Let $n$ be a positive integer with $d$ digits, all different from zero. For $k = 0,. . . , d - 1$, we define $n_k$ as the number obtained by moving the last $k$ digits of $n$ to the beginning. For example, if $n = 2184$ then $n_0 = 2184, n_1 = 4218, n_2 = 8421, n_3 = 1842$. For $m$ a positive integer, define $s_m(n)$ as the number of values $k$ ​​such that $n_k$ is a multiple of $m.$ Finally, define $a_d$ as the number of integers $n$ with $d$ digits all nonzero, for which $s_2 (n) + s_3 (n) + s_5 (n) = 2d.$ Find \[\lim_{d \to \infty} \frac{a_d}{5^d}.\]

Let $P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0$ and $Q(x)=b_nx^n+b_{n-1}x^{n-1}+\cdots+b_0$ be two polynomials with integral coefficients such that $a_n-b_n$ is a prime and $a_nb_0-a_0b_n\neq 0,$ and $a_{n-1}=b_{n-1}.$ Suppose that there exists a rational number $r$ such that $P(r)=Q(r)=0.$ Prove that $r\in\mathbb Z.$

A function $f(x)$ is such that for any integer $x$, $f(x)+xf(2-x)=6$. Compute $-2019f(2020)$.

A cat is tied to one corner of the base of a tower. The base forms an equilateral triangle of side length $4$ m, and the cat is tied with a leash of length $8$ m. Let $A$ be the area of the region accessible to the cat. If we write $A = \frac{m}{n} k - \sqrt{\ell}$, where $m,n, k, \ell$ are positive integers such that $m$ and $n$ are relatively prime, and $\ell$ is squarefree, what is the value of $m + n + k + \ell$ ?

Find the largest $r$ such that $4$ balls each of radius $r$ can be packed into a regular tetrahedron with side length $1$. In a packing, each ball lies outside every other ball, and every ball lies inside the boundaries of the tetrahedron. If $r$ can be expressed in the form $\frac{\sqrt{a}+b}{c}$ where $a, b, c$ are integers such that $\gcd(b, c) = 1$, what is $a + b + c$?

Determine if there is an infinite sequence $a_1,a_2,a_3,...,a_n$ of positive integers such that for all $n\ge 1$ the sum $a_1^2+a_2^2+a_3^2+...^2+a_n^2$ is a perfect square

Let $a, b, c, d$ be four positive real numbers. Prove that $$\frac{(a + b + c)^2}{a^2+b^2+c^2}+\frac{(b + c + d)^3}{b^3+c^3+d^3}+\frac{(c+d+a)^4}{c^4+d^4+a^4}+\frac{(d+a+b)^5}{d^5+a^5+b^5}\le 120$$

A hunter and an invisible rabbit play a game in the Euclidean plane. The rabbit's starting point, $A_0,$ and the hunter's starting point, $B_0$ are the same. After $n-1$ rounds of the game, the rabbit is at point $A_{n-1}$ and the hunter is at point $B_{n-1}.$ In the $n^{\text{th}}$ round of the game, three things occur in order: [list=i] [*]The rabbit moves invisibly to a point $A_n$ such that the distance between $A_{n-1}$ and $A_n$ is exactly $1.$ [*]A tracking device reports a point $P_n$ to the hunter. The only guarantee provided by the tracking device to the hunter is that the distance between $P_n$ and $A_n$ is at most $1.$ [*]The hunter moves visibly to a point $B_n$ such that the distance between $B_{n-1}$ and $B_n$ is exactly $1.$ [/list] Is it always possible, no matter how the rabbit moves, and no matter what points are reported by the tracking device, for the hunter to choose her moves so that after $10^9$ rounds, she can ensure that the distance between her and the rabbit is at most $100?$ [i]Proposed by Gerhard Woeginger, Austria[/i]

In a group of kids there are $2022$ boys and $2023$ girls. Every girl is a friend with exactly $2021$ boys. Friendship is a symmetric relation: if A is a friend of B, then B is also a friend of A. Prove that it is not possible that all boys have the same number of girl friends. [i]Proposed by the JMMO Problem Selection Committee[/i]

Let $m$ and $n$ be positive integers. A child walks the Cartesian plane taking a few steps. The child begins its journey at the point $(0, n)$ and ends at the point $(m, 0)$ in such a way that: $\bullet$ Each step has length $1$ and is parallel to either the $X$ or $Y$ axis. $\bullet$ For each point $(x, y)$ of its path it is true that $x\ge 0$ and $y\ge 0$. For each step of the child, the distance between the child and the axis to which said step is parallel is calculated. If the step causes the child to be further from the point $(0, 0)$ than before, we consider that distance as positive, otherwise, we consider that distance as negative. Prove that at the end of the boy's journey, the sum of all the distances is $0$.

Let $x, y, z$ be positive real numbers. Prove that: $\frac{x + 2y}{z + 2x + 3y}+\frac{y + 2z}{x + 2y + 3z}+\frac{z + 2x}{y + 2z + 3x} \le \frac{3}{2}$

The venusian prophet Zabruberson sent to his pupils a $ 10000$-letter word, each letter being $ A$ or $ E$: the [i]Zabrubic word[/i]. Their pupils consider then that for $ 1 \leq k \leq 10000$, each word comprised of $ k$ consecutive letters of the Zabrubic word is a [i]prophetic word[/i] of length $ k$. It is known that there are at most $ 7$ prophetic words of lenght $ 3$. Find the maximum number of prophetic words of length $ 10$.

