Found problems: 85335
2025 Serbia Team Selection Test for the BMO 2025, 6
Let $ABCD$ be a tangential and cyclic quadrilateral. Let $S$ be the intersection point of diagonals $AC$ and $BD$ of the quadrilateral. Let $I$, $I_1$, and $I_2$ be the incenters of quadrilateral $ABCD$ and triangles $ACD$ and $BCS$, respectively. Let the ray $II_2$ intersect the circumcircle of quadrilateral $ABCD$ at point $E$. Prove that the points $D$, $E$, $I_1$, and $I_2$ are collinear or concyclic.
[i]Proposed by Teodor von Burg[/i]
2022 Vietnam National Olympiad, 1
Consider 2 non-constant polynomials $P(x),Q(x)$, with nonnegative coefficients. The coefficients of $P(x)$ is not larger than $2021$ and $Q(x)$ has at least one coefficient larger than $2021$. Assume that $P(2022)=Q(2022)$ and $P(x),Q(x)$ has a root $\frac p q \ne 0 (p,q\in \mathbb Z,(p,q)=1)$. Prove that $|p|+n|q|\le Q(n)-P(n)$ for all $n=1,2,...,2021$
2019 Online Math Open Problems, 21
Let $p$ and $q$ be prime numbers such that $(p-1)^{q-1}-1$ is a positive integer that divides $(2q)^{2p}-1$. Compute the sum of all possible values of $pq$.
[i]Proposed by Ankit Bisain[/i]
1940 Moscow Mathematical Olympiad, 057
Draw a circle that has a given radius $R$ and is tangent to a given line and a given circle. How many solutions does this problem have?
2008 All-Russian Olympiad, 8
On the cartesian plane are drawn several rectangles with the sides parallel to the coordinate axes. Assume that any two rectangles can be cut by a vertical or a horizontal line. Show that it's possible to draw one horizontal and one vertical line such that each rectangle is cut by at least one of these two lines.
2014 Turkey EGMO TST, 4
Let $x,y,z$ be positive real numbers such that $x+y+z \ge x^2+y^2+z^2$. Show that;
$$\dfrac{x^2+3}{x^3+1}+\dfrac{y^2+3}{y^3+1}+\dfrac{z^2+3}{z^3+1}\ge6$$
2001 Turkey Team Selection Test, 3
Show that there is no continuous function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for every real number $x$
\[f(x-f(x)) = \dfrac x2.\]
2020 Canadian Junior Mathematical Olympiad, 5
There are finite many coins in David’s purse. The values of these coins are pair wisely distinct positive integers. Is that possible to make such a purse, such that David has exactly $2020$ different ways to select the coins in his purse and the sum of these selected coins is $2020$?
2011 Tournament of Towns, 1
The faces of a convex polyhedron are similar triangles. Prove that this polyhedron has two pairs of congruent faces.
2019 Iran MO (2nd Round), 3
$x_1,x_2,...,x_n>1$ are natural numbers and $n \geq 3$
Prove that : $(x_1x_2...x_n)^2 \ne x_1^3 + x_2^3 +...+x_n^3$
2011 Math Prize For Girls Problems, 19
If $-1 < x < 1$ and $-1 < y < 1$, define the "relativistic sum'' $x \oplus y$ to be
\[
x \oplus y = \frac{x + y}{1 + xy} \, .
\]
The operation $\oplus$ is commutative and associative. Let $v$ be the number
\[
v = \frac{\sqrt[7]{17} - 1}{\sqrt[7]{17} + 1} \, .
\]
What is the value of
\[
v \oplus v \oplus v \oplus v \oplus v
\oplus v \oplus v \oplus v \oplus v \oplus v
\oplus v \oplus v \oplus v \oplus v \, ?
\]
(In this expression, $\oplus$ appears 13 times.)
2005 Junior Balkan Team Selection Tests - Romania, 3
In a country 6 cities are connected two by two with round-trip air routes operated by exactly one of the two air companies in that country.
Prove that there exist 4 cities $A$, $B$, $C$ and $D$ such that each of the routes $A\leftrightarrow B$, $B\leftrightarrow C$, $C\leftrightarrow D$ and $D\leftrightarrow A$ are operated by the same company.
[i]Dan Schwartz[/i]
2020 Harvard-MIT Mathematics Tournament, 2
Let $n$ be a fixed positive integer. An $n$-staircase is a polyomino with $\frac{n(n+1)}{2}$ cells arranged in the shape of a staircase, with arbitrary size. Here are two examples of $5$-staircases:
[asy]
int s = 5;
for(int i = 0; i < s; i=i+1) { draw((0,0)--(0,i+1)--(s-i,i+1)--(s-i,0)--cycle); }
for(int i = 0; i < s; i=i+1) { draw((10,.67*s)--(10,.67*(s-i-1))--(.67*(s-i)+10,.67*(s-i-1))--(.67*(s-i)+10,.67*s)--cycle); }
[/asy]
Prove that an $n$-staircase can be dissected into strictly smaller $n$-staircases.
