Found problems: 85335
2007 JBMO Shortlist, 4
Let $a, b$ be two co-prime positive integers. A number is called [i]good [/i] if it can be written in the form $ax + by$ for non-negative integers $x, y$. Defi ne the function $f : Z\to Z $as $f(n) = n - n_a - n_b$, where $s_t$ represents the remainder of $s$ upon division by $t$. Show that an integer $n$ is [i]good [/i]if and only if the in finite sequence $n, f(n), f(f(n)), ...$ contains only non-negative integers.
ICMC 5, 1
Let $S$ be a set of $2022$ lines in the plane, no two parallel, no three concurrent. $S$ divides the plane into finite regions and infinite regions. Is it possible for all the finite regions to have integer area?
[i]Proposed by Tony Wang[/i]
LMT Team Rounds 2021+, 10
Let $\alpha = \cos^{-1} \left( \frac35 \right)$ and $\beta = \sin^{-1} \left( \frac35 \right) $.
$$\sum_{n=0}^{\infty}\sum_{m=0}^{\infty} \frac{\cos(\alpha n +\beta m)}{2^n3^m}$$
can be written as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A +B$.
2011 Middle European Mathematical Olympiad, 1
Initially, only the integer $44$ is written on a board. An integer a on the board can be re- placed with four pairwise different integers $a_1, a_2, a_3, a_4$ such that the arithmetic mean $\frac 14 (a_1 + a_2 + a_3 + a_4)$ of the four new integers is equal to the number $a$. In a step we simultaneously replace all the integers on the board in the above way. After $30$ steps we end up with $n = 4^{30}$ integers $b_1, b2,\ldots, b_n$ on the board. Prove that
\[\frac{b_1^2 + b_2^2+b_3^2+\cdots+b_n^2}{n}\geq 2011.\]
2020 AIME Problems, 10
Find the sum of all positive integers $n$ such that when $1^3+2^3+3^3+\cdots+n^3$ is divided by $n+5$, the remainder is $17.$
1989 Romania Team Selection Test, 2
The sequence ($a_n$) is defined by $a_1 = a_2 = 1, a_3 = 199$ and $a_{n+1} =\frac{1989+a_na_{n-1}}{a_{n-2}}$ for all $n \ge 3$. Prove that all terms of the sequence are positive integers
2007 Nicolae Păun, 3
Let $ M $ be a finite set of integers, and let be a function $ \varphi :\mathbb{Z}\longrightarrow\mathbb{Z} $ whose restriction to $ \mathbb{Z}\setminus M $ evaluates to a constant $ c, $ such that
$$ 2\le |\varphi (M)|=|M|\neq \frac{1}{c}\cdot \sum_{\iota \in \varphi (M) } \iota . $$
Prove that $ \varphi $ is not a sum between an injective function and a surjective function.
[i]Sorin Rădulescu[/i] and [i]Ion Savu[/i]
1965 All Russian Mathematical Olympiad, 063
Given $n^2$ numbers $x_{i,j}$ ($i,j=1,2,...,n$) satisfying the system of $n^3$ equations $$x_{i,j}+x_{j,k}+x_{k,i}=0 \,\,\, (i,j,k = 1,...,n)$$Prove that there exist such numbers $a_1,a_2,...,a_n$, that $x_{i,j}=a_i-a_j$ for all $i,j=1,...n$.
2022 Korea Winter Program Practice Test, 6
Determine all positive integers $(x_1,x_2,x_3,y_1,y_2,y_3)$ such that $y_1+ny_2^n+n^2y_3^{2n}$ divides $x_1+nx_2^n+n^2x_3^{2n}$ for all positive integer $n$.
2018 SIMO, Bonus
Anana has an ordered $n$-tuple $(a_1,a_2,...,a_n)$ if integers. Banana may make a guess on Anana's ordered integer $n$-tuple $(x_1,x_2,...,x_n)$, upon which Anana will reveal the product of differences $(a_1-x_1)(a_2-x_2)...(a_n-x_n)$. How many guesses does Banana need to figure out Anana's $n$-tuple for certain?
Russian TST 2022, P1
For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\] over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$
[i]Proposed by Shahjalal Shohag, Bangladesh[/i]
Swiss NMO - geometry, 2019.1
Let $A$ be a point and let k be a circle through $A$. Let $B$ and $C$ be two more points on $k$. Let $X$ be the intersection of the bisector of $\angle ABC$ with $k$. Let $Y$ be the reflection of $A$ wrt point $X$, and $D$ the intersection of the straight line $YC$ with $k$. Prove that point $D$ is independent of the choice of $B$ and $C$ on the circle $k$.
