Show that there are infinitely many polynomials P with real coefficients such that if x, y, and z are real numbers such that $x^2+y^2+z^2+2xyz=1$, then $$P\left(x\right)^2+P\left(y\right)^2+P\left(z\right)^2+2P\left(x\right)P\left(y\right)P\left(z\right) = 1$$

The 16 fields of a $4 \times 4$ checker board can be arranged in 18 lines as follows: the four lines, the four columns, the five diagonals from north west to south east and the five diagonals from north east to south west. These diagonals consists of 2,3 or 4 edge-adjacent fields of same colour; the corner fields of the chess board alone do not form a diagonal. Now, we put a token in 10 of the 16 fields. Each of the 18 lines contains an even number of tokens contains a point. What is the highest possible point number when can be achieved by optimal placing of the 10 tokens. Explain your answer.

Find all linear homogeneous differential equations with continuous coefficients (on the whole real line) such that for any solution $ f(t)$ and any real number $ c,f(t\plus{}c)$ is also a solution.

Real positive numbers $a, b, c$ satisfy $abc=1$. Prove the inequality $$\frac{a^2+b^2}{a^4+b^4}+\frac{b^2+c^2}{b^4+c^4}+\frac{c^2+a^2}{c^4+a^4}\leq a+b+c.$$

What is the smallest positive integer that cannot be written as the sum of two nonnegative palindromic integers? (An integer is [i]palindromic[/i] if the sequence of decimal digits are the same when read backwards.)

Find the sum of the squares of all real numbers satisfying the equation \[x^{256}-256^{32}=0.\] $\textbf{(A) }8\qquad\textbf{(B) }128\qquad\textbf{(C) }512\qquad\textbf{(D) }65,536\qquad\textbf{(E) }2(256^{32})$

\[\int_{-10}^{10}|4-|3-|2-|1-|x|||||\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

The figure below shows $13$ circles of radius $1$ within a larger circle. All the intersections occur at points of tangency. What is the area of the region, shaded in the figure, inside the larger circle but outside all the circles of radius $1 ?$ [asy]unitsize(20);filldraw(circle((0,0),2*sqrt(3)+1),rgb(0.5,0.5,0.5));filldraw(circle((-2,0),1),white);filldraw(circle((0,0),1),white);filldraw(circle((2,0),1),white);filldraw(circle((1,sqrt(3)),1),white);filldraw(circle((3,sqrt(3)),1),white);filldraw(circle((-1,sqrt(3)),1),white);filldraw(circle((-3,sqrt(3)),1),white);filldraw(circle((1,-1*sqrt(3)),1),white);filldraw(circle((3,-1*sqrt(3)),1),white);filldraw(circle((-1,-1*sqrt(3)),1),white);filldraw(circle((-3,-1*sqrt(3)),1),white);filldraw(circle((0,2*sqrt(3)),1),white);filldraw(circle((0,-2*sqrt(3)),1),white);[/asy] $\textbf{(A) } 4 \pi \sqrt{3} \qquad\textbf{(B) } 7 \pi \qquad\textbf{(C) } \pi(3\sqrt{3} +2) \qquad\textbf{(D) } 10 \pi (\sqrt{3} - 1) \qquad\textbf{(E) } \pi(\sqrt{3} + 6)$

The sides $AD$ and $BC$ of a convex quadrilateral $ABCD$ are extended to meet at $E$. Let $H$ and $G$ be the midpoints of $BD$ and $AC$, respectively. Find the ratio of the area of the triangle $EHG$ to that of the quadrilateral $ABCD$.

Ella the Witch was mixing a magic elixir which consisted of three components: $140$ ml of reindeer moss tea, $160$ ml of fly agaric extract, and $50$ ml of moonshine. She took an empty $350$ ml bottle, poured $140$ ml of reindeer moss tea into it and started adding fly agaric extract when she was disturbed by its black cat Mehsto. So she mistakenly poured too much fly agaric extract into the bottle and noticed her fault only later when the bottle Riled before all $50$ ml of moonshine was added. Ella made quick calculations, carefully shaked up the contents of the bottle, poured out some part of liquid and added some amount of mixture of reindeer moss tea and fly agaric extract taken in a certain proportion until the bottle was full again and the elixir had exactly the right compositsion. Which was the proportion of reindeer moss tea and fly agaric extract in the mixture that Ella added into the bottle?