Let $ a$, $ b$, and $ c$ be non-negative real numbers and let $ x$, $ y$, and $ z$ be positive real numbers such that $ a\plus{}b\plus{}c\equal{}x\plus{}y\plus{}z$. Prove that \[ \dfrac{a^3}{x^2}\plus{}\dfrac{b^3}{y^2}\plus{}\dfrac{c^3}{z^2} \ge a\plus{}b\plus{}c.\] [i]Hery Susanto, Malang[/i]

At the $69$ Thailand-Yaranaikian meeting attended by $96$ Thai delegates and a number (unknown) from the Yaranakian country. Some time after the meeting took place, the meeting also discovered something amazing that happened in this meeting!! That is, regardless of whether we select at least $69$ of Thai participants and select all the Yaranikian country participants who are known to Thais in the initial selection group, there is at least $1$ person fo form a minority. They found in that minority, there was always $1$ more Yaranikhians than Thais. Prove that there must be at least $28$ of the Yaranaikian attendees who know the Thai delegates. (Note: In this meeting, none of the attendees were half-breeds. Thai-Yara Nikian) [i](tatari/nightmare)[/i]

For a positive integer $n$, let $f(n)$ be the integer formed by reversing the digits of $n$ (and removing any leading zeroes). For example $f(14172)=27141$. Define a sequence of numbers $\{a_n\}_{n\ge 0}$ by $a_0=1$ and for all $i\ge 0$, $a_{i+1}=11a_i$ or $a_{i+1}=f(a_i)$ . How many possible values are there for $a_8$? [i]Proposed by James Lin[/i]

Given an integer $n\ge3$, prove that the set $X=\{1,2,3,\ldots,n^2-n\}$ can be divided into two non-intersecting subsets such that neither of them contains $n$ elements $a_1,a_2,\ldots,a_n$ with $a_1<a_2<\ldots<a_n$ and $a_k\le\frac{a_{k-1}+a_{k+1}}2$ for all $k=2,\ldots,n-1$.

Define a sequence of polynomials $F_n(x)$ by $F_0(x)=0, F_1(x)=x-1$, and for $n\geq 1$, $$F_{n+1}(x)=2xF_n(x)-F_{n-1}(x)+2F_1(x).$$ For each $n$, $F_n(x)$ can be written in the form $$F_n(x)=c_nP_1(x)P_2(x)\cdots P_{g(n)}(x)$$ where $c_n$ is a constant and $P_1(x),P_2(x)\cdots, P_{g(n)}(x)$ are non-constant polynomials with integer coefficients and $g(n)$ is as large as possible. For all $2< n< 101$, let $t$ be the minimum possible value of $g(n)$ in the above expression; for how many $k$ in the specified range is $g(k)=t$?

Let $D=\{ z \in \mathbb{C} : \operatorname{Im}(z) >0 , \operatorname{Re}(z) >0 \} $. Let $n \geq 1 $ and let $a_1,a_2,\dots a_n \in D$ be distinct complex numbers. Define $$f(z)=z \cdot \prod_{j=1}^{n} \frac{z-a_j}{z-\overline{a_j}}$$ Prove that $f'$ has at least one root in $D$. [i]Proposed by Géza Kós (Lorand Eotvos University, Budapest)[/i]

Let $ABC$ be a triangle with $AB = 5$, $AC = 4$, $BC = 6$. The angle bisector of $C$ intersects side $AB$ at $X$. Points $M$ and $N$ are drawn on sides $BC$ and $AC$, respectively, such that $\overline{XM} \parallel \overline{AC}$ and $\overline{XN} \parallel \overline{BC}$. Compute the length $MN$.

Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that the following two conditions hold: [b]1. [/b] For all real numbers $a$ and $b$ satisfying $a^2 + b^2 = 1$, We have $f(x) + f(y) \geq f(ax + by)$ for all real numbers $x, y$. [b]2.[/b] For all real numbers $x$ and $y$, there exist real numbers $a$ and $b$, such that $a^2 + b^2 = 1$ and $f(x) + f(y) = f(ax + by)$. [i]Proposed by Zaza Melikidze, Georgia[/i]

A circle is circumscribed about a triangle with sides $ 20$, $ 21$, and $ 29$, thus dividing the interior of the circle into four regions. Let $ A$, $ B$, and $ C$ be the areas of the non-triangular regions, with $ C$ being the largest. Then $ \textbf{(A)}\ A \plus{} B \equal{} C\qquad \textbf{(B)}\ A \plus{} B \plus{} 210 \equal{} C\qquad \textbf{(C)}\ A^2 \plus{} B^2 \equal{} C^2\qquad \\ \textbf{(D)}\ 20A \plus{} 21B \equal{} 29C\qquad \textbf{(E)}\ \frac{1}{A^2} \plus{} \frac{1}{B^2} \equal{} \frac{1}{C^2}$

Jenny and Kenny are walking in the same direction, Kenny at 3 feet per second and Jenny at 1 foot per second, on parallel paths that are 200 feet apart. A tall circular building 100 feet in diameter is centered midway between the paths. At the instant when the building first blocks the line of sight between Jenny and Kenny, they are 200 feet apart. Let $t$ be the amount of time, in seconds, before Jenny and Kenny can see each other again. If $t$ is written as a fraction in lowest terms, what is the sum of the numerator and denominator?

Find all integer numbers $a, b, m$ and $n$, such that the following two equalities are verified: $a^2+b^2=5mn$ and $m^2+n^2=5ab$

For prime numbers $p$ and $q$, natural numbers $n$, $k$, $r$, the equality $p^{2k}+q^{2n}=r^2$ holds. Prove that the number $r$ is prime.

$(a_n)$ is an arithmetic sequence, $(b_n)$ is a geometric sequence. If $b_1=a_1^2,b_2=a_2^2,b_3=a_3^2(a_1<a_2)$, and $\lim_{n\to\infty}(b_1+b_2+\cdots+b_n)=\sqrt2+1$, find $a_n$.