[i]Proposed by James Lin.[/i]
1997 Abels Math Contest (Norwegian MO), 2b
Let $A,B,C$ be different points on a circle such that $AB = AC$. Point $E$ lies on the segment $BC$, and $D \ne A$ is the intersection point of the circle and line $AE$. Show that the product $AE \cdot AD$ is independent of the choice of $E$.
CIME I 2018, 14
Let $\triangle ABC$ be a triangle with $AB=6, BC=8, AC=10$, and let $D$ be a point such that if $I_A, I_B, I_C, I_D$ are the incenters of the triangles $BCD,$ $ ACD,$ $ ABD,$ $ ABC$, respectively, the lines $AI_A, BI_B, CI_C, DI_D$ are concurrent. If the volume of tetrahedron $ABCD$ is $\frac{15\sqrt{39}}{2}$, then the sum of the distances from $D$ to $A,B,C$ can be expressed in the form $\frac{a}{b}$ for some positive relatively prime integers $a,b$. Find $a+b$.
[i]Proposed by [b]FedeX333X[/b][/i]
2008 IMC, 1
Let $ n, k$ be positive integers and suppose that the polynomial $ x^{2k}\minus{}x^k\plus{}1$ divides $ x^{2n}\plus{}x^n\plus{}1$. Prove that $ x^{2k}\plus{}x^k\plus{}1$ divides $ x^{2n}\plus{}x^n\plus{}1$.
2010 China Team Selection Test, 2
Given integer $a_1\geq 2$. For integer $n\geq 2$, define $a_n$ to be the smallest positive integer which is not coprime to $a_{n-1}$ and not equal to $a_1,a_2,\cdots, a_{n-1}$. Prove that every positive integer except 1 appears in this sequence $\{a_n\}$.
2019 Indonesia MO, 8
Let $n > 1$ be a positive integer and $a_1, a_2, \dots, a_{2n} \in \{ -n, -n + 1, \dots, n - 1, n \}$. Suppose
\[ a_1 + a_2 + a_3 + \dots + a_{2n} = n + 1 \]
Prove that some of $a_1, a_2, \dots, a_{2n}$ have sum 0.
1992 IMO Longlists, 46
Prove that the sequence $5, 12, 19, 26, 33,\cdots $ contains no term of the form $2^n -1.$
1998 Croatia National Olympiad, Problem 3
Let $A=\{1,2,\ldots,2n\}$ and let the function $g:A\to A$ be defined by $g(k)=2n-k+1$. Does there exist a function $f:A\to A$ such that $f(k)\ne g(k)$ and $f(f(f(k)))=g(k)$ for all $k\in A$, if (a) $n=999$; (b) $n=1000$?
2011 IMO, 1
Given any set $A = \{a_1, a_2, a_3, a_4\}$ of four distinct positive integers, we denote the sum $a_1 +a_2 +a_3 +a_4$ by $s_A$. Let $n_A$ denote the number of pairs $(i, j)$ with $1 \leq i < j \leq 4$ for which $a_i +a_j$ divides $s_A$. Find all sets $A$ of four distinct positive integers which achieve the largest possible value of $n_A$.
[i]Proposed by Fernando Campos, Mexico[/i]
1999 China Team Selection Test, 3
For every permutation $ \tau$ of $ 1, 2, \ldots, 10$, $ \tau \equal{} (x_1, x_2, \ldots, x_{10})$, define $ S(\tau) \equal{} \sum_{k \equal{} 1}^{10} |2x_k \minus{} 3x_{k \minus{} 1}|$. Let $ x_{11} \equal{} x_1$. Find
[b]I.[/b] The maximum and minimum values of $ S(\tau)$.
[b]II.[/b] The number of $ \tau$ which lets $ S(\tau)$ attain its maximum.
[b]III.[/b] The number of $ \tau$ which lets $ S(\tau)$ attain its minimum.
2014 Cuba MO, 5
The number 2013 is written on a blackboard. Two players participate, alternating in turns, in the next game. A movement consists in changing the number that is on the board for the difference of this number and one of its divisors. The player who writes a zero loses. Which of the two players can guarantee victory?
2023 Silk Road, 2
Let $n$ be a positive integer. Each cell of a $2n\times 2n$ square is painted in one of the $4n^2$ colors (with some colors may be missing). We will call any two-cell rectangle a [i]domino[/I], and a domino is called [i]colorful[/I] if its cells have different colors. Let $k$ be the total number of colorful dominoes in our square; $l$ be the maximum integer such that every partition of the square into dominoes contains at least $l$ colorful dominoes. Determine the maximum possible value of $4l-k$ over all possible colourings of the square.
2015 Thailand Mathematical Olympiad, 9
Determine all functions $f : R \to R$ satisfying $f(f(x) + 2y)= 6x + f(f(y) -x)$ for all real numbers $x,y$