Found problems: 85335
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 $m$.
2014 India IMO Training Camp, 1
Let $p$ be an odd prime and $r$ an odd natural number.Show that $pr+1$ does not divide $p^p-1$
2001 India IMO Training Camp, 2
A strictly increasing sequence $(a_n)$ has the property that $\gcd(a_m,a_n) = a_{\gcd(m,n)}$ for all $m,n\in \mathbb{N}$. Suppose $k$ is the least positive integer for which there exist positive integers $r < k < s$ such that $a_k^2 = a_ra_s$. Prove that $r | k$ and $k | s$.
2014 Contests, 2
Let $\mathbb{Z}$ be the set of integers. Find all functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that \[xf(2f(y)-x)+y^2f(2x-f(y))=\frac{f(x)^2}{x}+f(yf(y))\] for all $x, y \in \mathbb{Z}$ with $x \neq 0$.
2017 CMIMC Combinatorics, 6
Boris plays a game in which he rolls two standard four-sided dice independently and at random, and at the end of the game receives a number of dollars equal to the product of the two rolled numbers. After the initial roll of both dice, however, he can pay two dollars to reroll one die of choice, and he is allowed to pay to reroll as many times as he wishes. If Boris plays to maximize his expected gain, how much money, in dollars, can he expect to win by playing once?
2002 Stanford Mathematics Tournament, 2
Solve for all real $x$ that satisfy the equation $4^x=2^x+6$
2003 Junior Tuymaada Olympiad, 3
In the acute triangle $ ABC $, the point $ I $ is the center of the inscribed the circle, the point $ O $ is the center of the circumscribed circle and the point $ I_a $ is the center the excircle tangent to the side $ BC $ and the extensions of the sides $ AB $ and $ AC $. Point $ A'$ is symmetric to vertex $ A $ with respect to the line $ BC $. Prove that $ \angle IOI_a = \angle IA'I_a $.
2009 IMO, 2
Let $ ABC$ be a triangle with circumcentre $ O$. The points $ P$ and $ Q$ are interior points of the sides $ CA$ and $ AB$ respectively. Let $ K,L$ and $ M$ be the midpoints of the segments $ BP,CQ$ and $ PQ$. respectively, and let $ \Gamma$ be the circle passing through $ K,L$ and $ M$. Suppose that the line $ PQ$ is tangent to the circle $ \Gamma$. Prove that $ OP \equal{} OQ.$
[i]Proposed by Sergei Berlov, Russia [/i]
2014 District Olympiad, 2
Let $M$ be the set of palindromic integers of the form $5n+4$ where $n\ge 0$ is an integer.
[list=a]
[*]If we write the elements of $M$ in increasing order, what is the $50^{\text{th}}$ number?
[*]Among all numbers in $M$ with nonzero digits which sum up to $2014$ which is the largest and smallest one?[/list]
2008 ITest, 32
A right triangle has perimeter $2008$, and the area of a circle inscribed in the triangle is $100\pi^3$. Let $A$ be the area of the triangle. Compute $\lfloor A\rfloor$.
1993 Miklós Schweitzer, 4
Let f be a ternary operation on a set of at least four elements for which
(1) $f ( x , x , y ) \equiv f ( x , y , x ) \equiv f( x , y , y ) \equiv x$
(2) $f ( x , y , z ) = f ( y , z , x ) = f ( y , x , z ) \in \{ x , y , z \}$
for pairwise distinct x,y,z.
Prove that f is a nontrivial composition of g such that g is not a composition of f.
(The n-variable operation g is trivial if $g(x_1, ..., x_n) \equiv x_i$ for some i ($1 \leq i \leq n$) )
1976 IMO Longlists, 10
Show that the reciprocal of any number of the form $2(m^2+m+1)$, where $m$ is a positive integer, can be represented as a sum of consecutive terms in the sequence $(a_j)_{j=1}^{\infty}$
\[ a_j = \frac{1}{j(j + 1)(j + 2)}\]
2016 USAJMO, 4
Find, with proof, the least integer $N$ such that if any $2016$ elements are removed from the set ${1, 2,...,N}$, one can still find $2016$ distinct numbers among the remaining elements with sum $N$.
2012 VJIMC, Problem 4
Find all positive integers $n$ for which there exists a positive integer $k$ such that the decimal representation of $n^k$ starts and ends with the same digit.
2018 India PRMO, 12
Determine the number of $8$-tuples $(\epsilon_1, \epsilon_2,...,\epsilon_8)$ such that $\epsilon_1, \epsilon_2, ..., 8 \in \{1,-1\}$ and $\epsilon_1 + 2\epsilon_2 + 3\epsilon_3 +...+ 8\epsilon_8$ is a multiple of $3$.