\[ \left|x^{2}-4x-39601\right| \geq \left|x^{2}+4x-39601\right| \]

Answer: 199


Pyramid $OABCD$ has square base $ABCD,$ congruent edges $\overline{OA}, \overline{OB}, \overline{OC},$ and $\overline{OD},$ and $\angle AOB=45^\circ.$ Let $\theta$ be the measure of the dihedral angle formed by faces $OAB$ and $OBC.$ Given that $\cos \theta=m+\sqrt{n},$ where $m$ and $n$ are integers, find $m+n.$

Answer: 5


For an even integer positive integer $n$ Kevin has a tape of length $4 n$ with marks at $-2 n,-2 n+1, \ldots, 2 n-1,2 n$. He then randomly picks $n$ points in the set $-n,-n+1,-n+2, \ldots, n-1, n$, and places a stone on each of these points. We call a stone 'stuck' if it is on $2 n$ or $-2 n$, or either all the points to the right, or all the points to the left, all contain stones. Then, every minute, Kevin shifts the unstuck stones in the following manner: He picks an unstuck stone uniformly at random and then flips a fair coin. If the coin came up heads, he then moves that stone and every stone in the largest contiguous set containing that stone one point to the left. If the coin came up tails, he moves every stone in that set one point right instead. He repeats until all the stones are stuck. Let $p_{k}$ be the probability that at the end of the process there are exactly $k$ stones in the right half. Evaluate $$\frac{p_{n-1}-p_{n-2}+p_{n-3}-\ldots+p_{3}-p_{2}+p_{1}}{p_{n-1}+p_{n-2}+p_{n-3}+\ldots+p_{3}+p_{2}+p_{1}}$$ in terms of $n$.

Answer: \frac{1}{n-1}


The line $y = \frac{5}{3} x - \frac{17}{3}$ is to be parameterized using vectors.  Which of the following options are valid parameterizations?

(A) $\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 4 \\ 1 \end{pmatrix} + t \begin{pmatrix} -3 \\ -5 \end{pmatrix}$

(B) $\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 17 \\ 5 \end{pmatrix} + t \begin{pmatrix} 6 \\ 10 \end{pmatrix}$

(C) $\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 2 \\ -7/3 \end{pmatrix} + t \begin{pmatrix} 3/5 \\ 1 \end{pmatrix}$

(D) $\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 14/5 \\ -1 \end{pmatrix} + t \begin{pmatrix} 1 \\ 3/5 \end{pmatrix}$

(E) $\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 0 \\ -17/3 \end{pmatrix} + t \begin{pmatrix} 15 \\ -25 \end{pmatrix}$

Enter the letters of the correct options, separated by commas.

Answer: \text{A,C}


Let \( x_{1} = 1 \) and \( x_{n+1} = x_{n} + \left\lfloor \frac{x_{n}}{n} \right\rfloor + 2 \) for \( n = 1, 2, 3, \ldots \), where \( \lfloor x \rfloor \) denotes the largest integer not greater than \( x \). Determine \( x_{1997} \).

Answer: 23913


Doraemon and Nobita are playing the game "rock, paper, scissors." The rules state that the winner of each round receives two dorayakis, while the loser gets none. If there is a tie, each player receives one dorayaki. Nobita knows that Doraemon can only play "rock," but he still wants to share dorayakis with Doraemon. Therefore, he decides to play "scissors" once in every ten rounds and then play "rock" for the remaining rounds. After 20 rounds, all the dorayakis have been distributed, and Nobita has received 30 dorayakis. How many dorayakis did Doraemon receive?

Answer: 10


Triangle $ABC$ has $AB = 13, BC = 14$, and $AC = 15$. The points $D, E$, and $F$ are the midpoints of $\overline{AB}, \overline{BC}$, and $\overline{AC}$ respectively. Let $X \neq E$ be the intersection of the circumcircles of $\triangle BDE$ and $\triangle CEF$. What is $XA + XB + XC$?

Answer: \frac{195}{8}


Let rightangled  $\triangle ABC$  be given with right angle at vertex  $C$ . Let  $D$  be foot of altitude from  $C$  and let  $k$  be circle that touches  $BD$  at  $E$ ,  $CD$  at  $F$  and circumcircle of  $\triangle ABC$  at  $G$ .  $a.)$  Prove that points  $A$ ,  $F$  and  $G$  are collinear.  $b.)$  Express radius of circle  $k$  in terms of sides of  $\triangle ABC$ .

Answer: \frac{}{2}


Given six people are arranged in a row from left to right. Only A or B can be placed at the far left, and A cannot be placed at the far right, calculate the total number of different arrangements.

Answer: 216


A set of positive integers is said to be pilak if it can be partitioned into 2 disjoint subsets \(F\) and \(T\), each with at least 2 elements, such that the elements of \(F\) are consecutive Fibonacci numbers, and the elements of \(T\) are consecutive triangular numbers. Find all positive integers \(n\) such that the set containing all the positive divisors of \(n\) except \(n\) itself is pilak.

Answer: 30


Given an isosceles triangle \( A B C \) where \( A B = A C \) and \( \angle A B C = 53^\circ \). Point \( K \) is such that \( C \) is the midpoint of \( A K \). Point \( M \) is chosen such that:

- \( B \) and \( M \) are on the same side of the line \( A C \);
- \( K M = A B \);
- the angle \( \angle M A K \) is the largest possible.

What is the measure of the angle \( \angle B A M \) in degrees?

Answer: 44


For any three positive integers \( x, y, z \), let
\[
f(x, y, z) = [1+2+3+\cdots+(x+y-2)] - z .
\]

Find all positive integer tuples \( (a, b, c, d) \) such that
\[
f(a, b, c) = f(c, d, a) = 1993.

Answer: (23,42,23,42)


For which natural number \( n \) does the quantity \(\frac{n^2}{1.001^n}\) reach its maximum value?

Answer: 2001


Thirty-six hundredths is equal to
(A) 0.36
(B) 360
(C) 3.6
(D) 0.036
(E) 0.0036

Answer: 0.36


Two circular tracks $\alpha$ and $\beta$ of the same radius are tangent to each other. A car $A$ travels clockwise on track $\alpha$ and a car $B$ travels counterclockwise on track $\beta$. At the start, cars $A$ and $B$ are on the same line with the center of track $\alpha$, and this line is tangent to track $\beta$. After the start, the cars begin to approach the point of tangency of the tracks. Each car completes one full lap on its track in one hour (and never switches to the other track). For how much time during this hour will the distance between the cars be at least the diameter of each track?

Answer: 1/2


A leak formed in the hold of a ship. A pump was immediately switched on to remove the water, but it couldn't keep up, and after 10 minutes, the water level rose by 20 cm. Then, a second pump of equal power was turned on, and after 5 minutes, the water level dropped by 10 cm. The leak was then sealed.

How much time will it take for the pumps to remove the remaining water?

Answer: 1.25


Given a function defined on the set of positive integers as follows:
\[ f(n) = \begin{cases}
n - 3, & \text{if } n \geq 1000 \\
f[f(n + 7)], & \text{if } n < 1000
\end{cases} \]
Find the value of \( f(90) \).

Answer: 999


Let side $AD$ of convex quadrilateral $ABCD$ be extended through $D$, and let side $BC$ be extended through $C$, to meet in point $E.$ Let $S$ be the degree-sum of angles $CDE$ and $DCE$, and let $S'$ represent the degree-sum of angles $BAD$ and $ABC.$ If $r=S/S'$, then:

Answer: 1


Convert $115_{10}$ to base 11. Represent $10$ as $A$, if necessary.

Answer: \text{A5}_{11}


Denote $\mathbb{Z}_{>0}=\{1,2,3,...\}$ the set of all positive integers. Determine all functions $f:\mathbb{Z}_{>0}\rightarrow \mathbb{Z}_{>0}$ such that, for each positive integer $n$,
$\hspace{1cm}i) \sum_{k=1}^{n}f(k)$ is a perfect square, and
$\vspace{0.1cm}$
$\hspace{1cm}ii) f(n)$ divides $n^3$.

Answer: f(n) = n^3


Find the distance between the foci of the ellipse
\[\frac{x^2}{36} + \frac{y^2}{16} = 8.\]

Answer: 8\sqrt{10}


The percentage of seventh-grade students participating in the gymnastics section is between 2.9% and 3.1%. Determine the smallest possible number of students in this class.

Answer: 33


When the five numbers 10000, 1, 10, 100, and 1000 are arranged from largest to smallest, the middle number is
(A) 10000
(B) 1
(C) 10
(D) 100
(E) 1000

Answer: 100


If $a+b=1$, find the supremum of $$- \frac {1}{2a}- \frac {2}{b}.$$

Answer: - \frac {9}{2}


Let \( R \) be a point on the curve such that \( OMRN \) is a square. If \( r \) is the \( x \)-coordinate of \( R \), find the value of \( r \).

Answer: 1


Find all four-digit numbers that are 9 times larger than their reversed counterparts.

Answer: 9801


A circle is tangent to the extensions of two sides \(AB\) and \(AD\) of the square \(ABCD\), and the points of tangency cut off segments of length 4 cm from vertex \(A\). From point \(C\), two tangents are drawn to this circle. Find the side length of the square if the angle between the tangents is \(60^\circ\).

Answer: 4(\sqrt{2}-1)


In triangle \(ABC\), \(\angle ABC = 100^\circ\) and \(\angle ACB = 65^\circ\). On \(AB\) there is a point \(M\) such that \(\angle MCB = 55^\circ\), and on \(AC\) there is a point \(N\) such that \(\angle NBC = 80^\circ\). Find \(\angle NMC\).

Answer: 25


On side AC of triangle ABC with a 120-degree angle at vertex B, points D and E are marked such that AD = AB and CE = CB. A perpendicular DF is dropped from point D to line BE. Find the ratio BD / DF.

Answer: 2


In the 2013 Zhejiang College Entrance Examination, arrange the six letters A, B, C, D, E, F in a row, with both A and B on the same side of C. How many different arrangements are there? (Answer with a number.)

Answer: 480


All the squares of a board of  $(n+1)\times(n-1)$  squares
are painted with **three  colors** such that, for any two different
columns and any two different rows, the 4 squares in their
intersections they don't have all the same color. Find the
greatest possible value of  $n$ .

Answer: 5


All the complex roots of $(z + 1)^4 = 16z^4,$ when plotted in the complex plane, lie on a circle. Find the radius of this circle.

Answer: \frac{2}{3}


For the polynomial \(\left(x^{2}-x+1\right)^{100}\), find the sum of the coefficients of the even powers of \(x\).

Answer: \frac{1+3^{100}}{2}


Sierpinski's triangle is formed by taking a triangle, and drawing an upside down triangle inside each upright triangle that appears. A snake sees the fractal, but decides that the triangles need circles inside them. Therefore, she draws a circle inscribed in every upside down triangle she sees (assume that the snake can do an infinite amount of work). If the original triangle had side length  $1$ , what is the total area of all the individual circles?

*2015 CCA Math Bonanza Lightning Round #4.4*

Answer: \frac{\pi}{12}


When the integer  $ {\left(\sqrt{3} \plus{} 5\right)}^{103} \minus{} {\left(\sqrt{3} \minus{} 5\right)}^{103}$  is divided by 9, what is the remainder?

