Positive numbers are written in the squares of a 10 × 10 table. Frogs sit in five squares and cover the numbers in these squares. Kostya found the sum of all visible numbers and got 10. Then each frog jumped to an adjacent square and Kostya’s sum changed to $10^2$. Then the frogs jumped again, and the sum changed to $10^3$ and so on: every new sum was 10 times greater than the previous one. What maximum sum can Kostya obtain?

There are $n \geq 3$ islands in a city. Initially, the ferry company offers some routes between some pairs of islands so that it is impossible to divide the islands into two groups such that no two islands in different groups are connected by a ferry route. After each year, the ferry company will close a ferry route between some two islands $X$ and $Y$. At the same time, in order to maintain its service, the company will open new routes according to the following rule: for any island which is connected to a ferry route to exactly one of $X$ and $Y$, a new route between this island and the other of $X$ and $Y$ is added. Suppose at any moment, if we partition all islands into two nonempty groups in any way, then it is known that the ferry company will close a certain route connecting two islands from the two groups after some years. Prove that after some years there will be an island which is connected to all other islands by ferry routes.

We define a sequence $ \left(a_{1},a_{2},a_{3},\ldots \right)$ by \[ a_{n} \equal{} \frac {1}{n}\left(\left\lfloor\frac {n}{1}\right\rfloor \plus{} \left\lfloor\frac {n}{2}\right\rfloor \plus{} \cdots \plus{} \left\lfloor\frac {n}{n}\right\rfloor\right), \] where $\lfloor x\rfloor$ denotes the integer part of $x$. [b]a)[/b] Prove that $a_{n+1}>a_n$ infinitely often. [b]b)[/b] Prove that $a_{n+1}<a_n$ infinitely often. [i]Proposed by Johan Meyer, South Africa[/i]

Currently, Selena’s analog clock says $4{:}00$. Suddenly her clock breaks, so the hour hand moves $12$ times as fast as it normally does, but the minute hand stays the same speed. Find the degree measure of the smaller angle formed by the minute and the hour hand $2024$ minutes from now.

Let $a,b,c$ be given pairwise coprime positive integers where $a>bc$. Let $m<n$ be positive integers. We call $m$ to be a grandson of $n$ if and only if, for all possible piles of stones whose total mass adds up to $n$ and consist of stones with masses $a,b,c$, it's possible to take some of the stones out from this pile in a way that in the end, we can obtain a new pile of stones with total mass of $m$. Find the greatest possible number that doesn't have any grandsons.

We call a path Valid if i. It only comprises of the following kind of steps: A. $(x, y) \rightarrow (x + 1, y + 1)$ B. $(x, y) \rightarrow (x + 1, y - 1)$ ii. It never goes below the x-axis. Let $M(n)$ = set of all valid paths from $(0,0) $, to $(2n,0)$, where $n$ is a natural number. Consider a Valid path $T \in M(n)$. Denote $\phi(T) = \prod_{i=1}^{2n} \mu_i$, where $\mu_i$= a) $1$, if the $i^{th}$ step is $(x, y) \rightarrow (x + 1, y + 1)$ b) $y$, if the $i^{th} $ step is $(x, y) \rightarrow (x + 1, y - 1)$ Now Let $f(n) =\sum _{T \in M(n)} \phi(T)$. Evaluate the number of zeroes at the end in the decimal expansion of $f(2021)$

Let $a, b, c,x, y, z$ be positive reals such that $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1$. Prove that \[a^x+b^y+c^z\ge \frac{4abcxyz}{(x+y+z-3)^2}.\] [i]Proposed by Daniel Liu[/i]

We call an $m$-dimensional smooth manifold [i]parallelizable[/i] if it admits $m$ smooth tangent vector fields that are linearly independent at all points. Show that if $M$ is a closed orientable $2n$-dimensional smooth manifold of Euler characteristic $0$ that has an immersion into a parallelizable smooth $(2n+1)$-dimensional manifold $N$, then $M$ is itself parallelizable.

Let $n$ be a positive integer and let $d(n)$ denote the number of ordered pairs of positive integers $(x,y)$ such that $(x+1)^2-xy(2x-xy+2y)+(y+1)^2=n$. Find the smallest positive integer $n$ satisfying $d(n) = 61$. (Patrik Bak, Slovakia)

How many positive integers which divide $5n^{11}-2n^5-3n$ for all positive integers $n$ are there? $ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 18 $

Let $a,b,c$ be the lengths of the sides of a triangle, and $\alpha, \beta, \gamma$ respectively, the angles opposite these sides. Prove that if \[ a+b=\tan{\frac{\gamma}{2}}(a\tan{\alpha}+b\tan{\beta}) \] the triangle is isosceles.

Find all natural numbers which are the cardinal of a set of nonzero Euclidean vectors whose sum is $ 0, $ the sum of any two of them is nonzero, and their magnitudes are equal.

Find all primes $p$ and $q$ such that $p$ divides $q^2-4$ and $q$ divides $p^2-1$.

