Found problems: 85335
1972 IMO Longlists, 36
A finite number of parallel segments in the plane are given with the property that for any three of the segments there is a line intersecting each of them. Prove that there exists a line that intersects all the given segments.
2023 Chile Classification NMO Juniors, 4
In the convex quadrilateral $ABCD$, $M$ is the midpoint of side $AD$, $AD = BD$, lines $CM$ and $AB$ are parallel, and $3\angle LBAC = \angle LACD$. Find the measure of angle $\angle ACB$.
2008 iTest Tournament of Champions, 5
It is well-known that the $n^{\text{th}}$ triangular number can be given by the formula $n(n+1)/2$. A Pythagorean triple of $\textit{square numbers}$ is an ordered triple $(a,b,c)$ such that $a^2+b^2=c^2$. Let a Pythagorean triple of $\textit{triangular numbers}$ (a PTTN) be an ordered triple of positive integers $(a,b,c)$ such that $a\leq b<c$ and
\[\dfrac{a(a+1)}2+\dfrac{b(b+1)}2=\dfrac{c(c+1)}2.\]
For instance, $(3,5,6)$ is a PTTN ($6+15=21$). Here we call both $a$ and $b$ $\textit{legs}$ of the PTTN. Find the smallest natural number $n$ such that $n$ is a leg of $\textit{at least}$ six distinct PTTNs.
2015 Purple Comet Problems, 5
The diagram below shows a rectangle with one side divided into seven equal segments and the opposite
side divided in half. The rectangle has area 350. Find the area of the shaded region. For Diagram go to purplecomet.org/welcome/practice, the $2015$ middle school contest, and #5.
2018 APMO, 5
Find all polynomials $P(x)$ with integer coefficients such that for all real numbers $s$ and $t$, if $P(s)$ and $P(t)$ are both integers, then $P(st)$ is also an integer.
2011 All-Russian Olympiad Regional Round, 11.2
2011 non-zero integers are given. It is known that the sum of any one of them with the product of the remaining 2010 numbers is negative. Prove that if all numbers are split arbitrarily into two groups, the sum of the two products will also be negative.
(Authors: N. Agahanov & I. Bogdanov)
1991 Canada National Olympiad, 2
Let $n$ be a fixed positive integer. Find the sum of all positive integers with the property that in base $2$ each has exactly $2n$ digits, consisting of $n$ 1's and $n$ 0's. (The first digit cannot be $0$.)
2013 Purple Comet Problems, 24
Find the remainder when $333^{333}$ is divided by $33$.
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]