Answer: 1


The circles $\omega_{1}$ and $\omega_{2}$ with centers $O_{1}$ and $O_{2}$ respectively intersect at point $B$. The extension of segment $O_{2} B$ beyond point $B$ intersects circle $\omega_{1}$ at point $K$, while the extension of segment $O_{1} B$ beyond point $B$ intersects circle $\omega_{2}$ at point $L$. A line passing through point $B$ and parallel to $K L$ intersects circles $\omega_{1}$ and $\omega_{2}$ again at points $A$ and $C$ respectively. The rays $A K$ and $C L$ intersect at point $N$. Find the angle between lines $O_{1} N$ and $O_{2} B$.

Answer: 90


For $n$ a positive integer, let $R(n)$ be the sum of the remainders when $n$ is divided by $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, and $10$. For example, $R(15) = 1+0+3+0+3+1+7+6+5=26$. How many two-digit positive integers $n$ satisfy $R(n) = R(n+1)\,?$

Answer: 2


On a computer keyboard, the key for the digit 1 is not working. For example, if you try to type the number 1231234, only the number 23234 will actually print.

Sasha tried to type an 8-digit number, but only 202020 was printed. How many 8-digit numbers satisfy this condition?

Answer: 28


Let $E$ be the intersection point of the diagonals of a convex quadrilateral $ABCD$. Denote $F_{1}$, $F_{2}$, and $F_{3}$ as the areas of $\triangle ABE$, $\triangle CDE$, and the quadrilateral $ABCD$, respectively. Prove that $\sqrt{F_{1}} + \sqrt{F_{2}} \leq \sqrt{F}$ and determine when equality holds.

Answer: \sqrt{F_1}+\sqrt{F_2}\le\sqrt{F}


Given an ellipse with the equation $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ and an eccentricity of $\frac{1}{2}$. $F\_1$ and $F\_2$ are the left and right foci of the ellipse, respectively. A line $l$ passing through $F\_2$ intersects the ellipse at points $A$ and $B$. The perimeter of $\triangle F\_1AB$ is $8$.

(I) Find the equation of the ellipse.
(II) If the slope of line $l$ is $0$, and its perpendicular bisector intersects the $y$-axis at $Q$, find the range of the $y$-coordinate of $Q$.
(III) Determine if there exists a point $M(m, 0)$ on the $x$-axis such that the $x$-axis bisects $\angle AMB$. If it exists, find the value of $m$; otherwise, explain the reason.

Answer: m = 4


Let $a, b, c$ be the sides of triangle $\triangle ABC$, and let $S$ be its area. Prove that
\[a^{2} + b^{2} + c^{2} \geq 4 \sqrt{3} S + (a - b)^{2} + (b - c)^{2} + (c - a)^{2},\]
with equality if and only if $a = b = c$.

Answer: ^2+b^2+^2\ge4\sqrt{3}S+(b)^2+()^2+()^2


Let \(p\) and \(q\) be relatively prime positive integers such that \(\dfrac pq = \dfrac1{2^1} + \dfrac2{4^2} + \dfrac3{2^3} + \dfrac4{4^4} + \dfrac5{2^5} + \dfrac6{4^6} + \cdots\), where the numerators always increase by 1, and the denominators alternate between powers of 2 and 4, with exponents also increasing by 1 for each subsequent term. Compute \(p+q\).

Answer: 169


Given a positive integer \( n \), there are \( 3n \) numbers that satisfy:
\[
0 \leq a_{1} \leq a_{2} \leq \cdots \leq a_{3n},
\]
and
\[
\left(\sum_{i=1}^{3n} a_{i}\right)^{3} \geq k_{n}\left(\sum_{i=1}^{n} a_{i} a_{n+i} a_{2n+i}\right) \text{ always holds. }
\]

Find the best possible value of \( k_{n} \) (expressed in terms of \( n \)).

Answer: 27n^2


Find the largest integer  $ n$  satisfying the following conditions:
(i)  $ n^2$  can be expressed as the difference of two consecutive cubes;
(ii)  $ 2n\plus{}79$  is a perfect square.

Answer: 181


For each positive integer $n$ let $a_n$ be the least positive integer multiple of $23$ such that $a_n \equiv 1 \pmod{2^n}.$ Find the number of positive integers $n$ less than or equal to $1000$ that satisfy $a_n = a_{n+1}.$

Answer: 363


In triangle \( ABC \), median \( AM \) is drawn. Circle \(\omega\) passes through point \( A \), touches line \( BC \) at point \( M \), and intersects sides \( AB \) and \( AC \) at points \( D \) and \( E \) respectively. On arc \( AD \), not containing point \( E \), a point \( F \) is chosen such that \(\angle BFE = 72^{\circ}\). It turns out that \(\angle DEF = \angle ABC\). Find the angle \(\angle CME\).

Answer: 36


Let  $n$  be a positive integer,  $n\geq 2$ . For each  $t\in \mathbb{R}$ ,  $t\neq k\pi$ ,  $k\in\mathbb{Z}$ , we consider the numbers
\[ x_n(t) = \sum_{k=1}^n k(n-k)\cos{(tk)} \textrm{ and } y_n(t) =  \sum_{k=1}^n k(n-k)\sin{(tk)}. \]
Prove that if  $x_n(t) = y_n(t) =0$  if and only if   $\tan {\frac {nt}2} = n \tan {\frac t2}$ .

*Constantin Buse*

Answer: \tan(\frac{nt}{2})=n\tan(\frac{}{2})


Let  $\Omega$  be a unit circle and  $A$  be a point on  $\Omega$ . An angle  $0 < \theta < 180^\circ$  is chosen uniformly at random, and  $\Omega$  is rotated  $\theta$  degrees clockwise about  $A$ . What is the expected area swept by this rotation?

Answer: 2\pi


Given the function \( f(x) = x^2 + x + \sqrt{3} \), if for all positive numbers \( a, b, c \), the inequality \( f\left(\frac{a+b+c}{3} - \sqrt[3]{abc}\right) \geq f\left(\lambda \left(\frac{a+b}{2} - \sqrt{ab}\right)\right) \) always holds, find the maximum value of the positive number \( \lambda \).

Answer: \frac{2}{3}


Given that the random variable $X$ follows a normal distribution $N(2,\sigma^{2})$, and its normal distribution density curve is the graph of the function $f(x)$, and $\int_{0}^{2} f(x)dx=\dfrac{1}{3}$, calculate $P(x > 4)$.

Answer: \dfrac{1}{3}


In an acute-angled triangle \( ABC \), the angle bisector \( AN \), the altitude \( BH \), and the line perpendicular to side \( AB \) passing through its midpoint intersect at a single point. Find angle \( BAC \).

Answer: 60


$x$ is a positive rational number, and $(x)$ represents the number of prime numbers less than or equal to $x$. For instance, $(5)=3$, meaning there are three prime numbers (2, 3, and 5) less than or equal to 5. Thus, $(x)$ defines an operation on $x$. Find the value of $((20) \times (1) + (7))$.

Answer: 2


Person A and person B start from locations A and B respectively, walking towards each other at speeds of 65 meters per minute and 55 meters per minute simultaneously. They meet after 10 minutes. What is the distance between A and B in meters? Also, what is the distance of the meeting point from the midpoint between A and B in meters?

Answer: 50


A dormitory is installing a shower room for 100 students. How many shower heads are economical if the boiler preheating takes 3 minutes per shower head, and it also needs to be heated during the shower? Each group is allocated 12 minutes for showering.

Answer: 20


Given a positive integer \( n \) and a sequence of real numbers \( a_{1}, a_{2}, \cdots, a_{n} \) such that for each \( m \leq n \), it holds that \( \left|\sum_{k=1}^{m} \frac{a_{k}}{k}\right| \leq 1 \), find the maximum value of \( \left|\sum_{k=1}^{n} a_{k}\right| \).

Answer: 2n-1


Let \( S = \{ 1, 2, \cdots, 2005 \} \). Find the smallest number \( n \) such that in any subset of \( n \) pairwise coprime numbers from \( S \), there is at least one prime number.

Answer: 16


Let  $P_n$  be the number of permutations  $\pi$  of  $\{1,2,\dots,n\}$  such that \[|i-j|=1\text{ implies }|\pi(i)-\pi(j)|\le 2\] for all  $i,j$  in  $\{1,2,\dots,n\}.$  Show that for  $n\ge 2,$  the quantity \[P_{n+5}-P_{n+4}-P_{n+3}+P_n\] does not depend on  $n,$  and find its value.

Answer: 4


An isosceles triangle with a base \( a \) and a base angle \( \alpha \) is inscribed in a circle. Additionally, a second circle is constructed, which is tangent to both of the triangle's legs and the first circle. Find the radius of the second circle.

Answer: \frac{}{2\sin\alpha(1+\cos\alpha)}


A theater box office sells $2n$ tickets at 5 cents each, with each person limited to buying one ticket. Initially, the box office has no money, and among the first $2n$ people in line, half have 5 cent coins while the other half only have one dollar bills. How many different ways can these $2n$ ticket buyers line up so that the box office does not experience any difficulty in providing change?

Answer: \frac{1}{n+1}\binom{2n}{n}


A point \( C \) is taken on the segment \( AB \). A line passing through point \( C \) intersects the circles with diameters \( AC \) and \( BC \) at points \( K \) and \( L \), and also intersects the circle with diameter \( AB \) at points \( M \) and \( N \). Prove that \( KM = LN \).

Answer: KM=LN


Find the number of ordered quadruples of positive integers \((a, b, c, d)\) such that \(a, b, c\), and \(d\) are all (not necessarily distinct) factors of 30 and \(a b c d > 900\).

Answer: 1940


In a $3 \times 3$ grid (each cell is a $1 \times 1$ square), place two identical chess pieces, with at most one piece per cell. There are ___ different ways to arrange the pieces (if two arrangements can overlap by rotation, they are considered the same arrangement).

Answer: 10


Parallelogram  $ABCD$  is given such that  $\angle ABC$  equals  $30^o$  . Let  $X$  be the foot of the perpendicular from  $A$  onto  $BC$ , and  $Y$  the foot of the perpendicular from  $C$  to  $AB$ . If  $AX = 20$  and  $CY = 22$ , find the area of the parallelogram.

Answer: 880


Let \\(n\\) be a positive integer, and \\(f(n) = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n}\\). It is calculated that \\(f(2) = \frac{3}{2}\\), \\(f(4) > 2\\), \\(f(8) > \frac{5}{2}\\), and \\(f(16) > 3\\). Observing the results above, according to the pattern, it can be inferred that \\(f(128) > \_\_\_\_\_\_\_\_．

Answer: \frac{9}{2}


Given the function $f(x)=2\ln x - ax^2 + 3$,
(1) Discuss the monotonicity of the function $y=f(x)$;
(2) If there exist real numbers $m, n \in [1, 5]$ such that $f(m)=f(n)$ holds when $n-m \geq 2$, find the maximum value of the real number $a$.

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


Consider  $ \triangle ABC$  and points  $ M \in (AB)$ ,  $ N \in (BC)$ ,  $ P \in (CA)$ ,  $ R \in (MN)$ ,  $ S \in (NP)$ ,  $ T \in (PM)$  such that  $ \frac {AM}{MB} \equal{} \frac {BN}{NC} \equal{} \frac {CP}{PA} \equal{} k$  and  $ \frac {MR}{RN} \equal{} \frac {NS}{SP} \equal{} \frac {PT}{TN} \equal{} 1 \minus{} k$  for some  $ k \in (0, 1)$ . Prove that  $ \triangle STR \sim \triangle ABC$  and, furthermore, determine  $ k$  for which the minimum of  $ [STR]$  is attained.