A rectangular box has width $12$ inches, length $16$ inches, and height $\tfrac{m}{n}$ inches, where $m$ and $n$ are relatively prime positive integers. Three rectangular sides of the box meet at a corner of the box. The center points of those three rectangular sides are the vertices of a triangle with area $30$ square inches. Find $m + n.$

Let $P_1$, $P_2$, $\dots$, $P_{2n}$ be $2n$ distinct points on the unit circle $x^2+y^2=1$, other than $(1,0)$. Each point is colored either red or blue, with exactly $n$ red points and $n$ blue points. Let $R_1$, $R_2$, $\dots$, $R_n$ be any ordering of the red points. Let $B_1$ be the nearest blue point to $R_1$ traveling counterclockwise around the circle starting from $R_1$. Then let $B_2$ be the nearest of the remaining blue points to $R_2$ travelling counterclockwise around the circle from $R_2$, and so on, until we have labeled all of the blue points $B_1, \dots, B_n$. Show that the number of counterclockwise arcs of the form $R_i \to B_i$ that contain the point $(1,0)$ is independent of the way we chose the ordering $R_1, \dots, R_n$ of the red points.

For a positive integer $s$, denote with $v_2(s)$ the maximum power of $2$ that divides $s$. Prove that for any positive integer $m$ that: $$v_2\left(\prod_{n=1}^{2^m}\binom{2n}{n}\right)=m2^{m-1}+1.$$ (FYROM)

Consider the obtuse-angled triangle $\triangle ABC$ and its side lengths $a,b,c$. Prove that $a^3\cos\angle A +b^3\cos\angle B + c^3\cos\angle C < abc$.

Find all natural numbers $x$ and $y$ such that $x^y-y^x=1$ .

A rectangular piece of paper has corners labeled $A, B, C$, and $D$, with $BC = 80$ and $CD = 120$. Let $M$ be the midpoint of side $AB$. The corner labeled $A$ is folded along line $MD$ and the corner labeled $B$ is folded along line $MC$ until the segments $AM$ and $MB$ coincide. Let $S$ denote the point in space where $A$ and $B$ meet. If $H$ is the foot of the perpendicular from $S$ to the original plane of the paper, find $HM$.

Let $ABC$ be a triangle, and $A_1$, $B_1$, $C_1$ points on the sides $BC$, $CA$, $AB$, respectively, such that the triangle $A_1B_1C_1$ is equilateral. Let $I_1$ and $\omega_1$ be the incenter and the incircle of $AB_1C_1$. Define $I_2$, $\omega_2$ and $I_3$, $\omega_3$ similarly, with respect to the triangles $BA_1C_1$ and $CA_1B_1$, respectively. Let $l_1 \neq BC$ be the external tangent line to $\omega_2$ and $\omega_3$. Define $l_2$ and $l_3$ similarly, with respect to the pairs $\omega_1$, $\omega_3$ and $\omega_1$, $\omega_2$. Knowing that $A_1I_2 = A_1I_3$, show that the lines $l_1$, $l_2$, $l_3$ are concurrent.

Let $Q$ be a set of prime numbers, not necessarily finite. For a positive integer $n$ consider its prime factorization: define $p(n)$ to be the sum of all the exponents and $q(n)$ to be the sum of the exponents corresponding only to primes in $Q$. A positive integer $n$ is called [i]special[/i] if $p(n)+p(n+1)$ and $q(n)+q(n+1)$ are both even integers. Prove that there is a constant $c>0$ independent of the set $Q$ such that for any positive integer $N>100$, the number of special integers in $[1,N]$ is at least $cN$. (For example, if $Q=\{3,7\}$, then $p(42)=3$, $q(42)=2$, $p(63)=3$, $q(63)=3$, $p(2022)=3$, $q(2022)=1$.)

Let two bijective and continuous functions$f,g: \mathbb{R}\to\mathbb{R}$ such that : $\left(f\circ g^{-1}\right)(x)+\left(g\circ f^{-1}\right)(x)=2x$ for any real $x$. Show that If we have a value $x_{0}\in\mathbb{R}$ such that $f(x_{0})=g(x_{0})$, then $f=g$.

Given an integer $ n > 3.$ Let $ a_{1},a_{2},\cdots,a_{n}$ be real numbers satisfying $ min |a_{i} \minus{} a_{j}| \equal{} 1, 1\le i\le j\le n.$ Find the minimum value of $ \sum_{k \equal{} 1}^n|a_{k}|^3.$

For real numbers $ x$ and $ y$, define $ x\spadesuit y \equal{} (x \plus{} y)(x \minus{} y)$. What is $ 3\spadesuit(4\spadesuit 5)$? $ \textbf{(A) } \minus{} 72 \qquad \textbf{(B) } \minus{} 27 \qquad \textbf{(C) } \minus{} 24 \qquad \textbf{(D) } 24 \qquad \textbf{(E) } 72$

Consider equilateral triangle $ABC$ and $M,N\in (BC)$, $P,Q\in (CA)$, $R,S\in (AB)$ such that $MN=PQ=RS$ and $M\in (BN)$, $P\in(CQ)$, $R\in(AS)$. Prove that there exist three noncollinear points inside hexagon $MNPQRS$ with the same sum of distances to the sides of the hexagon if and only if triangles $ARQ$, $BMS$ and $CPN$ are congruent. [i]Vasile Pop[/i]