The incircle of the triangle $ABC$ touches the sides $BC$, $CA$, and $AB$ in the points $D$, $E$ and $F$, respectively. Let $K$ be the point symmetric to $D$ with respect to the incenter. The lines $DE$ and $FK$ intersect at $S$. Prove that $AS$ is parallel to $BC$. [i](4th Middle European Mathematical Olympiad, Team Competition, Problem 5)[/i]

For any natural number $n{}$ and positive integer $k{}$, we say that $n{}$ is $k-good$ if there exist non-negative integers $a_1,\ldots, a_k$ such that $$n=a_1^2+a_2^4+a_3^8+\ldots+a_k^{2^k}.$$ Is there a positive integer $k{}$ for which every natural number is $k-good$?

The origami club meets once a week at a fixed time, but this week, the club had to reschedule the meeting to a different time during the same day. However, the room that they usually meet has $5$ available time slots, one of which is the original time the origami club meets. If at any given time slot, there is a $30$ percent chance the room is not available, what is the probability the origami club will be able to meet at that day?

Aaron has a coin that is slightly unbalanced. The odds of getting heads are $60\%$. What are the odds that if he flips it endlessly, at some point during his flipping he has a total of three more tails than heads?

Let $n$ be a positive integer with the following property: $2^n-1$ divides a number of the form $m^2+81$, where $m$ is a positive integer. Find all possible $n$.

Bob, a spherical person, is floating around peacefully when Dave the giant orange fish launches him straight up 23 m/s with his tail. If Bob has density 100 $\text{kg/m}^3$, let $f(r)$ denote how far underwater his centre of mass plunges underwater once he lands, assuming his centre of mass was at water level when he's launched up. Find $\lim_{r\to0} \left(f(r)\right) $. Express your answer is meters and round to the nearest integer. Assume the density of water is 1000 $\text{kg/m}^3$. [i](B. Dejean, 6 points)[/i]

The adjacent map is part of a city: the small rectangles are rocks, and the paths in between are streets. Each morning, a student walks from intersection A to intersection B, always walking along streets shown, and always going east or south. For variety, at each intersection where he has a choice, he chooses with probability $\frac{1}{2}$ whether to go east or south. Find the probability that through any given morning, he goes through $C$. [asy] defaultpen(linewidth(0.7)+fontsize(8)); size(250); path p=origin--(5,0)--(5,3)--(0,3)--cycle; path q=(5,19)--(6,19)--(6,20)--(5,20)--cycle; int i,j; for(i=0; i<5; i=i+1) { for(j=0; j<6; j=j+1) { draw(shift(6*i, 4*j)*p); }} clip((4,2)--(25,2)--(25,21)--(4,21)--cycle); fill(q^^shift(18,-16)*q^^shift(18,-12)*q, black); label("A", (6,19), SE); label("B", (23,4), NW); label("C", (23,8), NW); draw((26,11.5)--(30,11.5), Arrows(5)); draw((28,9.5)--(28,13.5), Arrows(5)); label("N", (28,13.5), N); label("W", (26,11.5), W); label("E", (30,11.5), E); label("S", (28,9.5), S);[/asy] $\textbf {(A) } \frac{11}{32} \qquad \textbf {(B) } \frac 12 \qquad \textbf {(C) } \frac 47 \qquad \textbf {(D) } \frac{21}{32} \qquad \textbf {(E) } \frac 34$

Players $1,2,\ldots n$ are seated around a table, and each has a single penny. Player $1$ passes a penny to Player $2$, who then passes two pennies to Player $3$, who then passes one penny to player $4$, who then passes two pennies to Player $5$ and so on, players alternately pass one or two pennies to the next player who still has some pennies. The player who runs out of pennies drops out of the game and leaves the table. Find an infinite set of numbers $n$ for which some player ends up with all the $n$ pennies.

Consider an acute triangle $ABC$ of area $S$. Let $CD \perp AB$ ($D \in AB$), $DM \perp AC$ ($M \in AC$) and $DN \perp BC$ ($N \in BC$). Denote by $H_1$ and $H_2$ the orthocentres of the triangles $MNC$, respectively $MND$. Find the area of the quadrilateral $AH_1BH_2$ in terms of $S$.

Let $S_n$ denote the set of permutations of the sequence $(1,2,\dots, n)$. For every permutation $\pi=(\pi_1, \dots, \pi_n)\in S_n$, let $\mathrm{inv}(\pi)$ be the number of pairs $1\le i < j \le n$ with $\pi_i>\pi_j$; i. e. the number of inversions in $\pi$. Denote by $f(n)$ the number of permutations $\pi\in S_n$ for which $\mathrm{inv}(\pi)$ is divisible by $n+1$. Prove that there exist infinitely many primes $p$ such that $f(p-1)>\frac{(p-1)!}{p}$, and infinitely many primes $p$ such that $f(p-1)<\frac{(p-1)!}{p}$. (Proposed by Fedor Petrov, St. Petersburg State University)

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 $T_4$.