Answer: \frac{1}{2}


Given the sample 7, 8, 9, x, y has an average of 8, and xy=60, then the standard deviation of this sample is \_\_\_\_\_\_.

Answer: \sqrt{2}


A pedestrian is moving in a straight line towards a crosswalk at a constant speed of 3.6 km/h. Initially, the distance from the pedestrian to the crosswalk is 40 meters. The length of the crosswalk is 6 meters. What distance from the crosswalk will the pedestrian be after two minutes?

Answer: 74


The figure shows a square in the interior of a regular hexagon.  The square and regular hexagon share a common side. What is the degree measure of $\angle ABC$? [asy]
size(150);
pair A, B, C, D, E, F, G, H;
A=(0,.866);
B=(.5,1.732);
C=(1.5,1.732);
D=(2,.866);
E=(1.5,0);
F=(.5,0);
G=(.5,1);
H=(1.5,1);
draw(A--B);
draw(B--C);
draw(C--D);
draw(D--E);
draw(E--F);
draw(F--A);
draw(F--G);
draw(G--H);
draw(H--E);
draw(D--H);
label("A", C, N);
label("B", D, E);
label("C", H, N);
[/asy]

Answer: 45


In a regular 1976-gon, the midpoints of all sides and the midpoints of all diagonals are marked. What is the maximum number of marked points that lie on one circle?

Answer: 1976


Given two points A and B on a number line, their distance is 2, and the distance between point A and the origin O is 3. Then, the sum of all possible distances between point B and the origin O equals to     .

Answer: 12


Circles \(\omega_{1}, \omega_{2},\) and \(\omega_{3}\) are centered at \(M, N,\) and \(O\), respectively. The points of tangency between \(\omega_{2}\) and \(\omega_{3}\), \(\omega_{3}\) and \(\omega_{1}\), and \(\omega_{1}\) and \(\omega_{2}\) are \(A, B,\) and \(C\), respectively. Line \(MO\) intersects \(\omega_{3}\) and \(\omega_{1}\) again at \(P\) and \(Q\), respectively, and line \(AP\) intersects \(\omega_{2}\) again at \(R\). Given that \(ABC\) is an equilateral triangle of side length 1, compute the area of \(PQR\).

Answer: 2\sqrt{3}


Find the smallest natural number \( N \) that is divisible by \( p \), ends with \( p \), and has a digit sum equal to \( p \), given that \( p \) is a prime number and \( 2p+1 \) is a cube of a natural number.

Answer: 11713


\( M \) is the midpoint of side \( BC \) of triangle \( ABC \). Let \( r_1 \) and \( r_2 \) be the radii of the circles inscribed in triangles \( ABM \) and \( ACM \), respectively. Prove that \( r_1 < 2r_2 \).

Answer: r_1<2r_2


Let  $p(x)$  and  $q(x)$  be two cubic polynomials such that  $p(0)=-24$ ,  $q(0)=30$ , and \[p(q(x))=q(p(x))\] for all real numbers  $x$ .  Find the ordered pair  $(p(3),q(6))$ .

Answer: (3,-24)


Alice starts with the number 0. She can apply 100 operations on her number. In each operation, she can either add 1 to her number or square her number. After applying all operations, her score is the minimum distance from her number to any perfect square. What is the maximum score she can attain?

Answer: 94


Let $\triangle{PQR}$ be a right triangle with $PQ = 90$, $PR = 120$, and $QR = 150$. Let $C_{1}$ be the inscribed circle. Construct $\overline{ST}$ with $S$ on $\overline{PR}$ and $T$ on $\overline{QR}$, such that $\overline{ST}$ is perpendicular to $\overline{PR}$ and tangent to $C_{1}$. Construct $\overline{UV}$ with $U$ on $\overline{PQ}$ and $V$ on $\overline{QR}$ such that $\overline{UV}$ is perpendicular to $\overline{PQ}$ and tangent to $C_{1}$. Let $C_{2}$ be the inscribed circle of $\triangle{RST}$ and $C_{3}$ the inscribed circle of $\triangle{QUV}$. The distance between the centers of $C_{2}$ and $C_{3}$ can be written as $\sqrt {10n}$. What is $n$?

Answer: 725


Given set $A = \{1, 2, 3\}$, and functions $f$ and $g$ are from set $A$ to set $A$. Find the number of function pairs $(f, g)$ such that the intersection of the images of $f$ and $g$ is empty.

Answer: 42


In the store "Third is not superfluous," there is a promotion: if a customer presents three items at the checkout, the cheapest one of them is free. Ivan wants to buy 11 items costing $100, 200, 300, ..., 1100 rubles. What is the minimum amount of money he can spend to buy these items?

Answer: 4800


Let  $ABC$  be an isosceles triangle with  $AB = AC = 4$  and  $BC = 5$ . Two circles centered at  $B$  and  $C$  each have radius  $2$ , and the line through the midpoint of  $\overline{BC}$  perpendicular to  $\overline{AC}$  intersects the two circles in four different points. If the greatest possible distance between any two of those four points can be expressed as  $\frac{\sqrt{a}+b\sqrt{c}}{d}$  for positive integers  $a$ ,  $b$ ,  $c$ , and  $d$  with gcd $(b, d) = 1$  and  $a$  and  $c$  each not divisible by the square of any prime, find  $a + b + c + d$ .

Answer: 451


From the vertex \( A \) of triangle \( ABC \), the perpendiculars \( AM \) and \( AN \) are dropped onto the bisectors of the external angles at vertices \( B \) and \( C \) respectively. Prove that the segment \( MN \) is equal to the semiperimeter of triangle \( ABC \).

Answer: \frac{}{2}


In a right square prism \( P-ABCD \) with side edges and base edges both equal to 4, determine the total length of all the curves formed on its surface by points that are 3 units away from vertex \( P \).

Answer: 6\pi


The 200-digit number \( M \) is composed of 200 ones. What is the sum of the digits of the product \( M \times 2013 \)?

Answer: 1200


Define  $f\left(n\right)=\textrm{LCM}\left(1,2,\ldots,n\right)$ . Determine the smallest positive integer  $a$  such that  $f\left(a\right)=f\left(a+2\right)$ .

*2017 CCA Math Bonanza Lightning Round #2.4*

Answer: 13


A unit cube has vertices $P_1,P_2,P_3,P_4,P_1',P_2',P_3',$ and $P_4'$. Vertices $P_2$, $P_3$, and $P_4$ are adjacent to $P_1$, and for $1\le i\le 4,$ vertices $P_i$ and $P_i'$ are opposite to each other. A regular octahedron has one vertex in each of the segments $\overline{P_1P_2}$, $\overline{P_1P_3}$, $\overline{P_1P_4}$, $\overline{P_1'P_2'}$, $\overline{P_1'P_3'}$, and $\overline{P_1'P_4'}$.  Find the side length of the octahedron.

[asy]
import three;

size(5cm);
triple eye = (-4, -8, 3);
currentprojection = perspective(eye);

triple[] P = {(1, -1, -1), (-1, -1, -1), (-1, 1, -1), (-1, -1, 1), (1, -1, -1)}; // P[0] = P[4] for convenience
triple[] Pp = {-P[0], -P[1], -P[2], -P[3], -P[4]};

// draw octahedron
triple pt(int k){ return (3*P[k] + P[1])/4; }
triple ptp(int k){ return (3*Pp[k] + Pp[1])/4; }
draw(pt(2)--pt(3)--pt(4)--cycle, gray(0.6));
draw(ptp(2)--pt(3)--ptp(4)--cycle, gray(0.6));
draw(ptp(2)--pt(4), gray(0.6));
draw(pt(2)--ptp(4), gray(0.6));
draw(pt(4)--ptp(3)--pt(2), gray(0.6) + linetype("4 4"));
draw(ptp(4)--ptp(3)--ptp(2), gray(0.6) + linetype("4 4"));

// draw cube
for(int i = 0; i < 4; ++i){
draw(P[1]--P[i]); draw(Pp[1]--Pp[i]);
for(int j = 0; j < 4; ++j){
if(i == 1 || j == 1 || i == j) continue;
draw(P[i]--Pp[j]); draw(Pp[i]--P[j]);
}
dot(P[i]); dot(Pp[i]);
dot(pt(i)); dot(ptp(i));
}

label("$P_1$", P[1], dir(P[1]));
label("$P_2$", P[2], dir(P[2]));
label("$P_3$", P[3], dir(-45));
label("$P_4$", P[4], dir(P[4]));
label("$P'_1$", Pp[1], dir(Pp[1]));
label("$P'_2$", Pp[2], dir(Pp[2]));
label("$P'_3$", Pp[3], dir(-100));
label("$P'_4$", Pp[4], dir(Pp[4]));
[/asy]

Answer: \frac{3 \sqrt{2}}{4}


Fill the numbers $1, 2, \cdots, 9$ into a $3 \times 3$ grid such that each cell contains one number, each row's numbers increase from left to right, and each column's numbers decrease from top to bottom. How many different ways are there to achieve this?
(A) 12
(B) 24
(C) 42
(D) 48

Answer: 42


Six chairs sit in a row.  Six people randomly seat themselves in the chairs. Each person randomly chooses either to set their feet on the floor, to cross their legs to the right, or to cross their legs to the left.  There is only a problem if two people sitting next to each other have the person on the right crossing their legs to the left and the person on the left crossing their legs to the right.  The probability that this will **not** happen is given by  $\frac{m}{n}$  where  $m$  and  $n$  are relatively prime positive integers. Find  $m+n$ .

Answer: 1106


In a two-story house that is inhabited in both floors as well as on the ground floor, 35 people live above someone and 45 people live below someone. One third of all the people living in the house live on the first floor.

How many people live in the house in total?

Answer: 60


A secret agent is trying to decipher a passcode. So far, he has obtained the following information:

- It is a four-digit number.
- It is not divisible by seven.
- The digit in the tens place is the sum of the digit in the units place and the digit in the hundreds place.
- The number formed by the first two digits of the code (in this order) is fifteen times the last digit of the code.
- The first and last digits of the code (in this order) form a prime number.

Does the agent have enough information to decipher the code? Justify your conclusion.

Answer: 4583


In Mr. Jacob's class, $12$ of the $20$ students received a 'B' on the latest exam. If the same proportion of students received a 'B' in Mrs. Cecilia's latest exam, and Mrs. Cecilia originally had $30$ students, but $6$ were absent during the exam, how many students present for Mrs. Cecilia’s exam received a 'B'?

Answer: 14


Let  $ABCD$  be a unit square. For any interior points  $M,N$  such that the line  $MN$  does not contain a vertex of the square, we denote by  $s(M,N)$  the least area of the triangles having their vertices in the set of points  $\{ A,B,C,D,M,N\}$ . Find the least number  $k$  such that  $s(M,N)\le k$ , for all points  $M,N$ .

*Dinu Șerbănescu*

Answer: 1/8


In the Cartesian coordinate system $(xOy)$, the parametric equations of curve $C_{1}$ are given by $\begin{cases}x=2t-1 \\ y=-4t-2\end{cases}$ $(t$ is the parameter$)$, and in the polar coordinate system with the coordinate origin $O$ as the pole and the positive half of the $x$-axis as the polar axis, the polar equation of curve $C_{2}$ is $\rho= \frac{2}{1-\cos \theta}$.

