Find all polynomials \(P(x)\) with real coefficients such that for all nonzero real numbers \(x\), \[P(x)+P\left(\frac1x\right) =\frac{P\left(x+\frac1x\right) +P\left(x-\frac1x\right)}2.\] [i]Proposed by Holden Mui[/i]

Estimate the value of the $2020$th prime number $p$ such that $p + 2$ is also prime. If $E > 0$ is your estimate and $A$ is the correct answer, you will receive $25 \min \left( \frac{E}{A}, \frac{A}{E}\right)^2$ points, rounded to the nearest integer. (An estimate less than or equal to $0$ will receive $0$ points.

Consider the following graphs of position [i]vs.[/i] time. [asy] size(500); picture pic; // Rectangle draw(pic,(0,0)--(20,0)--(20,15)--(0,15)--cycle); label(pic,"0",(0,0),S); label(pic,"2",(4,0),S); label(pic,"4",(8,0),S); label(pic,"6",(12,0),S); label(pic,"8",(16,0),S); label(pic,"10",(20,0),S); label(pic,"-15",(0,2),W); label(pic,"-10",(0,4),W); label(pic,"-5",(0,6),W); label(pic,"0",(0,8),W); label(pic,"5",(0,10),W); label(pic,"10",(0,12),W); label(pic,"15",(0,14),W); label(pic,rotate(90)*"x (m)",(-2,7),W); label(pic,"t (s)",(11,-2),S); // Tick Marks draw(pic,(4,0)--(4,0.3)); draw(pic,(8,0)--(8,0.3)); draw(pic,(12,0)--(12,0.3)); draw(pic,(16,0)--(16,0.3)); draw(pic,(20,0)--(20,0.3)); draw(pic,(4,15)--(4,14.7)); draw(pic,(8,15)--(8,14.7)); draw(pic,(12,15)--(12,14.7)); draw(pic,(16,15)--(16,14.7)); draw(pic,(20,15)--(20,14.7)); draw(pic,(0,2)--(0.3,2)); draw(pic,(0,4)--(0.3,4)); draw(pic,(0,6)--(0.3,6)); draw(pic,(0,8)--(0.3,8)); draw(pic,(0,10)--(0.3,10)); draw(pic,(0,12)--(0.3,12)); draw(pic,(0,14)--(0.3,14)); draw(pic,(20,2)--(19.7,2)); draw(pic,(20,4)--(19.7,4)); draw(pic,(20,6)--(19.7,6)); draw(pic,(20,8)--(19.7,8)); draw(pic,(20,10)--(19.7,10)); draw(pic,(20,12)--(19.7,12)); draw(pic,(20,14)--(19.7,14)); // Path add(pic); path A=(0,14)--(20,14); draw(A); label("I.",(8,-4),3*S); path B=(0,6)--(20,6); picture pic2=shift(30*right)*pic; draw(shift(30*right)*B); label("II.",(38,-4),3*S); add(pic2); path C=(0,12)--(20,14); picture pic3=shift(60*right)*pic; draw(shift(60*right)*C); label("III.",(68,-4),3*S); add(pic3); [/asy] Which of the graphs could be the motion of a particle in the given potential? (A) $\text{I}$ (B) $\text{III}$ (C) $\text{I and II}$ (D) $\text{I and III}$ (E) $\text{I, II, and III}$

The natural numbers $X$ and $Y$ are obtained from each other by permuting the digits. Prove that the sums of the digits of the numbers $5X$ and $5Y$ coincide.

A value of $ x$ satisfying the equation $ x^2 \plus{} b^2 \equal{} (a \minus{} x)^2$ is: $ \textbf{(A)}\ \frac{b^2 \plus{} a^2}{2a}\qquad \textbf{(B)}\ \frac{b^2 \minus{} a^2}{2a}\qquad \textbf{(C)}\ \frac{a^2 \minus{} b^2}{2a}\qquad \textbf{(D)}\ \frac{a \minus{} b}{2}\qquad \textbf{(E)}\ \frac{a^2 \minus{} b^2}{2}$

There are $110$ guinea pigs for each of the $110$ species, arranging as a $110\times 110$ array. Find the maximum integer $n$ such that, no matter how the guinea pigs align, we can always find a column or a row of $110$ guinea pigs containing at least $n$ different species.

Find all functions $f :\mathbb{N} \rightarrow \mathbb{N}$ such that $f(m + f(n)f(m)) = nf(m) + m$ holds for all $m,n \in \mathbb{N}$.

Let $I$ be the center of the circle of the inscribed triangle $ABC$ and $M$ be the center of its side $BC$. If $|AI| = |MI|$, prove that there are two of the sides of triangle $ABC$, of which one is twice of the other.

Three lines were drawn through the point $X$ in space. These lines crossed some sphere at six points. It turned out that the distances from point $X$ to some five of them are equal to $2$ cm, $3$ cm, $4$ cm, $5$ cm, $6$ cm. What can be the distance from point $X$ to the sixth point? (Alexey Panasenko)

Let $d$ be a positive integer. The seqeunce $a_1, a_2, a_3,...$ of positive integers is defined by $a_1 = 1$ and $a_{n + 1} = n\left \lfloor \frac{a_n}{n} \right \rfloor+ d$ for $n = 1,2,3, ...$ . Prove that there exists a positive integer $N$ so that the terms $a_N,a_{N + 1}, a_{N + 2},...$ form an arithmetic progression. Note: If $x$ is a real number, $\left \lfloor x \right \rfloor $ denotes the largest integer that is less than or equal to $x$.

