Let $P(x) = x^3 - 6x^2 - 5x + 4$. Suppose that $y$ and $z$ are real numbers such that \[ zP(y) = P(y - n) + P(y + n) \] for all reals $n$. Evaluate $P(y)$.

(B.Frenkin) The product of two sides in a triangle is equal to $ 8Rr$, where $ R$ and $ r$ are the circumradius and the inradius of the triangle. Prove that the angle between these sides is less than $ 60^{\circ}$.

Let $b\geq2$ and $w\geq2$ be fixed integers, and $n=b+w$. Given are $2b$ identical black rods and $2w$ identical white rods, each of side length 1. We assemble a regular $2n-$gon using these rods so that parallel sides are the same color. Then, a convex $2b$-gon $B$ is formed by translating the black rods, and a convex $2w$-gon $W$ is formed by translating the white rods. An example of one way of doing the assembly when $b=3$ and $w=2$ is shown below, as well as the resulting polygons $B$ and $W$. [asy]size(10cm); real w = 2*Sin(18); real h = 0.10 * w; real d = 0.33 * h; picture wht; picture blk; draw(wht, (0,0)--(w,0)--(w+d,h)--(-d,h)--cycle); fill(blk, (0,0)--(w,0)--(w+d,h)--(-d,h)--cycle, black); // draw(unitcircle, blue+dotted); // Original polygon add(shift(dir(108))*blk); add(shift(dir(72))*rotate(324)*blk); add(shift(dir(36))*rotate(288)*wht); add(shift(dir(0))*rotate(252)*blk); add(shift(dir(324))*rotate(216)*wht); add(shift(dir(288))*rotate(180)*blk); add(shift(dir(252))*rotate(144)*blk); add(shift(dir(216))*rotate(108)*wht); add(shift(dir(180))*rotate(72)*blk); add(shift(dir(144))*rotate(36)*wht); // White shifted real Wk = 1.2; pair W1 = (1.8,0.1); pair W2 = W1 + w*dir(36); pair W3 = W2 + w*dir(108); pair W4 = W3 + w*dir(216); path Wgon = W1--W2--W3--W4--cycle; draw(Wgon); pair WO = (W1+W3)/2; transform Wt = shift(WO)*scale(Wk)*shift(-WO); draw(Wt * Wgon); label("$W$", WO); /* draw(W1--Wt*W1); draw(W2--Wt*W2); draw(W3--Wt*W3); draw(W4--Wt*W4); */ // Black shifted real Bk = 1.10; pair B1 = (1.5,-0.1); pair B2 = B1 + w*dir(0); pair B3 = B2 + w*dir(324); pair B4 = B3 + w*dir(252); pair B5 = B4 + w*dir(180); pair B6 = B5 + w*dir(144); path Bgon = B1--B2--B3--B4--B5--B6--cycle; pair BO = (B1+B4)/2; transform Bt = shift(BO)*scale(Bk)*shift(-BO); fill(Bt * Bgon, black); fill(Bgon, white); label("$B$", BO);[/asy] Prove that the difference of the areas of $B$ and $W$ depends only on the numbers $b$ and $w$, and not on how the $2n$-gon was assembled. [i]Proposed by Ankan Bhattacharya[/i]

Let $a = \frac{x}{y} +\frac{y}{z} +\frac{z}{x}$, let $b = \frac{x}{z} +\frac{y}{x} +\frac{z}{y}$ and let $c = \left(\frac{x}{y} +\frac{y}{z} \right)\left(\frac{y}{z} +\frac{z}{x} \right)\left(\frac{z}{x} +\frac{x}{y} \right)$. The value of $|ab-c|$ is:

2. The quadrilateral $A B C D$ is convex. Its sides $A B$ and $C D$ are parallel. It is known that the angles $D A C$ and $A B D$ are equal. Furthermore the angles $C A B$ and $D B C$ are equal. Is $A B C D$ necessarily a square? Alexandr Terteryan