(1) Write the Cartesian equation of curve $C_{2}$;

(2) Let $M_{1}$ be a point on curve $C_{1}$, and $M_{2}$ be a point on curve $C_{2}$. Find the minimum value of $|M_{1}M_{2}|$.

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


Shanille O'Keal shoots free throws on a basketball court. She hits the first and misses the second, and thereafter the probability that she hits the next shot is equal to the proportion of shots she has hit so far. What is the probability she hits exactly  $50$  of her first  $100$  shots?

Answer: \frac{1}{99}


There exist constants $b_1, b_2, b_3, b_4, b_5, b_6, b_7$ such that
\[
\cos^7 \theta = b_1 \cos \theta + b_2 \cos 2 \theta + b_3 \cos 3 \theta + b_4 \cos 4 \theta + b_5 \cos 5 \theta + b_6 \cos 6 \theta + b_7 \cos 7 \theta
\]
for all angles $\theta$. Find $b_1^2 + b_2^2 + b_3^2 + b_4^2 + b_5^2 + b_6^2 + b_7^2$.

Answer: \frac{429}{1024}


The sequence $\mathrm{Az}\left(a_{n}\right)$ is defined as follows:

$$
a_{0}=a_{1}=1, \quad a_{n+1}=a_{n}+\frac{1}{a_{n-1}}
$$

Show that $a_{n} \geq \sqrt{n}$.

Answer: a_n\ge\sqrt{n}


Let $\omega$ be a circle, and let $ABCD$ be a quadrilateral inscribed in $\omega$. Suppose that $BD$ and $AC$ intersect at a point $E$. The tangent to $\omega$ at $B$ meets line $AC$ at a point $F$, so that $C$ lies between $E$ and $F$. Given that $AE=6, EC=4, BE=2$, and $BF=12$, find $DA$.

Answer: 2 \sqrt{42}


Find the largest integer $n$ satisfying the following conditions:
(i) $n^2$ can be expressed as the difference of two consecutive cubes;
(ii) $2n + 79$ is a perfect square.

Answer: 181


Given a square with four vertices and its center, find the probability that the distance between any two of these five points is less than the side length of the square.

Answer: \frac{2}{5}


Suppose that $(a_1,b_1),$ $(a_2,b_2),$ $\dots,$ $(a_{100},b_{100})$ are distinct ordered pairs of nonnegative integers. Let $N$ denote the number of pairs of integers $(i,j)$ satisfying $1\leq i<j\leq 100$ and $|a_ib_j-a_jb_i|=1$. Determine the largest possible value of $N$ over all possible choices of the $100$ ordered pairs.

Answer: 197


Compute $\arccos(\cos 9).$ All functions are in radians.

Answer: 9 - 2\pi


In the diagram, there are more than three triangles. If each triangle has the same probability of being selected, what is the probability that a selected triangle has all or part of its interior shaded? Express your answer as a common fraction.

[asy]
draw((0,0)--(1,0)--(0,1)--(0,0)--cycle,linewidth(1));
draw((0,0)--(.5,0)--(.5,.5)--(0,0)--cycle,linewidth(1));

label("A",(0,1),NW);
label("B",(.5,.5),NE);
label("C",(1,0),SE);
label("D",(.5,0),S);
label("E",(0,0),SW);

filldraw((.5,0)--(1,0)--(.5,.5)--(.5,0)--cycle,gray,black);[/asy]

Answer: \frac{3}{5}


Three people are sitting in a row of eight seats. If there must be empty seats on both sides of each person, then the number of different seating arrangements is.

Answer: 24


The vertex of the parabola $y = x^2 - 8x + c$ will be a point on the $x$-axis if the value of $c$ is:
$\textbf{(A)}\ - 16 \qquad  \textbf{(B) }\ - 4 \qquad  \textbf{(C) }\ 4 \qquad  \textbf{(D) }\ 8 \qquad  \textbf{(E) }\ 16$

Answer: \textbf{(E)}\16


Show that if the numbers \( a_{1}, a_{2}, \ldots \) are not all zeros and satisfy the relation \( a_{n+2} = \left|a_{n+1}\right| - a_{n} \), then from some point on they are periodic and the smallest period is 9.

Answer: 9


On the clock tower of the train station, there is an electronic timing clock. On the boundary of the circular clock face, there is a small color light at each minute mark. At 9:37:20 PM, there are ____ small color lights within the acute angle formed between the minute hand and the hour hand.

Answer: 11


Triangle $ABC$ has sides $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$ of length 43, 13, and 48, respectively. Let $\omega$ be the circle circumscribed around $\triangle ABC$ and let $D$ be the intersection of $\omega$ and the perpendicular bisector of $\overline{AC}$ that is not on the same side of $\overline{AC}$ as $B$. The length of $\overline{AD}$ can be expressed as $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find the greatest integer less than or equal to $m + \sqrt{n}$.

Answer: 12


For which real \( k \) do we have \( \cosh x \leq \exp(k x^2) \) for all real \( x \)?

Answer: \frac{1}{2}


If positive numbers \(a, b, c\) and constant \(k\) satisfy the inequality \(\frac{k a b c}{a+b+c} \leq (a+b)^2 + (a+b+4c)^2\), find the maximum value of the constant \(k\).

Answer: 100


Given six balls numbered 1, 2, 3, 4, 5, 6 and boxes A, B, C, D, each to be filled with one ball, with the conditions that ball 2 cannot be placed in box B and ball 4 cannot be placed in box D, determine the number of different ways to place the balls into the boxes.

Answer: 252


For every odd number $p>1$ we have:
$\textbf{(A)}\ (p-1)^{\frac{1}{2}(p-1)}-1 \; \text{is divisible by} \; p-2\qquad \textbf{(B)}\ (p-1)^{\frac{1}{2}(p-1)}+1 \; \text{is divisible by} \; p\\ \textbf{(C)}\ (p-1)^{\frac{1}{2}(p-1)} \; \text{is divisible by} \; p\qquad \textbf{(D)}\ (p-1)^{\frac{1}{2}(p-1)}+1 \; \text{is divisible by} \; p+1\\ \textbf{(E)}\ (p-1)^{\frac{1}{2}(p-1)}-1 \; \text{is divisible by} \; p-1$

Answer: \textbf{(A)}


In a tournament, any two players play against each other. Each player earns one point for a victory, 1/2 for a draw, and 0 points for a loss. Let \( S \) be the set of the 10 lowest scores. We know that each player obtained half of their score by playing against players in \( S \).

a) What is the sum of the scores of the players in \( S \)?

b) Determine how many participants are in the tournament.

Note: Each player plays only once with each opponent.

Answer: 25


Given that the total amount of money originally owned by Moe, Loki, and Nick was $72, and each of Loki, Moe, and Nick gave Ott$\, 4, determine the fractional part of the group's money that Ott now has.

Answer: \frac{1}{6}


In the Cartesian coordinate system $(xOy)$, the equation of circle $C$ is $((x-4)^{2}+y^{2}=1)$. If there exists at least one point on the line $y=kx-3$ such that a circle with this point as the center and $2$ as the radius intersects with circle $C$, then the maximum value of $k$ is _____.

Answer: \frac{24}{7}


Two circles touch in  $M$ , and lie inside a rectangle  $ABCD$ . One of them touches the sides  $AB$  and  $AD$ , and the other one touches  $AD,BC,CD$ . The radius of the second circle is four times that of the first circle. Find the ratio in which the common tangent of the circles in  $M$  divides  $AB$  and  $CD$ .

Answer: 1:1


Two types of anti-inflammatory drugs must be selected from $X_{1}$, $X_{2}$, $X_{3}$, $X_{4}$, $X_{5}$, with the restriction that $X_{1}$ and $X_{2}$ must be used together, and one type of antipyretic drug must be selected from $T_{1}$, $T_{2}$, $T_{3}$, $T_{4}$, with the further restriction that $X_{3}$ and $T_{4}$ cannot be used at the same time. Calculate the number of different test schemes.

Answer: 14


A semicircle has diameter  $XY$ . A square  $PQRS$  with side length 12 is inscribed in the semicircle with  $P$  and  $S$  on the diameter. Square  $STUV$  has  $T$  on  $RS$ ,  $U$  on the semicircle, and  $V$  on  $XY$ . What is the area of  $STUV$ ?

Answer: 36


Prove that the reciprocal of the real root of the equation \( x^3 - x + 1 = 0 \) satisfies the equation \( x^5 + x + 1 = 0 \). Find these roots to one decimal place of accuracy.

Answer: -0.8


Prove that the feet of the perpendiculars dropped from the intersection point of the diagonals of a cyclic quadrilateral to its sides form a quadrilateral into which a circle can be inscribed. Find the radius of this circle if the diagonals of the given cyclic quadrilateral are perpendicular, the radius of the circumscribed circle is $R$, and the distance from its center to the intersection point of the diagonals is $d$.

Answer: \frac{R^2-^2}{2R}


In a competition there are  $18$  teams and in each round  $18$  teams are divided into  $9$  pairs where the  $9$  matches are played coincidentally. There are  $17$  rounds, so that each pair of teams play each other exactly once. After  $n$  rounds, there always exists  $4$  teams such that there was exactly one match played between these teams in those  $n$  rounds. Find the maximum value of  $n$ .

Answer: 7


Given $2n$ people, some of whom are friends with each other. In this group, each person has exactly $k(k \geqslant 1)$ friends. Find the positive integer $k$ such that the $2n$ people can always be divided into two groups of $n$ people each, and every person in each group has at least one friend in their own group.

Answer: 2


What is the largest possible number of subsets of the set \(\{1, 2, \ldots, 2n + 1\}\) such that the intersection of any two subsets consists of one or several consecutive integers?

Answer: (n+1)^2


Let \( k>1 \) be a real number. Calculate:

(a) \( L = \lim _{n \rightarrow \infty} \int_{0}^{1}\left(\frac{k}{\sqrt[n]{x} + k - 1}\right)^{n} \, \mathrm{d}x \).

(b) \( \lim _{n \rightarrow \infty} n\left[L - \int_{0}^{1}\left(\frac{k}{\sqrt[n]{x} + k - 1}\right)^{n} \, \mathrm{d}x\right] \).

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


Suppose  $\overline{AB}$  is a segment of unit length in the plane. Let  $f(X)$  and  $g(X)$  be functions of the plane such that  $f$  corresponds to rotation about  $A$   $60^\circ$  counterclockwise and  $g$  corresponds to rotation about  $B$   $90^\circ$  clockwise. Let  $P$  be a point with  $g(f(P))=P$ ; what is the sum of all possible distances from  $P$  to line  $AB$ ?

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


A function  $f$  defined on integers such that $f (n) =n + 3$  if  $n$  is odd $f (n) = \frac{n}{2}$  if  $n$  is even

If  $k$  is an odd integer, determine the values for which  $f (f (f (k))) = k$ .

Answer: 1


An ant crawls along the edges of a cube with side length 1 unit. Starting from one of the vertices, in each minute the ant travels from one vertex to an adjacent vertex. After crawling for 7 minutes, the ant is at a distance of \(\sqrt{3}\) units from the starting point. Find the number of possible routes the ant has taken.

