Found problems: 85335
2025 Malaysian IMO Team Selection Test, 4
Let $ABC$ be a triangle, with incenter $I$ and $A$-excenter $J$. The lines $BI$, $CI$, $BJ$ and $CJ$ intersect the circumcircle of $ABC$ at $P$, $Q$, $R$ and $S$ respectively. Let $IM$, $JN$ be diameters in the circumcircles of triangles $IPQ$ and $JRS$ respectively.
Prove that $\angle BAM+\angle CAN=180^{\circ}$.
[i]Proposed by Ivan Chan Kai Chin[/i]
2006 Bulgaria National Olympiad, 1
Let $p$ be a prime such that $p^2$ divides $2^{p-1}-1$. Prove that for all positive integers $n$ the number $\left(p-1\right)\left(p!+2^n\right)$ has at least $3$ different prime divisors.
[i]Aleksandar Ivanov[/i]
2010 IMO Shortlist, 5
$n \geq 4$ players participated in a tennis tournament. Any two players have played exactly one game, and there was no tie game. We call a company of four players $bad$ if one player was defeated by the other three players, and each of these three players won a game and lost another game among themselves. Suppose that there is no bad company in this tournament. Let $w_i$ and $l_i$ be respectively the number of wins and losses of the $i$-th player. Prove that \[\sum^n_{i=1} \left(w_i - l_i\right)^3 \geq 0.\]
[i]Proposed by Sung Yun Kim, South Korea[/i]
2017 Iran Team Selection Test, 5
$k,n$ are two arbitrary positive integers. Prove that there exists at least $(k-1)(n-k+1)$ positive integers that can be produced by $n$ number of $k$'s and using only $+,-,\times, \div$ operations and adding parentheses between them, but cannot be produced using $n-1$ number of $k$'s.
[i]Proposed by Aryan Tajmir[/i]
1970 AMC 12/AHSME, 7
Inside square $ABCD$ with side $s$, quarter-circle arcs with radii $s$ and centers at $A$ and $B$ are drawn. These arcs intersect at point $X$ inside the square. How far is $X$ from side $CD$?
$\textbf{(A) }\frac{1}{2}s(\sqrt{3}+4)\qquad\textbf{(B) }\frac{1}{2}s\sqrt{3}\qquad\textbf{(C) }\frac{1}{2}s(1+\sqrt{3})\qquad$
$\textbf{(D) }\frac{1}{2}s(\sqrt{3}-1)\qquad \textbf{(E) }\frac{1}{2}s(2-\sqrt{3})$
2005 Georgia Team Selection Test, 3
Let $ x,y,z$ be positive real numbers,satisfying equality $ x^{2}\plus{}y^{2}\plus{}z^{2}\equal{}25$. Find the minimal possible value of the expression $ \frac{xy}{z} \plus{} \frac{yz}{x} \plus{} \frac{zx}{y}$.
2012 Belarus Team Selection Test, 1
For $n$ positive integers $a_1,...,a_n$ consider all their pairwise products $a_ia_j$, $1 \le i < j \le n$. Let $N$ be the number of those products which are the cubes of positive integers.
Find the maximal possible value of $N$ if it is known that none of $a_j$ is a cube of an integer.
(S. Mazanik)
2001 Hungary-Israel Binational, 6
Let be given $32$ positive integers with the sum $120$, none of which is greater than $60.$ Prove that these integers can be divided into two disjoint subsets with the same sum of elements.
TNO 2023 Junior, 1
In the convex quadrilateral $ABCD$, it is given that $\angle BAD = \angle DCB = 90^\circ$, $AB = 7$, $CD = 11$, and that $BC$ and $AD$ are integers greater than 11. Determine the values of $BC$ and $AD$.
2014 PUMaC Algebra A, 3
A function $f$ has its domain equal to the set of integers $0$, $1$, $\ldots$, $11$, and $f(n)\geq 0$ for all such $n$, and $f$ satisfies
[list]
[*]$f(0)=0$
[*]$f(6)=1$
[*]If $x\geq 0$, $y\geq 0$, and $x+y\leq 11$, then $f(x+y)=\tfrac{f(x)+f(y)}{1-f(x)f(y)}$.[/list]
Find $f(2)^2+f(10)^2$.
2013 F = Ma, 14
A cart of mass $m$ moving at $12 \text{ m/s}$ to the right collides elastically with a cart of mass $4.0 \text{ kg}$ that is originally at rest. After the collision, the cart of mass $m$ moves to the left with a velocity of $6.0 \text{ m/s}$. Assuming an elastic collision in one dimension only, what is the velocity of the center of mass ($v_{\text{cm}}$) of the two carts before the collision?
$\textbf{(A) } v_{\text{cm}} = 2.0 \text{ m/s}\\
\textbf{(B) } v_{\text{cm}}=3.0 \text{ m/s}\\
\textbf{(C) } v_{\text{cm}}=6.0 \text{ m/s}\\
\textbf{(D) } v_{\text{cm}}=9.0 \text{ m/s}\\
\textbf{(E) } v_{\text{cm}}=18.0 \text{ m/s}$
2021 Indonesia MO, 8
On a $100 \times 100$ chessboard, the plan is to place several $1 \times 3$ boards and $3 \times 1$ board, so that
[list]
[*] Each tile of the initial chessboard is covered by at most one small board.