The kindergarten group is standing in the column of pairs. The number of boys equals the number of girls in each of the two columns. The number of mixed (boy and girl) pairs equals to the number of the rest pairs. Prove that the total number of children in the group is divisible by eight.

There are $24$ different pencils, $4$ different colors, and $6$ pencils of each color. They were given to $6$ children in such a way that each got $4$ pencils. What is the least number of children that you can randomly choose so that you can guarantee that you have pencils of all colors. P.S. for 10 grade gives same problem with $40$ pencils, $10$ of each color and $10$ children.

An acute scalene triangle $ABC$ with circumcircle $\Omega$ is given. The altitude from $B$ intersects side $AC$ at $B_1$ and circle $\Omega$ at $B_2$. The circle with diameter $B_1B_2$ intersects circle $\Omega$ again at $B_3$. Similarly, the altitude from $C$ intersects side $AB$ at $C_1$ and circle $\Omega$ at $C_2$. The circle with diameter $C_1C_2$ intersects circle $\Omega$ again at $C_3$. Let $X$ be the intersection of lines $B_1B_3$ and $C_1C_3$, and let $Y$ be the intersection of lines $B_3C$ and $C_3B$. Prove that line $XY$ bisects side $BC$.

Let $ABCD$ be a unit square, and let there be two unit circles centered at $C$ and $D$. Let $P$ be the point of intersection of the two circles inside the square. Compute $\angle APB$ in degrees.

Denote $S$ as the set of prime divisors of all integers of form $2^{n^2+1} - 3^n, n \in Z^+$. Prove that $S$ and $P-S$ both contain infinitely many elements (where $P$ is set of prime numbers).

Prove that $\lfloor \sqrt[3]{n}+\sqrt[3]{n+1}+\sqrt[3]{n+2}\rfloor =\lfloor \sqrt[3]{27n+26}\rfloor$ for all positive integers $n$.

Each of six fruit baskets contains pears, plums and apples. The number of plums in each basket equals the total number of apples in all other baskets combined while the number of apples in each basket equals the total number of pears in all other baskets combined. Prove that the total number of fruits is a multiple of $31$.

An ant, which is on an edge of a cube of side $8$, must travel on the surface and return to the starting point. It's path must contain interior points of the six faces of the cube and should visit only once each face of the cube. Find the length of the path that the ant can carry out and justify why it is the shortest path.

Consider all triangles $ABC$ in the cartesian plane whose vertices are at lattice points (i.e. with integer coordinates) and which contain exactly one lattice point (to be denoted $P$) in its interior. Let the line $AP$ meet $BC$ at $E$. Determine the maximum possible value of the ratio $\frac{AP}{PE}$.

Each of $1000$ elves has a hat, red on the inside and blue on the outside or vise versa. An elf with a hat that is red outside can only lie, and an elf with a hat that is blue outside can only tell the truth. One day every elf tells every other elf, “Your hat is red on the outside.” During that day, some of the elves turn their hats inside out at any time during the day. (An elf can do that more than once per day.) Find the smallest possible number of times any hat is turned inside out.

Determine all pairs $(a,b)$ of positive integers for which there exist positive integers $g$ and $N$ such that $$\gcd (a^n+b,b^n+a)=g$$ holds for all integers $n\geqslant N.$ (Note that $\gcd(x, y)$ denotes the greatest common divisor of integers $x$ and $y.$) [i]Proposed by Valentio Iverson, Indonesia[/i]

A [i]composite[/i] (positive integer) is a product $ab$ with $a$ and $b$ not necessarily distinct integers in $\{2,3,4,\dots\}$. Show that every composite is expressible as $xy+xz+yz+1$, with $x,y,z$ positive integers.

Let $\ell$ be a positive integer, and let $m,n$ be positive integers with $m\geq n$, such that $A_1,A_2,\cdots,A_m,B_1,\cdots,B_m$ are $m+n$ pairwise distinct subsets of the set $\{1,2,\cdots,\ell\}$. It is known that $A_i\Delta B_j$ are pairwise distinct, $1\leq i\leq m, 1\leq j\leq n$, and runs over all nonempty subsets of $\{1,2,\cdots,\ell\}$. Find all possible values of $m,n$.

[i](9 pts)[/i] Let $ABC$ be a triangle with circumcenter $O$. The interior bisector of $\angle BAC$ intersects $BC$ at $D$. Circle $\omega_A$ is tangent to segments $AB$ and $AC$ and internally tangent to the circumcircle of $ABC$ at the point $P$. Let $E$ and $F$ be the respective points at which the $B$-excircle and $C$-excircle of $ABC$ are tangent to $AC$ and $AB$. Suppose that lines $BE$ and $CF$ pass through a common point $N$ on the circumcircle of $AEF$. [i]Note: for a triangle $ABC$, the $A$-excircle is the circle lying outside triangle $ABC$ that is tangent to side $BC$ and the extensions of sides $AB, AC$. The $B, C$-excircles are defined similarly.[/i] (a) [i](7 pts)[/i] Prove that the circumcircle of $PDO$ passes through $N$. (b) [i](2 pts)[/i] Suppose that $\frac{PD}{BC} = \frac{2}{7}$. Find, with proof, the value of $\cos (\angle BAC)$.

The inscribed and exscribed sphere of a triangular pyramid $ABCD$ touch her face $BCD$ at different points $X$ and $Y$. Prove that the triangle $AXY$ is obtuse triangle.