Answer: 546


In the diagram shown, $\overrightarrow{OA}\perp\overrightarrow{OC}$ and $\overrightarrow{OB}\perp\overrightarrow{OD}$. If $\angle{AOD}$ is 3.5 times $\angle{BOC}$, what is $\angle{AOD}$? [asy]
unitsize(1.5cm);
defaultpen(linewidth(.7pt)+fontsize(10pt));
dotfactor=4;

pair O=(0,0), A=dir(0), B=dir(50), C=dir(90), D=dir(140);
pair[] dots={O,A,B,C,D};

dot(dots);
draw(O--1.2*D,EndArrow(4));
draw(O--1.2*B,EndArrow(4));
draw(O--1.2*C,EndArrow(4));
draw(O--1.2*A,EndArrow(4));

label("$D$",D,SW);
label("$C$",C,W);
label("$B$",B,E);
label("$A$",A,N);
label("$O$",O,S);
[/asy]

Answer: 140\text{ degrees}


In a class of  $ n\geq 4$  some students are friends. In this class any  $ n \minus{} 1$  students can be seated in a round table such that every student is sitting next to a friend of him in both sides, but  $ n$  students can not be seated in that way. Prove that the minimum value of  $ n$  is  $ 10$ .

Answer: 10


Given an angle and a fixed point \( P \) on the angle bisector. The line \( e_{1} \) passing through \( P \) intersects the sides of the angle at segment lengths \( a_{1} \) and \( b_{1} \), while the line \( e_{2} \) passing through \( P \) intersects the sides of the angle at segment lengths \( a_{2} \) and \( b_{2} \). Prove that

$$
\frac{1}{a_{1}}+\frac{1}{b_{1}}=\frac{1}{a_{2}}+\frac{1}{b_{2}}
$$

Answer: \frac{1}{a_{1}}+\frac{1}{b_{1}}=\frac{1}{a_{2}}+\frac{1}{b_{2}}


In the following diagram, \(ABCD\) is a square, \(BD \parallel CE\) and \(BE = BD\). Let \(\angle E = x^{\circ}\). Find \(x\).

Answer: 30


Given an isosceles trapezoid $\mathrm{ABCE}$ with bases $\mathrm{BC}$ and $\mathrm{AE}$, where $\mathrm{BC}$ is smaller than $\mathrm{AE}$ with lengths 3 and 4 respectively. The smaller lateral side $\mathrm{AB}$ is equal to $\mathrm{BC}$. Point $\mathrm{D}$ lies on $\mathrm{AE}$ such that $\mathrm{AD}$ : $\mathrm{DE}=3:1$. Point $\mathrm{F}$ lies on $\mathrm{AD}$ such that $\mathrm{AF}$ : $\mathrm{FD}=2:1$. Point $\mathrm{G}$ lies on $\mathrm{BD}$ such that $\mathrm{BG}$ : $\mathrm{GD}=1:2$. Determine the angle measure $\angle \mathrm{CFG}$.

Answer: 45


Let \( a, b, c \) be positive real numbers such that
\[
\begin{cases}
a^{2} + ab + b^{2} = 25 \\
b^{2} + bc + c^{2} = 49 \\
c^{2} + ca + a^{2} = 64
\end{cases}
\]

Find \( (a+b+c)^{2} \).

Answer: 129


Determine the minimum of the expression

$$
\frac{2}{|a-b|}+\frac{2}{|b-c|}+\frac{2}{|c-a|}+\frac{5}{\sqrt{ab+bc+ca}}
$$

under the conditions that \(ab + bc + ca > 0\), \(a + b + c = 1\), and \(a, b, c\) are distinct.

Answer: 10\sqrt{6}


Find the largest real number \( m \) such that for all positive numbers \( a, b, \) and \( c \) satisfying \( a + b + c = 1 \),
$$
10\left(a^{3}+b^{3}+c^{3}\right)-m\left(a^{5}+b^{5}+c^{5}\right) \geqslant 1.
$$

Answer: 9


For every integer  $n > 1$ , the sequence  $\left( {{S}_{n}} \right)$  is defined by  ${{S}_{n}}=\left\lfloor {{2}^{n}}\underbrace{\sqrt{2+\sqrt{2+...+\sqrt{2}}}}_{n\ radicals} \right\rfloor $ where   $\left\lfloor x \right\rfloor$   denotes the floor function of  $x$ . Prove that  ${{S}_{2001}}=2\,{{S}_{2000}}+1$ .
 .

Answer: S_{2001}=2S_{2000}+1


On  $5\times 5$  squares, we cover the area with several S-Tetrominos (=Z-Tetrominos) along the square so that in every square, there are two or fewer tiles covering that (tiles can be overlap). Find the maximum possible number of squares covered by at least one tile.

Answer: 24


The locus of the centers of all circles of given radius $a$, in the same plane, passing through a fixed point, is:
$\textbf{(A) }\text{a point}\qquad \textbf{(B) }\text{a straight line}\qquad \textbf{(C) }\text{two straight lines}\qquad \textbf{(D) }\text{a circle}\qquad  \textbf{(E) }\text{two circles}$

Answer: \textbf{(D)}


A farmer sold domestic rabbits. By the end of the market, he sold exactly one-tenth as many rabbits as the price per rabbit in forints. He then distributed the revenue between his two sons. Starting with the older son, the boys alternately received one-hundred forint bills, but at the end, the younger son received only a few ten-forint bills. The father then gave him his pocket knife and said that this made their shares equal in value. How much was the pocket knife worth?

Answer: 40


A jobber buys an article at $$24$ less $12\frac{1}{2}\%$. He then wishes to sell the article at a gain of $33\frac{1}{3}\%$ of his cost
after allowing a $20\%$ discount on his marked price. At what price, in dollars, should the article be marked?
$\textbf{(A)}\ 25.20 \qquad \textbf{(B)}\ 30.00 \qquad \textbf{(C)}\ 33.60 \qquad \textbf{(D)}\ 40.00 \qquad \textbf{(E)}\ \text{none of these}$

Answer: \textbf{(E)}\


Let \( O \) be the circumcenter of triangle \( ABC \), and let us draw the reflections of \( O \) with respect to the sides of the triangle. Show that the resulting triangle \( A_1 B_1 C_1 \) is congruent to triangle \( ABC \). Determine the orthocenter of triangle \( A_1 B_1 C_1 \).

Answer: O


Denote by \( f(n) \) the integer obtained by reversing the digits of a positive integer \( n \). Find the greatest integer that is certain to divide \( n^{4} - f(n)^{4} \) regardless of the choice of \( n \).

Answer: 99


Determine the number of pairs of integers, \((a, b)\), with \(1 \leq a \leq 100\) so that the line with equation \(b=ax-4y\) passes through point \((r, 0)\), where \(r\) is a real number with \(0 \leq r \leq 3\), and passes through point \((s, 4)\), where \(s\) is a real number with \(2 \leq s \leq 4\).

Answer: 6595


How many integers  $n>1$  are there such that  $n$  divides  $x^{13}-x$  for every positive integer  $x$ ?

Answer: 31


Two circles of radius $r$ are externally tangent to each other and internally tangent to the ellipse $x^2 + 4y^2 = 5$. Find $r$.

Answer: \frac{\sqrt{15}}{4}


Given that the sequence $\{a\_n\}$ is an arithmetic sequence with all non-zero terms, $S\_n$ denotes the sum of its first $n$ terms, and satisfies $a\_n^2 = S\_{2n-1}$ for all positive integers $n$. If the inequality $\frac{λ}{a\_{n+1}} \leqslant \frac{n + 8 \cdot (-1)^n}{2n}$ holds true for any positive integer $n$, determine the maximum value of the real number $λ$.

Answer: -\frac{21}{2}


Given the medians $A A_{1}, B B_{1}$, and $C C_{1}$ of triangle $A B C$, triangle $K M N$ is formed. From the medians $K K_{1}, M M_{1}$, and $N N_{1}$ of triangle $K M N$, triangle $P Q R$ is formed. Prove that the third triangle is similar to the first one and find the similarity ratio.

Answer: \frac{3}{4}


In a convex quadrilateral \(ABCD\) with internal angles \(<180^\circ\), point \(E\) is the intersection of the diagonals. Let \(F_1\) and \(F_2\) be the areas of triangles \(\triangle ABE\) and \(\triangle CDE\) respectively, and let \(F\) be the area of quadrilateral \(ABCD\). Prove that \(\sqrt{F_1} + \sqrt{F_2} \leq \sqrt{F}\). In what case is equality possible? (6 points)

Answer: \sqrt{F_1}+\sqrt{F_2}\le\sqrt{F}


Find a nonzero monic polynomial $P(x)$ with integer coefficients and minimal degree such that $P(1-\sqrt[3]{2}+\sqrt[3]{4})=0$. (A polynomial is called monic if its leading coefficient is 1.)

Answer: x^{3}-3x^{2}+9x-9


A positive integer is said to be "nefelibata" if, upon taking its last digit and placing it as the first digit, keeping the order of all the remaining digits intact (for example, 312 -> 231), the resulting number is exactly double the original number. Find the smallest possible nefelibata number.

Answer: 105263157894736842


There are 50,000 employees at an enterprise. For each employee, the sum of the number of their direct supervisors and their direct subordinates is 7. On Monday, each employee issues an order and gives a copy of this order to each of their direct subordinates (if any). Then, every day, each employee takes all orders received the previous day and either distributes copies of them to all their direct subordinates or, if they do not have any subordinates, executes the orders themselves. It turns out that by Friday, no orders are being distributed within the institution. Prove that there are at least 97 supervisors who do not have any higher supervisors.

Answer: 97


Circle $C$ has its center at $C(5, 5)$ and has a radius of 3 units. Circle $D$ has its center at $D(14, 5)$ and has a radius of 3 units. What is the area of the gray region bound by the circles and the $x$-axis?
```asy
import olympiad; size(150); defaultpen(linewidth(0.8));
xaxis(0,18,Ticks("%",1.0));
yaxis(0,9,Ticks("%",1.0));
fill((5,5)--(14,5)--(14,0)--(5,0)--cycle,gray(0.7));
filldraw(circle((5,5),3),fillpen=white);
filldraw(circle((14,5),3),fillpen=white);
dot("$C$",(5,5),S); dot("$D$",(14,5),S);
```

Answer: 45 - \frac{9\pi}{2}


A repunit is a positive integer, all of whose digits are 1s. Let $a_{1}<a_{2}<a_{3}<\ldots$ be a list of all the positive integers that can be expressed as the sum of distinct repunits. Compute $a_{111}$.

Answer: 1223456


Given that $b$ is an odd multiple of 9, find the greatest common divisor of $8b^2 + 81b + 289$ and $4b + 17$.

Answer: 17


On graph paper, a square measuring $11 \times 11$ cells is drawn. It is required to mark the centers of some cells in such a way that the center of any other cell lies on the segment connecting any two marked points that are on the same vertical or horizontal line. What is the minimum number of cells that need to be marked?

Answer: 22


Find a polynomial $ p\left(x\right)$ with real coefficients such that

$ \left(x\plus{}10\right)p\left(2x\right)\equal{}\left(8x\minus{}32\right)p\left(x\plus{}6\right)$

for all real $ x$ and $ p\left(1\right)\equal{}210$.