[*] The boards cover the entire chessboard tile, except for one tile.
[*] The sides of the board are placed parallel to the chessboard.
[/list]
Suppose that to carry out the instructions above, it takes $H$ number of $1 \times 3$ boards and $V$ number of $3 \times 1$ boards. Determine all possible pairs of $(H,V)$.
[i]Proposed by Muhammad Afifurrahman, Indonesia[/i]
2005 iTest, 1
During the $2005$ iTest, you will be introduced to Joe and Kathryn, two high school seniors. If $J$ is the number of distinct permutations of $JOE$, and $K$ is the number of distinct permutations of $KATHRYN$, find $K -J$.
2008 Flanders Math Olympiad, 1
Determine all natural numbers $n$ of $4$ digits whose quadruple minus the number formed by the digits of $n$ in reverse order equals $30$.
2012 Waseda University Entrance Examination, 1
Answer the following questions:
(1) For complex numbers $\alpha ,\ \beta$, if $\alpha \beta =0$, then prove that $\alpha =0$ or $\beta =0$.
(2) For complex number $\alpha$, if $\alpha^2$ is a positive real number, then prove that $\alpha$ is a real number.
(3) For complex numbers $\alpha_1,\ \alpha_2,\ \cdots,\ \alpha_{2n+1}\ (n=1,\ 2,\ \cdots)$, assume that $\alpha_1\alpha_2,\ \cdots ,\ \alpha_k\alpha_{k+1},\ \cdots,\ \alpha_{2n}\alpha_{2n+1}$ and $\alpha_{2n+1}\alpha_1$ are all positive real numbers. Prove that $\alpha_1,\ \alpha_2,\ \cdots,\ \alpha_{2n+1}$ are all real numbers.
2006 Thailand Mathematical Olympiad, 5
Let $f : Z_{\ge 0} \to Z_{\ge 0}$ satisfy the functional equation $$f(m^2 + n^2) =(f(m) - f(n))^2 + f(2mn)$$ for all nonnegative integers $m, n$. If $8f(0) + 9f(1) = 2006$, compute $f(0)$.
2024 Iranian Geometry Olympiad, 3
Inside a convex quadrilateral $ABCD$ with $BC>AD$, a point $T$ is chosen. $S$ lies on the segment $AT$ such that $DT = BC, \angle TSD = 90^\circ$.
Prove that if $\angle DTA + \angle TAB + \angle ABC = 180^\circ$, then $AB + ST \geqslant CD + AS$.
[i]Proposed by Alexander Tereshin - Russia[/i]
2006 Junior Tuymaada Olympiad, 3
Given a convex $ n $-gon ($ n \geq 5 $). Prove that the number of triangles of area $1$ with vertices at the vertices of the $ n $-gon does not exceed $ \frac{1}{3} n (2n-5) $.
1989 Swedish Mathematical Competition, 3
Find all positive integers $n$ such that $n^3 - 18n^2 + 115n - 391$ is the cube of a positive intege
2011 Today's Calculation Of Integral, 692
Evaluate $\int_0^{\frac{\pi}{12}} \frac{\tan ^ 2 x-3}{3\tan ^ 2 x-1}dx$.
created by kunny
2013 Greece Team Selection Test, 2
Let $ABC$ be a non-isosceles,aqute triangle with $AB<AC$ inscribed in circle $c(O,R)$.The circle $c_{1}(B,AB)$ crosses $AC$ at $K$ and $c$ at $E$.
$KE$ crosses $c$ at $F$ and $BO$ crosses $KE$ at $L$ and $AC$ at $M$ while $AE$ crosses $BF$ at $D$.Prove that:
i)$D,L,M,F$ are concyclic.
ii)$B,D,K,M,E$ are concyclic.
2006 AMC 10, 3
A football game was played between two teams, the Cougars and the Panthers. The two teams scored a total of 34 points, and the Cougars won by a margin of 14 points. How many points did the Panthers score?
$ \textbf{(A) } 10 \qquad \textbf{(B) } 14 \qquad \textbf{(C) } 17 \qquad \textbf{(D) } 20 \qquad \textbf{(E) } 24$
ICMC 8, 6
A set of points in the plane is called rigid if each point is equidistant from the three (or more) points nearest to it.
(a) Does there exist a rigid set of $9$ points?
(b) Does there exist a rigid set of $11$ points?
2018 PUMaC Algebra A, 1
Let
$$a_k = 0.\overbrace{0 \ldots 0}^{k - 1 \: 0's} 1 \overbrace{0 \ldots 0}^{k - 1 \: 0's} 1$$
The value of $\sum_{k = 1}^\infty a_k$ can be expressed as a rational number $\frac{p}{q}$ in simplest form. Find $p + q$.
2005 CentroAmerican, 6
Let $n$ be a positive integer and $p$ a fixed prime. We have a deck of $n$ cards, numbered $1,\ 2,\ldots,\ n$ and $p$ boxes for put the cards on them. Determine all posible integers $n$ for which is possible to distribute the cards in the boxes in such a way the sum of the numbers of the cards in each box is the same.