2002 AMC 10, 7
If an arc of $ 45^\circ$ on circle $ A$ has the same length as an arc of $ 30^\circ$ on circle $ B$, then the ratio of the area of circle $ A$ to the area of circle $ B$ is
$ \textbf{(A)}\ \frac {4}{9} \qquad \textbf{(B)}\ \frac {2}{3} \qquad \textbf{(C)}\ \frac {5}{6} \qquad \textbf{(D)}\ \frac {3}{2} \qquad \textbf{(E)}\ \frac {9}{4}$
2023 Romania EGMO TST, P1
In town $ A,$ there are $ n$ girls and $ n$ boys, and each girl knows each boy. In town $ B,$ there are $ n$ girls $ g_1, g_2, \ldots, g_n$ and $ 2n \minus{} 1$ boys $ b_1, b_2, \ldots, b_{2n\minus{}1}.$ The girl $ g_i,$ $ i \equal{} 1, 2, \ldots, n,$ knows the boys $ b_1, b_2, \ldots, b_{2i\minus{}1},$ and no others. For all $ r \equal{} 1, 2, \ldots, n,$ denote by $ A(r),B(r)$ the number of different ways in which $ r$ girls from town $ A,$ respectively town $ B,$ can dance with $ r$ boys from their own town, forming $ r$ pairs, each girl with a boy she knows. Prove that $ A(r) \equal{} B(r)$ for each $ r \equal{} 1, 2, \ldots, n.$
2022 IFYM, Sozopol, 2
Finding all quads of integers $(a, b, c, p)$ where $p \ge 5$ is prime number such that the remainders of the numbers $am^3 + bm^2 + cm$, $m = 0, 1, . . . , p - 1$, upon division of $p$ are two by two different..
2018 Mathematical Talent Reward Programme, MCQ: P 7
$A=\{1,2,3,4,5,6,7,8\} .$ How many functions $f: A \rightarrow A$ are there such that $f(1)<f(2)<f(3)$
[list=1]
[*] ${{8}\choose{3}}$
[*] ${{8}\choose{3}}5^{8}$
[*] ${{8}\choose{3}} 8^{5}$
[*] $\frac{8 !}{3 !} $
[/list]
2011 Kyiv Mathematical Festival, 2
Find maximum of the expression $(a -b^2)(b - a^2)$, where $0 \le a,b \le 1$.
1967 IMO Longlists, 19
The $n$ points $P_1,P_2, \ldots, P_n$ are placed inside or on the boundary of a disk of radius 1 in such a way that the minimum distance $D_n$ between any two of these points has its largest possible value $D_n.$ Calculate $D_n$ for $n = 2$ to 7. and justify your answer.
2015 NIMO Problems, 2
Let $ABCD$ be a square with side length $100$. Denote by $M$ the midpoint of $AB$. Point $P$ is selected inside the square so that $MP = 50$ and $PC = 100$. Compute $AP^2$.
[i]Based on a proposal by Amogh Gaitonde[/i]
2008 China Team Selection Test, 3
Let $ 0 < x_{1}\leq\frac {x_{2}}{2}\leq\cdots\leq\frac {x_{n}}{n}, 0 < y_{n}\leq y_{n \minus{} 1}\leq\cdots\leq y_{1},$ Prove that $ (\sum_{k \equal{} 1}^{n}x_{k}y_{k})^2\leq(\sum_{k \equal{} 1}^{n}y_{k})(\sum_{k \equal{} 1}^{n}(x_{k}^2 \minus{} \frac {1}{4}x_{k}x_{k \minus{} 1})y_{k}).$ where $ x_{0} \equal{} 0.$
2023 Yasinsky Geometry Olympiad, 3
Let $I$ be the center of the inscribed circle of the triangle $ABC$. The inscribed circle is tangent to sides $BC$ and $AC$ at points $K_1$ and $K_2$ respectively. Using a ruler and a compass, find the center of excircle for triangle $CK_1K_2$ which is tangent to side $CK_2$, in at most $4$ steps (each step is to draw a circle or a line).
(Hryhorii Filippovskyi, Volodymyr Brayman)
2023 Federal Competition For Advanced Students, P2, 2
Given is a triangle $ABC$ with circumcentre $O$. The circumcircle of triangle $AOC$ intersects side $BC$ at $D$ and side $AB$ at $E$. Prove that the triangles $BDE$ and $AOC$ have circumradiuses of equal length.
2013 IFYM, Sozopol, 8
The irrational numbers $\alpha ,\beta ,\gamma ,\delta$ are such that $\forall$ $n\in \mathbb{N}$ :
$[n\alpha ].[n\beta ]=[n\gamma ].[n\delta ]$.
Is it true that the sets $\{ \alpha ,\beta \}$ and $\{ \gamma ,\delta \}$ are equal?
2016 JBMO Shortlist, 2
The natural numbers from $1$ to $50$ are written down on the blackboard. At least how many of them should be deleted, in order that the sum of any two of the remaining numbers is not a prime?
2023 Baltic Way, 19
Show that $S(2^{2^{2 \cdot 2023}})>2023$, where $S(m)$ denotes the digit sum of $m$.