Answer: 2(x + 4)(x - 4)(x - 8)


The  $n^{th}$  derivative of a function  $f(x)$  (if it exists) is denoted by  $f^{(n)}(x) $ . Let  $f(x)=\frac{e^x}{x}$ .
Suppose  $f$  is differentiable infinitely many times in  $(0,\infty) $ . Then find  $\lim_{n \to \infty}\frac{f^{(2n)}1}{(2n)!}$

Answer: 1


Given the function $$f(x)=\cos 2x + 2\sqrt{3}\sin x\cos x$$
(1) Find the range of the function $f(x)$ and write down the intervals where $f(x)$ is monotonically increasing;
(2) If $$0 < \theta < \frac{\pi}{6}$$ and $$f(\theta) = \frac{4}{3}$$, calculate the value of $\cos 2\theta$.

Answer: \frac{\sqrt{15} + 2}{6}


On a line, 2022 points are marked such that every two neighboring points are at an equal distance from each other. Half of the points are painted red, and the other half are painted blue. Can the sum of the lengths of all possible segments where the left end is red and the right end is blue be equal to the sum of the lengths of all segments where the left end is blue and the right end is red? (The endpoints of the segments under consideration do not have to be neighboring marked points.)

Answer: No


In the right-angled triangle $ABC$, the midpoint of the leg $CB$ is connected to the incenter of the triangle. This line intersects the leg $CA$ at point $D$. Express the ratio $AD:DC$ as a function of the angle at vertex $A$.

Answer: \tan(\frac{\alpha}{2})


Integers $x$ and $y$ with $x>y>0$ satisfy $x+y+xy=104$. What is $x$?

Answer: 34


Janne buys a camera which costs $200.00 without tax. If she pays 15% tax on this purchase, how much tax does she pay?
(A) $30.00
(B) $18.00
(C) $20.00
(D) $15.00
(E) $45.00

Answer: 30


Let $S=\{1,2, \ldots 2016\}$, and let $f$ be a randomly chosen bijection from $S$ to itself. Let $n$ be the smallest positive integer such that $f^{(n)}(1)=1$, where $f^{(i)}(x)=f\left(f^{(i-1)}(x)\right)$. What is the expected value of $n$?

Answer: \frac{2017}{2}


Define a monic irreducible polynomial with integral coefficients to be a polynomial with leading coefficient 1 that cannot be factored, and the prime factorization of a polynomial with leading coefficient 1 as the factorization into monic irreducible polynomials. How many not necessarily distinct monic irreducible polynomials are there in the prime factorization of $\left(x^{8}+x^{4}+1\right)\left(x^{8}+x+1\right)$ (for instance, $(x+1)^{2}$ has two prime factors)?

Answer: 5


Compute the value of the infinite series
$$
\sum_{n=2}^{\infty} \frac{n^{4}+3 n^{2}+10 n+10}{2^{n} \cdot \left(n^{4}+4\right)}
$$

Answer: \frac{11}{10}


Given an ellipse $M$ with its axes of symmetry being the coordinate axes, and its eccentricity is $\frac{\sqrt{2}}{2}$, and one of its foci is at $(\sqrt{2}, 0)$.

$(1)$ Find the equation of the ellipse $M$;
$(2)$ Suppose a line $l$ intersects the ellipse $M$ at points $A$ and $B$, and a parallelogram $OAPB$ is formed with $OA$ and $OB$ as adjacent sides, where point $P$ is on the ellipse $M$ and $O$ is the origin. Find the minimum distance from point $O$ to line $l$.

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


Answer: 6


Find the smallest positive number $\lambda$ such that for any triangle with side lengths $a, b, c$, given $a \geqslant \frac{b+c}{3}$, it holds that
$$
a c + b c - c^{2} \leqslant \lambda\left(a^{2} + b^{2} + 3 c^{2} + 2 a b - 4 b c\right).
$$

Answer: \frac{2\sqrt{2} + 1}{7}


Given the function $f(x)=2\sin (\\omega x)$, where $\\omega > 0$.

(1) When $ \\omega =1$, determine the even-odd property of the function $F(x)=f(x)+f(x+\\dfrac{\\mathrm{ }\\!\\!\\pi\\!\\!{ }}{2})$ and explain the reason.

(2) When $ \\omega =2$, the graph of the function $y=f(x)$ is translated to the left by $ \\dfrac{\\mathrm{ }\\!\\!\\pi\\!\\!{ }}{6}$ unit, and then translated upward by 1 unit to obtain the graph of the function $y=g(x)$. Find all possible values of the number of zeros of $y=g(x)$ in the interval $[a,a+10π]$ for any $a∈R$.

Answer: 20


Given vectors $\overrightarrow{a}=(1, \sqrt {3})$ and $\overrightarrow{b}=(-2,2 \sqrt {3})$, calculate the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$.

Answer: \dfrac {\pi}{3}


Let \( a_{1}, a_{2}, \ldots \) be an arithmetic sequence and \( b_{1}, b_{2}, \ldots \) be a geometric sequence. Suppose that \( a_{1} b_{1}=20 \), \( a_{2} b_{2}=19 \), and \( a_{3} b_{3}=14 \). Find the greatest possible value of \( a_{4} b_{4} \).

Answer: \frac{37}{4}


Calculate the lengths of the arcs of the curves given by the equations in polar coordinates.

$$
\rho = 8(1 - \cos \varphi), \quad -\frac{2 \pi}{3} \leq \varphi \leq 0
$$

Answer: 16


There is a table of numbers with 20 rows and 15 columns. Let \( A_{1}, \ldots, A_{20} \) be the sums of the numbers in each row, and \( B_{1}, \ldots, B_{15} \) be the sums of the numbers in each column.

a) Is it possible that \( A_{1}=\cdots=A_{20}=B_{1}=\cdots=B_{15} \)?

b) If the equalities in part (a) hold, what is the value of the sum \( A_{1} + \cdots + A_{20} + B_{1} + \cdots + B_{15} \)?

Answer: 0


Answer: -8388607.99609375


Let  $ABCD$  be an inscribed trapezoid such that the sides  $[AB]$  and  $[CD]$  are parallel. If  $m(\widehat{AOD})=60^\circ$  and the altitude of the trapezoid is  $10$ , what is the area of the trapezoid?

Answer: 100\sqrt{3}


For what single digit $n$ does 91 divide the 9-digit number $12345 n 789$?

Answer: 7


Given the hyperbola $C\_1$: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > b > 0)$ with left and right foci $F\_1$ and $F\_2$, respectively, and hyperbola $C\_2$: $\frac{x^2}{16} - \frac{y^2}{4} = 1$, determine the length of the major axis of hyperbola $C\_1$ given that point $M$ lies on one of the asymptotes of hyperbola $C\_1$, $OM \perp MF\_2$, and the area of $\triangle OMF\_2$ is $16$.

Answer: 16


Two congruent squares, $ABCD$ and $JKLM$, each have side lengths of 12 units. Square $JKLM$ is placed such that its center coincides with vertex $C$ of square $ABCD$. Determine the area of the region covered by these two squares in the plane.

Answer: 216


Each person in their heart silently remembers two non-zero numbers. Calculate the square of the sum of these two numbers, and record the result as "Sum". Calculate the square of the difference of these two numbers, and record the result as "Diff". Then calculate the product of these two numbers, and record the result as "Prod." Use "Sum", "Diff", and "Prod" to compute the following expression:
\[
\left(\frac{\text{Sum} - \text{Diff}}{\text{Prod}}\right)^{2} = ?
\]

Answer: 16


Yesterday Han drove 1 hour longer than Ian at an average speed 5 miles per hour faster than Ian. Jan drove 2 hours longer than Ian at an average speed 10 miles per hour faster than Ian. Han drove 70 miles more than Ian. How many more miles did Jan drive than Ian?
$\mathrm{(A)}\ 120\qquad\mathrm{(B)}\ 130\qquad\mathrm{(C)}\ 140\qquad\mathrm{(D)}\ 150\qquad\mathrm{(E)}\ 160$

Answer: \mathrm{(D)}


In the quadrilateral \(ABCD\), \(AB = 1\), \(BC = 2\), \(CD = \sqrt{3}\), \(\angle ABC = 120^\circ\), and \(\angle BCD = 90^\circ\). What is the exact length of side \(AD\)?

Answer: \sqrt{7}


Answer: 4


Consider polynomial functions  $ax^2 -bx +c$  with integer coefficients which have two distinct zeros in the open interval  $(0,1).$  Exhibit with proof the least positive integer value of  $a$  for which such a polynomial exists.

Answer: 5


Let \( M = \{1, 2, \cdots, 65\} \) and \( A \subseteq M \) be a subset. If \( |A| = 33 \) and there exist \( x, y \in A \) with \( x < y \) such that \( x \mid y \), then \( A \) is called a "good set". Find the largest \( a \in M \) such that every 33-element subset containing \( a \) is a good set.

Answer: 21


The ratio of the volume of a cone to the volume of a sphere inscribed in it is $k$. Find the angle between the slant height and the base of the cone and the permissible values of $k$.

Answer: k\ge2


Compute the number of tuples \(\left(a_{0}, a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\right)\) of (not necessarily positive) integers such that \(a_{i} \leq i\) for all \(0 \leq i \leq 5\) and
\[ a_{0} + a_{1} + \cdots + a_{5} = 6. \]

Answer: 2002


Using equal-length toothpicks to form a rectangular diagram as shown, if the length of the rectangle is 20 toothpicks long and the width is 10 toothpicks long, how many toothpicks are used?

Answer: 430


An isosceles trapezoid \( ABCD \) with bases \( BC \) and \( AD \) is such that \( \angle ADC = 2 \angle CAD = 82^\circ \). Inside the trapezoid, a point \( T \) is chosen such that \( CT = CD \) and \( AT = TD \). Find \( \angle TCD \). Give the answer in degrees.

Answer: 38


On bookshelf A, there are 4 English books and 2 Chinese books, while on bookshelf B, there are 2 English books and 3 Chinese books.
$(Ⅰ)$ Without replacement, 2 books are taken from bookshelf A, one at a time. Find the probability of getting an English book on the first draw and still getting an English book on the second draw.
$(Ⅱ)$ First, 2 books are randomly taken from bookshelf B and placed on bookshelf A. Then, 2 books are randomly taken from bookshelf A. Find the probability of getting 2 English books from bookshelf A.

Answer: \frac{93}{280}


Given $$\alpha \in \left( \frac{5}{4}\pi, \frac{3}{2}\pi \right)$$ and it satisfies $$\tan\alpha + \frac{1}{\tan\alpha} = 8$$, then $\sin\alpha\cos\alpha = \_\_\_\_\_\_$; $\sin\alpha - \cos\alpha = \_\_\_\_\_\_$.

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


Sasha has  $10$  cards with numbers  $1, 2, 4, 8,\ldots, 512$ . He writes the number  $0$  on the board and invites Dima to play a game. Dima tells the integer  $0 < p < 10, p$  can vary from round to round. Sasha chooses  $p$  cards before which he puts a “ $+$ ” sign, and before the other cards he puts a “ $-$ " sign. The obtained number is calculated and added to the number on the board. Find the greatest absolute value of the number on the board Dima can get on the board after several rounds regardless Sasha’s moves.

Answer: 1023


There are $N$ points marked on a plane. Any three of them form a triangle whose angles are expressible in degrees as natural numbers. What is the maximum possible $N$ for which this is possible?