The game Battleship is played on a $10\times10$ grid. A [i]fleet[/i] consists of 10 ships: one occupying $4$ cells, two occupying $3$ cells each, three occupying $2$ cells each and four occupying $1$ cell each (see figure). [asy] size(10cm); // Function to draw a square centered at a given position void drawSquare(pair center, real sideLength) { real halfSide = sideLength / 2; draw(shift(center) * box((-halfSide, -halfSide), (halfSide, halfSide))); } // Side length of each square real sideLength = 1; // Coordinates for the squares pair[] positions = { // Top row remains the same (0, 0), (1, 0), (3, 0), (4, 0), (6, 0), (7, 0), (9, 0), (11, 0), (13, 0), (15, 0), // Bottom row moved one square (1 unit) to the right (2, 2), (3, 2), (4, 2), (5, 2), (7, 2), (8, 2), (9, 2), (11, 2), (12, 2), (13, 2) }; // Draw all squares for (pair pos : positions) { drawSquare(pos, sideLength); } [/asy] Ships can be placed either horizontally or vertically, but they must not touch each other, not even at a vertex. Is it possible to place two fleets on the same board according to these rules?

Compute the remainder of $3^{2021}$ mod $15$

The real numbers $ a,b$ fulfil the conditions (i) $ 0<a<a\plus{}\frac12\le b$; (ii) $ a^{40}\plus{}b^{40}\equal{}1$. Prove that $ b$ has the first 12 digits after the decimal point equal to 9. [i]Mircea Fianu[/i]

Find the sum of all $\left\lfloor x\right\rfloor$ such that $x^2-15\left\lfloor x\right\rfloor+36=0$. $\text{(A) }15\qquad\text{(B) }26\qquad\text{(C) }45\qquad\text{(D) }49\qquad\text{(E) }75$

We are given a set $S=\{z_1,z_2,\cdots ,z_{1993}\}$, where $z_1,z_2,\cdots ,z_{1993}$ are nonzero complex numbers (also viewed as nonzero vectors in the plane). Prove that we can divide $S$ into some groups such that the following conditions are satisfied: (1) Each element in $S$ belongs and only belongs to one group; (2) For any group $p$, if we use $T(p)$ to denote the sum of all memebers in $p$, then for any memeber $z_i (1\le i \le 1993)$ of $p$, the angle between $z_i$ and $T(p)$ does not exceed $90^{\circ}$; (3) For any two groups $p$ and $q$, the angle between $T(p)$ and $T(q)$ exceeds $90^{\circ}$ (use the notation introduced in (2)).

1. Every square of a $8 \times 8$ board is filled with a positive integer, such that the following condition holds: if a chess knight can move from some square to another then the ratio of numbers from these two squares is a prime number. Is it possible that some square is filled with 5 , and another one with 6 ? Egor Bakaev

The number $2021$ can be written as the sum of $2021$ consecutive integers. What is the largest term in the sequence of $2021$ consecutive integers? [i]Proposed by Taiki Aiba[/i]

Let $S=\{(x,y) : x \in \{0,1,2,3,4\}, y \in \{0,1,2,3,4,5\}$, and $(x,y) \neq (0,0) \}$. Let $T$ be the set of all right triangles whose vertices are in $S$. For every right triangle $t=\triangle ABC$ with vertices $A$, $B$, and $C$ in counter-clockwise order and right angle at $A$, let $f(t)= \tan (\angle CBA)$. What is \[ \displaystyle \prod_{t \in T} f(t) \text{?} \] [asy] size((120)); dot((1,0)); dot((2,0)); dot((3,0)); dot((4,0)); dot((0,1)); dot((0,2)); dot((0,3)); dot((0,4)); dot((0,5)); dot((1,1)); dot((1,2)); dot((1,3)); dot((1,4)); dot((1,5)); dot((2,1)); dot((2,2)); dot((2,3)); dot((2,4)); dot((2,5)); dot((3,1)); dot((3,2)); dot((3,3)); dot((3,4)); dot((3,5)); dot((4,1)); dot((4,2)); dot((4,3)); dot((4,4)); dot((4,5)); label("$\circ$", (0,0)); label("$S$", (-.7,2.5)); [/asy] $\textbf{(A)}\ 1 \qquad \textbf{(B)}\ \frac{625}{144} \qquad \textbf{(C)}\ \frac{125}{24} \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ \frac{625}{24}$