A $300\times 300$ board is arbitrarily filled with $2\times 1$ dominoes with no overflow, underflow, or overlap. (Tokens can be placed vertically or horizontally.) Decide if it is possible to paint the tiles with three different colors, so that the following conditions are met: $\bullet$ Each token is painted in one and only one of the colors. $\bullet$ The same number of tiles are painted in each color. $\bullet$ No piece is a neighbor of more than two pieces of the same color. Note: Two dominoes are [i]neighbors [/i]if they share an edge.

Let $ABC$ be a triangle with $\angle B=120^{o}$. Let $D$ be point on the $B$-angle bisector, such that $\angle ADB=2\angle ACB$. Point $E$ lies on the segment $AB$, so that $AE=AD$. Show that $EC=ED$.

Find all pairs $ (a,b)$ of positive integers that satisfy the equation: $ a^{b^2} \equal{} b^a$.

Given a triangle $ABC$, points $K,L,M$ are the midpoints of the sides $BC,CA,AB$, and points $X,Y,Z$ are the midpoints of the arcs $BC,CA,AB$ of the circumcircle not containing $A,B,C$ respectively. If $R$ denotes the circumradius and $r$ the inradius of the triangle, show that $r+KX+LY+MZ=2R$.

Find all sequences $a_1$, $a_2$, $\dots$ of nonnegative integers such that for all positive integers $n$, the polynomial \[1+x^{a_1}+x^{a_2}+\dots+x^{a_n}\] has at least one integer root. (Here $x^0=1$.) [i]Kornpholkrit Weraarchakul[/i]

Kikoriki live on the shores of a pond in the form of an equilateral triangle with a side of $600$ m, Krash and Wally live on the same shore, $300$ m from each other. In summer, Dokko to Krash walk $900$ m, and Wally to Rosa - also $900$ m. Prove that in winter, when the pond freezes and it will be possible to walk directly on the ice, Dokko will walk as many meters to Krash as Wally to Rosa. [url=https://en.wikipedia.org/wiki/Kikoriki]about Kikoriki/GoGoRiki / Smeshariki [/url]

(I.Bogdanov) A section of a regular tetragonal pyramid is a regular pentagon. Find the ratio of its side to the side of the base of the pyramid.

Let $d(n)$ denote the number of positive divisors of $n$. For positive integer $n$ we define $f(n)$ as $$f(n) = d\left(k_1\right) + d\left(k_2\right)+ \cdots + d\left(k_m\right),$$ where $1 = k_1 < k_2 < \cdots < k_m = n$ are all divisors of the number $n$. We call an integer $n > 1$ [i]almost perfect[/i] if $f(n) = n$. Find all almost perfect numbers. [i]Paulius Ašvydis[/i]

A regular tetrahedron $A_1B_1C_1D_1$ is inscribed in a regular tetrahedron $ABCD$, where $A_1$ lies in the plane $BCD$, $B_1$ in the plane $ACD$, etc. Prove that $A_1B_1 \ge\frac{ AB}{3}$.

Consider collections of unordered pairs of $V$ different objects $a$, $b$, $c$, $\ldots$, $k$. Three pairs such as $ab$, $bc$, $ab$ are said to form a triangle. Prove that, if $4E\leq V^2$, it is possible to choose $E$ pairs so that no triangle is formed.

Find all $f: R \longrightarrow R$ such that \[f(xy+f(x))=xf(y)+f(x)\] for every pair of real numbers $x,y$.

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

Two circles $C_{1}$ and $C_{2}$ intersect at points $A$ and $B$. Let $P$, $Q$ be points on circles $C_{1}$, $C_{2}$ respectively, such that $|AP| = |AQ|$. The segment $PQ$ intersects circles $C_{1}$ and $C_{2}$ in points $M$, $N$ respectively. Let $C$ be the center of the arc $BP$ of $C_{1}$ which does not contain point $A$ and let $D$ be the center of arc $BQ$ of $C_{2}$ which does not contain point $A$ Let $E$ be the intersection of $CM$ and $DN$. Prove that $AE$ is perpendicular to $CD$. Proposed by Steve Dinh

Let $x, y$ and $z$ be real numbers such that $x + y + z = 0$. Show that $x^3 + y^3 + z^3 = 3xyz$.

An umbrella seller has umbrellas of $7$ different colours. He has a total of $2023$ umbrellas in stock but because of the plastic packaging, the colours are not visible. What is the minimum number of umbrellas that one must buy in order to ensure that at least $23$ umbrellas are of the same colour ?