Answer: 180


A student, Ellie, was supposed to calculate $x-y-z$, but due to a misunderstanding, she computed $x-(y+z)$ and obtained 18. The actual answer should have been 6. What is the value of $x-y$?

Answer: 12


Define a "digitized number" as a ten-digit number  $a_0a_1\ldots a_9$  such that for  $k=0,1,\ldots, 9$ ,  $a_k$  is equal to the number of times the digit  $k$  occurs in the number.  Find the sum of all digitized numbers.

Answer: 6210001000


What is the largest four-digit negative integer congruent to $2 \pmod{17}$?

Answer: -1001


Let \( P \) be a polynomial with integer coefficients such that \( P(0)+P(90)=2018 \). Find the least possible value for \( |P(20)+P(70)| \).

Answer: 782


A cake has a shape of triangle with sides  $19,20$  and  $21$ . It is allowed to cut it it with a line into two pieces and put them on a round plate such that pieces don't overlap each other and don't  stick out of the plate. What is the minimal diameter of the plate?

Answer: 21


Given the function $f(x)=2\sin ωx (ω > 0)$, if the minimum value of $f(x)$ in the interval $\left[ -\frac{\pi}{3}, \frac{\pi}{4} \right]$ is $-2$, find the minimum value of $ω$.

Answer: \frac{3}{2}


In an arithmetic sequence $\{a + n b \mid n = 1, 2, \cdots\}$ that contains an infinite geometric sequence, find the necessary and sufficient conditions for the real numbers $a$ and $b$ (where $a, b \neq 0$).

Answer: \frac{}{b}\in\mathbb{Q}


Three numbers, whose sum is 114, are, on one hand, three consecutive terms of a geometric progression, and on the other hand, the first, fourth, and twenty-fifth terms of an arithmetic progression, respectively. Find these numbers.

Answer: 2,14,98


A hotel has three types of rooms available: a triple room, a double room, and a single room (each type of room can only accommodate the corresponding number of people). There are 4 adult men and 2 little boys looking for accommodation. The little boys should not stay in a room by themselves (they must be accompanied by an adult). If all three rooms are occupied, the number of different arrangements for accommodation is $\boxed{36}$.

Answer: 36


You are given a positive integer  $k$  and not necessarily distinct positive integers  $a_1, a_2 , a_3 , \ldots,
a_k$ . It turned out that for any coloring of all positive integers from  $1$  to  $2021$  in one of the  $k$  colors so that there are exactly  $a_1$  numbers of the first color,  $a_2$  numbers of the second color,  $\ldots$ , and  $a_k$  numbers of the  $k$ -th color, there is always a number  $x \in \{1, 2, \ldots, 2021\}$ , such that the total number of numbers colored in the same color as  $x$  is exactly  $x$ . What are the possible values of  $k$ ?

*Proposed by Arsenii Nikolaiev*

Answer: 2021


Given that Kira needs to store 25 files onto disks, each with 2.0 MB of space, where 5 files take up 0.6 MB each, 10 files take up 1.0 MB each, and the rest take up 0.3 MB each, determine the minimum number of disks needed to store all 25 files.

Answer: 10


In the plane $Oxy$, the coordinates of point $A$ are defined by the equation $5a^{2} + 12ax + 4ay + 8x^{2} + 8xy + 4y^{2} = 0$, and a parabola with vertex at point $B$ is defined by the equation $ax^{2} - 2a^{2}x - ay + a^{3} + 4 = 0$. Find all values of the parameter $a$ for which points $A$ and $B$ lie on opposite sides of the line $y - 3x = 4$ (points $A$ and $B$ do not lie on this line).

Answer: (-\infty,-2)\cup(0,\frac{2}{3})\cup(\frac{8}{7},+\infty)


Answer: 4\midn


Find the  $2019$ th strictly positive integer  $n$  such that  $\binom{2n}{n}$  is not divisible by  $5$ .

Answer: 37805


A strip of width 1 is being tightly arranged with non-overlapping rectangular plates of width 1 and lengths \( a_1, a_2, a_3, \ldots \) (where \( a_1 \neq 1 \)). Starting from the second rectangular plate, each plate is similar but not congruent to the rectangle formed by the previously placed plates. After placing the first \( n \) rectangular plates, let \( S_n \) denote the length of the strip covered. Does there exist a real number that \( S_n \) does not exceed? Prove your conclusion.

Answer: Nosuchrealexists


Let  $P(x) = x^3 + ax^2 + bx + 1$  be a polynomial with real coefficients and three real roots  $\rho_1$ ,  $\rho_2$ ,  $\rho_3$  such that  $|\rho_1| < |\rho_2| < |\rho_3|$ . Let  $A$  be the point where the graph of  $P(x)$  intersects  $yy'$  and the point  $B(\rho_1, 0)$ ,  $C(\rho_2, 0)$ ,  $D(\rho_3, 0)$ . If the circumcircle of  $\vartriangle ABD$  intersects  $yy'$  for a second time at  $E$ , find the minimum value of the length of the segment  $EC$  and the polynomials for which this is attained.

*Brazitikos Silouanos, Greece*

Answer: \sqrt{2}


Let \( M \) be a set of \( n \) points on a plane, satisfying the following conditions:
1. There are 7 points in \( M \) that are the vertices of a convex heptagon.
2. For any 5 points in \( M \), if these 5 points are the vertices of a convex pentagon, then the interior of this convex pentagon contains at least one point from \( M \).

Find the minimum value of \( n \).

Answer: 11


Given $\tan (\alpha-\beta)= \frac {1}{2}$, $\tan \beta=- \frac {1}{7}$, and $\alpha$, $\beta\in(0,\pi)$, find the value of $2\alpha-\beta$.

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


Positive numbers  $x, y, z$  satisfy  $x^2+y^2+z^2+xy+yz+zy \le 1$ .
Prove that  $\big( \frac{1}{x}-1\big) \big( \frac{1}{y}-1\big)\big( \frac{1}{z}-1\big)  \ge 9 \sqrt6 -19$ .

Answer: 9\sqrt{6}-19


Let \( n \) be a positive integer greater than 3, such that \((n, 3) = 1\). Find the value of \(\prod_{k=1}^{m}\left(1+2 \cos \frac{2 a_{k} \pi}{n}\right)\), where \(a_{1}, a_{2}, \cdots, a_{m}\) are all positive integers less than or equal to \( n \) that are relatively prime to \( n \).

Answer: 1


It is currently $3\!:\!00\!:\!00 \text{ p.m.}$ What time will it be in $6666$ seconds?  (Enter the time in the format "HH:MM:SS", without including "am" or "pm".)

Answer: 4\!:\!51\!:\!06 \text{ p.m.}


Let $n$ be a positive integer. Find all $n \times n$ real matrices $A$ with only real eigenvalues satisfying $$A+A^{k}=A^{T}$$ for some integer $k \geq n$.

Answer: A = 0


In how many ways can the sequence $1,2,3,4,5$ be rearranged so that no three consecutive terms are increasing and no three consecutive terms are decreasing?
$\textbf{(A)} ~10\qquad\textbf{(B)} ~18\qquad\textbf{(C)} ~24 \qquad\textbf{(D)} ~32 \qquad\textbf{(E)} ~44$

Answer: \textbf{(D)}~32


The median \(AD\) of an acute-angled triangle \(ABC\) is 5. The orthogonal projections of this median onto the sides \(AB\) and \(AC\) are 4 and \(2\sqrt{5}\), respectively. Find the side \(BC\).

Answer: 2 \sqrt{10}


Altitudes $\overline{AP}$ and $\overline{BQ}$ of an acute triangle $\triangle ABC$ intersect at point $H$.  If $HP=5$ while $HQ=2$, then calculate $(BP)(PC)-(AQ)(QC)$. [asy]
size(150); defaultpen(linewidth(0.8));
pair B = (0,0), C = (3,0), A = (2,2), P = foot(A,B,C), Q = foot(B,A,C),H = intersectionpoint(B--Q,A--P);
draw(A--B--C--cycle);
draw(A--P^^B--Q);
label("$A$",A,N); label("$B$",B,W); label("$C$",C,E); label("$P$",P,S); label("$Q$",Q,E); label("$H$",H,NW);
[/asy]

Answer: 21


Given a bicycle's front tire lasts for 5000km and the rear tire lasts for 3000km, determine the maximum distance the bicycle can travel if the tires are swapped reasonably during use.

Answer: 3750


There are 300 non-zero integers written in a circle such that each number is greater than the product of the three numbers that follow it in the clockwise direction. What is the maximum number of positive numbers that can be among these 300 written numbers?

Answer: 200


Let $R$  be a set of nine distinct integers. Six of the elements are $2$, $3$, $4$, $6$, $9$, and $14$. What is the number of possible values of the median of $R$?

Answer: 7


When the two-digit integer \( XX \), with equal digits, is multiplied by the one-digit integer \( X \), the result is the three-digit integer \( PXQ \). What is the greatest possible value of \( PXQ \) if \( PXQ \) must start with \( P \) and end with \( X \)?

Answer: 396


How many three-digit whole numbers have at least one 7 or at least one 9 as digits?

Answer: 452


A bitstring of length  $\ell$  is a sequence of  $\ell$   $0$ 's or  $1$ 's in a row. How many bitstrings of length  $2014$  have at least  $2012$  consecutive  $0$ 's or  $1$ 's?

Answer: 16


An isosceles right triangle with side lengths in the ratio 1:1:\(\sqrt{2}\) is inscribed in a circle with a radius of \(\sqrt{2}\). What is the area of the triangle and the circumference of the circle?

Answer: 2\pi\sqrt{2}


Rotate a square with a side length of 1 around a line that contains one of its sides. The lateral surface area of the resulting solid is \_\_\_\_\_\_.

Answer: 2\pi


Calculate the car's average miles-per-gallon for the entire trip given that the odometer readings are $34,500, 34,800, 35,250$, and the gas tank was filled with $8, 10, 15$ gallons of gasoline.

Answer: 22.7


Let's call any natural number "very prime" if any number of consecutive digits (in particular, a digit or number itself) is a prime number. For example,  $23$  and  $37$  are "very prime" numbers, but  $237$  and  $357$  are not. Find the largest "prime" number (with justification!).

Answer: 373


If $\{a_1,a_2,a_3,\ldots,a_n\}$ is a set of real numbers, indexed so that $a_1 < a_2 < a_3 < \cdots < a_n,$ its complex power sum is defined to be $a_1i + a_2i^2+ a_3i^3 + \cdots + a_ni^n,$ where $i^2 = - 1.$ Let $S_n$ be the sum of the complex power sums of all nonempty subsets of $\{1,2,\ldots,n\}.$ Given that $S_8 = - 176 - 64i$ and $S_9 = p + qi,$ where $p$ and $q$ are integers, find $|p| + |q|.$

Answer: 368


In a certain land, all Arogs are Brafs, all Crups are Brafs, all Dramps are Arogs, and all Crups are Dramps. Which of the following statements is implied by these facts?
$\textbf{(A) } \text{All Dramps are Brafs and are Crups.}\\ \textbf{(B) } \text{All Brafs are Crups and are Dramps.}\\ \textbf{(C) } \text{All Arogs are Crups and are Dramps.}\\ \textbf{(D) } \text{All Crups are Arogs and are Brafs.}\\ \textbf{(E) } \text{All Arogs are Dramps and some Arogs may not be Crups.}$

Answer: \textbf{(D)}


Let \(ABCDEFGH\) be a rectangular cuboid. How many acute-angled triangles are formed by joining any three vertices of the cuboid?