Let $S$ be the set of non-negative integer powers of $3$ and $5$, $S = \{1, 3, 5, 3^2, 5^2, \ldots \}$. For every $a$ and $b$ in $S$ satisfying $$ \left| \pi - \frac{a}{b} \right| < 0.1 $$ Find the minimum value of $ab$. [i]Proposed by Irakli Shalibashvili, Georgia [/i]

In an art museum, $n$ paintings are exhibited, where $n \geq 33.$ In total, $15$ colors are used for these paintings such that any two paintings have at least one common color, and no two paintings have exactly the same colors. Determine all possible values of $n \geq 33$ such that regardless of how we color the paintings with the given properties, we can choose four distinct paintings, which we can label as $T_1, T_2, T_3,$ and $T_4,$ such that any color that is used in both $T_1$ and $T_2$ can also be found in either $T_3$ or $T_4$.

In a sequence of $71$ nonzero real numbers, each number (apart from the fitrst one and the last one) is one less than the product of its two neighbors. Prove that the first and the last number are equal. [i](Josef Tkadlec)[/i]

Determine all natural numbers $n$ such that for each natural number $a$ relatively prime with $n$ and $a \le 1 + \left\lfloor \sqrt{n} \right\rfloor$ there exists some integer $x$ with $a \equiv x^2 \mod n$. Remark: "Natural numbers" is the set of positive integers.

Let $ a $ be an odd natural that is not a perfect square, and $ m,n\in\mathbb{N} . $ Then [b]a)[/b] $ \left\{ m\left( a+\sqrt a \right) \right\}\neq\left\{ n\left( a-\sqrt a \right) \right\} $ [b]b)[/b] $ \left[ m\left( a+\sqrt a \right) \right]\neq\left[ n\left( a-\sqrt a \right) \right] $ Here, $ \{\},[] $ denotes the fractionary, respectively the integer part.

There are $n$ bus stops placed around the circular lake. Each bus stop is connected by a road to the two adjacent stops (we call a [i]segment [/i] the entire road between two stops). Determine the number of bus routes that start and end in the fixed bus stop A, pass through each bus stop at least once and travel through exactly $n+1$ [i]segments[/i].

Let $ABC$ be a triangle. Circle $\Gamma$ passes through $A$, meets segments $AB$ and $AC$ again at points $D$ and $E$ respectively, and intersects segment $BC$ at $F$ and $G$ such that $F$ lies between $B$ and $G$. The tangent to circle $BDF$ at $F$ and the tangent to circle $CEG$ at $G$ meet at point $T$. Suppose that points $A$ and $T$ are distinct. Prove that line $AT$ is parallel to $BC$. (Nigeria)

Find the sum of the digits in the answer to $\underbrace{9999\cdots 99}_{94\text{ nines}} \times \underbrace{4444\cdots 44}_{94\text{ fours}}$ where a string of $94$ nines is multiplied by a string of $94$ fours. $\text{(A)}\ 846 \qquad \text{(B)}\ 855 \qquad \text{(C)}\ 945 \qquad \text{(D)}\ 954 \qquad \text{(E)}\ 1072$

An isosceles triangle has angles of $50^\circ,x^\circ,$ and $y^\circ$. Find the maximum possible value of $x-y$. [i]Proposed by Nathan Ramesh

Let $f(x)$ be a quadratic function with integer coefficients. If we know that $f(0)$, $f(3)$ and $f(4)$ are all different and elements of the set $\{2, 20, 202, 2022\}$, determine all possible values of $f(1)$.

$AH$ is the altitude of the acute triangle $ABC$, $K$ and $L$ are the feet of the perpendiculars, from point $H$ on sides $AB$ and $AC$ respectively. Prove that the angles $BKC$ and $BLC$ are equal.

An acute triangle $ABC$ is surrounded by equilateral triangles $KLM$ and $PQR$ such that its vertices lie on the sides of these equilateral triangle as shown on the picture. Lines $PK$ and $QL$ intersect at point $D$. Prove that $\angle ABC + \angle PDQ = 120^o$. (Yurii Biletskyi) [img]https://cdn.artofproblemsolving.com/attachments/4/6/32d3f74f07ca6a8edcabe4a08aa321eb3a5010.png[/img]