Answer: 8


Let  $p$  be an odd prime of the form  $p=4n+1$ . [list=a][*] Show that  $n$  is a quadratic residue  $\pmod{p}$ . [*] Calculate the value  $n^{n}$    $\pmod{p}$ . [/list]

Answer: 1


A company gathered at a meeting. Let's call a person sociable if, in this company, they have at least 20 acquaintances, with at least two of those acquaintances knowing each other. Let's call a person shy if, in this company, they have at least 20 non-acquaintances, with at least two of those non-acquaintances not knowing each other. It turned out that in the gathered company, there are neither sociable nor shy people. What is the maximum number of people that can be in this company?

Answer: 40


In triangle \(ABC\), it is given that \(|AB| = |BC|\) and \(\angle ABC = 20^\circ\). Point \(M\) is taken on \(AB\) such that \(\angle MCA = 60^\circ\), and point \(N\) is taken on \(CB\) such that \(\angle NAC = 50^\circ\). Find \(\angle NMA\).

Answer: 30


The inclination angle $\alpha$ of the line $l: \sqrt{3}x+3y+1=0$ is $\tan^{-1}\left( -\frac{\sqrt{3}}{3} \right)$. Calculate the value of the angle $\alpha$.

Answer: \frac{5\pi}{6}


A point inside a triangle is connected to the vertices by three line segments. What is the maximum number of these line segments that can be equal to the opposite side?

Answer: 1


How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\times 5$ square array of dots?

Answer: 100


A positive integer $n$ is loose if it has six positive divisors and satisfies the property that any two positive divisors $a<b$ of $n$ satisfy $b \geq 2 a$. Compute the sum of all loose positive integers less than 100.

Answer: 512


Given that \( a \) and \( d \) are non-negative numbers, \( b \) and \( c \) are positive numbers, and \( b+c \geq a+d \), find the minimum value of the following expression:
\[
\frac{b}{c+d} + \frac{c}{a+b}.
\]

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


Find the differentials of the functions:
1) \( y = x^3 - 3^x \);
2) \( F(\varphi) = \cos \frac{\varphi}{3} + \sin \frac{3}{\varphi} \);
3) \( z = \ln \left(1 + e^{10x}\right) + \operatorname{arcctg} e^{5x} \); calculate \(\left.dz\right|_{x=0; dx=0.1}\).

Answer: 0.25


How many positive integers \(N\) possess the property that exactly one of the numbers \(N\) and \((N+20)\) is a 4-digit number?

Answer: 40


Isosceles triangles \(ABC\) (\(AB = BC\)) and \(A_1B_1C_1\) (\(A_1B_1 = B_1C_1\)) are similar, and \(AC : A_1C_1 = 5 : \sqrt{3}\).

Vertices \(A_1\) and \(B_1\) are located on sides \(AC\) and \(BC\), respectively, and vertex \(C_1\) is on the extension of side \(AB\) beyond point \(B\), with \(A_1B_1 \perp BC\). Find the angle \(B\).

Answer: 120


An electronic watch displays the time as $09:15:12$ at 9:15:12 AM and 13:11:29 at 1:11:29 PM. How many times in a 24-hour day does the six digits of the time form a symmetric sequence (i.e., the time reads the same forwards and backwards, such as 01:33:10)?

Answer: 96


The function \( f(x) \) satisfies for all real numbers \( x \):
\[ f(2-x) = f(2+x) \]
\[ f(5+x) = f(5-x) \]
and \( f(0) = 0 \).

Determine the minimum number of zeros of \( f(x) \) on the interval \([-21, 21]\).

Answer: 14


Binbin's height is 1.46 meters, his father is 0.32 meters taller than Binbin, and his mother's height is 1.5 meters.
(1) How tall is Binbin's father?
(2) How much shorter is Binbin's mother than his father?

Answer: 0.28


What fraction of the Earth's volume lies above the 45 degrees north parallel? You may assume the Earth is a perfect sphere. The volume in question is the smaller piece that we would get if the sphere were sliced into two pieces by a plane.

Answer: \frac{8-5\sqrt{2}}{16}


A boy named Vasya looked at a clock and saw a time where the number of hours was greater than zero. He read the digits as a three- or four-digit number (for example, 5:07 would become 507). Then he calculated the number of minutes that had passed since midnight. Could it be that the doubled second number he got is divisible by the first number?

Answer: No


In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given vectors $\vec{m}=(a,c)$ and $\vec{n}=(\cos C,\cos A)$.
1. If $\vec{m}\parallel \vec{n}$ and $a= \sqrt {3}c$, find angle $A$;
2. If $\vec{m}\cdot \vec{n}=3b\sin B$ and $\cos A= \frac {3}{5}$, find the value of $\cos C$.

Answer: \frac {4-6 \sqrt {2}}{15}


Five positive consecutive integers starting with $a$ have average $b$. What is the average of 5 consecutive integers that start with $b$?

Answer: $a+4$


Let \( a \in \mathbf{R}_{+} \). If the function
\[
f(x)=\frac{a}{x-1}+\frac{1}{x-2}+\frac{1}{x-6} \quad (3 < x < 5)
\]
achieves its maximum value at \( x=4 \), find the value of \( a \).

Answer: -\frac{9}{2}


Let $P$ be an interior point of triangle $ABC$ and extend lines from the vertices through $P$ to the opposite sides. Let $a$, $b$, $c$, and $d$ denote the lengths of the segments indicated in the figure. Find the product $abc$ if $a + b + c = 43$ and $d = 3$.

Answer: 441


If five geometric means are inserted between $8$ and $5832$, the fifth term in the geometric series:
$\textbf{(A)}\ 648\qquad\textbf{(B)}\ 832\qquad\textbf{(C)}\ 1168\qquad\textbf{(D)}\ 1944\qquad\textbf{(E)}\ \text{None of these}$

Answer: \textbf{(A)}\648


The integer part and decimal part of $(\sqrt{10}+3)^{2n+1}$ for $n \in \mathbf{N}$ are denoted as $I$ and $F$, respectively. What is the value of $P(I + F)$?

Answer: 1


Given real numbers \( x, y, z \) satisfying
\[
\frac{y}{x-y}=\frac{x}{y+z}, \quad z^{2}=x(y+z)-y(x-y)
\]
find the value of
\[
\frac{y^{2}+z^{2}-x^{2}}{2 y z}.
\]

Answer: \frac{1}{2}


Find all positive integers  $k$  satisfying: there is only a finite number of positive integers  $n$ , such that the positive integer solution  $x$  of  $xn+1\mid n^2+kn+1$  is not unique.

Answer: k\ne2


Ten positive integers are arranged around a circle. Each number is one more than the greatest common divisor of its two neighbors. What is the sum of the ten numbers?

Answer: 28


Find all 10-digit numbers \( a_0 a_1 \ldots a_9 \) such that for each \( k \), \( a_k \) is the number of times that the digit \( k \) appears in the number.

Answer: 6210001000


Let QR = x, PR = y, and PQ = z.  Given that the area of the square on side QR is 144 = x^2 and the area of the square on side PR is 169 = y^2, find the area of the square on side PQ.

Answer: 25


A rhombus \(ABCD\) and a triangle \(ABC\), containing its longer diagonal, are inscribed with circles. Find the ratio of the radii of these circles, if the acute angle of the rhombus is \(\alpha\).

Answer: 1+\cos\frac{\alpha}{2}


Given the set of 10 integers {1, 2, 3, ..., 9, 10}, choose any 3 distinct numbers to be the coefficients of the quadratic function f(x) = ax^2 + bx + c. Determine the number of ways to choose the coefficients such that f(1)/3 is an integer.

Answer: 252


In rectangle $ABCD,$ $P$ is a point on side $\overline{BC}$ such that $BP = 12$ and $CP = 4.$ If $\tan \angle APD = 2,$ then find $AB.$

Answer: 12


On a flat plane in Camelot, King Arthur builds a labyrinth  $\mathfrak{L}$  consisting of  $n$  walls, each of which is an infinite straight line. No two walls are parallel, and no three walls have a common point. Merlin then paints one side of each wall entirely red and the other side entirely blue.

At the intersection of two walls there are four corners: two diagonally opposite corners where a red side and a blue side meet, one corner where two red sides meet, and one corner where two blue sides meet. At each such intersection, there is a two-way door connecting the two diagonally opposite corners at which sides of different colours meet.

After Merlin paints the walls, Morgana then places some knights in the labyrinth. The knights can walk through doors, but cannot walk through walls.

Let  $k(\mathfrak{L})$  be the largest number  $k$  such that, no matter how Merlin paints the labyrinth  $\mathfrak{L},$  Morgana can always place at least  $k$  knights such that no two of them can ever meet. For each  $n,$  what are all possible values for  $k(\mathfrak{L}),$  where  $\mathfrak{L}$  is a labyrinth with  $n$  walls?

Answer: n+1


If \( a, b, c \) are the three real roots of the equation
\[ x^{3} - x^{2} - x + m = 0, \]
then the minimum value of \( m \) is _____.

Answer: -\frac{5}{27}


If P and Q are points on the line y = 1 - x and the curve y = -e^x, respectively, find the minimum value of |PQ|.

Answer: \sqrt{2}


A certain pharmaceutical company has developed a new drug to treat a certain disease, with a cure rate of $p$. The drug is now used to treat $10$ patients, and the number of patients cured is denoted as $X$.
$(1)$ If $X=8$, two patients are randomly selected from these $10$ people for drug interviews. Find the distribution of the number of patients cured, denoted as $Y$, among the selected patients.
$(2)$ Given that $p\in \left(0.75,0.85\right)$, let $A=\{k\left|\right.$ probability $P\left(X=k\right)$ is maximum$\}$, and $A$ contains only two elements. Find $E\left(X\right)$.

Answer: \frac{90}{11}


The lateral sides of a right trapezoid are 10 and 8. The diagonal of the trapezoid, drawn from the vertex of the acute angle, bisects this angle. Find the area of the trapezoid.

Answer: 104


Determine by how many times the number \((2014)^{2^{2014}} - 1\) is greater than the number written in the following form:
\[
\left(\left((2014)^{2^0} + 1\right) \cdot \left((2014)^{2^1} + 1\right) \cdot \left((2014)^{2^2} + 1\right) \ldots \cdot \left((2014)^{2^{2013}} + 1\right)\right) + 1.
\]

Answer: 2013


On the AMC 8 contest Billy answers 13 questions correctly, answers 7 questions incorrectly and doesn't answer the last 5. What is his score?
$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 13\qquad\textbf{(D)}\ 19\qquad\textbf{(E)}\ 26$

Answer: \textbf{(C)}\13


Given the function \( f(x) = \sqrt{x+2} + k \), and that there exist \( a, b \) (\(a < b\)) such that the range of \( f(x) \) on \([a, b]\) is \([a, b]\), find the range of values for the real number \( k \).

Answer: (-\frac{9}{4},-2]


A machine that records the number of visitors to a museum shows 1,879,564. Note that this number has all distinct digits. What is the minimum number of additional visitors needed for the machine to register another number that also has all distinct digits?
(a) 35
(b) 36
(c) 38
(d) 47
(e) 52

Answer: 38