Found problems: 85335
2012 Cuba MO, 6
Let $ABC$ be a right triangle at $A$, and let $AD$ be the relative height to the hypotenuse. Let $N$ be the intersection of the bisector of the angle of vertex $C$ with $AD$. Prove that $$AD \cdot BC = AB \cdot DC + BD \cdot AN.$$
2009 Switzerland - Final Round, 4
Let $n$ be a natural number. Each cell of a $n \times n$ square contains one of $n$ different symbols, such that each of the symbols is in exactly $n$ cells. Show that a row or a column exists that contains at least \sqrt{n} different symbols.
2019 Harvard-MIT Mathematics Tournament, 8
In triangle $ABC$ with $AB < AC$, let $H$ be the orthocenter and $O$ be the circumcenter. Given that the midpoint of $OH$ lies on $BC$, $BC = 1$, and the perimeter of $ABC$ is 6, find the area of $ABC$.
1997 Akdeniz University MO, 3
$(x_n)$ be a sequence with $x_1=0$,
$$x_{n+1}=5x_n + \sqrt{24x_n^2+1}$$.
Prove that for $k \geq 2$ $x_k$ is a natural number.
2006 AMC 12/AHSME, 22
Suppose $ a, b,$ and $ c$ are positive integers with $ a \plus{} b \plus{} c \equal{} 2006$, and $ a!b!c! \equal{} m\cdot10^n$, where $ m$ and $ n$ are integers and $ m$ is not divisible by 10. What is the smallest possible value of $ n$?
$ \textbf{(A) } 489 \qquad \textbf{(B) } 492 \qquad \textbf{(C) } 495 \qquad \textbf{(D) } 498 \qquad \textbf{(E) } 501$
2021 IMO Shortlist, C8
Determine the largest integer $N$ for which there exists a table $T$ of integers with $N$ rows and $100$ columns that has the following properties:
$\text{(i)}$ Every row contains the numbers $1$, $2$, $\ldots$, $100$ in some order.
$\text{(ii)}$ For any two distinct rows $r$ and $s$, there is a column $c$ such that $|T(r,c) - T(s, c)|\geq 2$. (Here $T(r,c)$ is the entry in row $r$ and column $c$.)
2009 Argentina Iberoamerican TST, 3
Within a group of $ 2009$ people, every two people has exactly one common friend. Find the least value of the difference between the person with maximum number of friends and the person with minimum number of friends.
1962 AMC 12/AHSME, 31
The ratio of the interior angles of two regular polygons with sides of unit length is $ 3: 2$. How many such pairs are there?
$ \textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ 3 \qquad
\textbf{(D)}\ 4 \qquad
\textbf{(E)}\ \text{infinitely many}$
1992 Miklós Schweitzer, 5
Prove that if the $a_i$'s are different natural numbers, then $\sum_ {j = 1}^n a_j ^ 2 \prod_{k \neq j} \frac{a_j + a_k}{a_j-a_k}$ is a square number.
Estonia Open Junior - geometry, 2012.1.3
A rectangle $ABEF$ is drawn on the leg $AB$ of a right triangle $ABC$, whose apex $F$ is on the leg $AC$. Let $X$ be the intersection of the diagonal of the rectangle $AE$ and the hypotenuse $BC$ of the triangle. In what ratio does point $X$ divide the hypotenuse $BC$ if it is known that $| AC | = 3 | AB |$ and $| AF | = 2 | AB |$?
1991 Vietnam National Olympiad, 1
$1991$ students sit around a circle and play the following game. Starting from some student $A$ and counting clockwise, each student on turn says a number. The numbers are $1,2,3,1,2,3,...$ A student who says $2$ or $3$ must leave the circle. The game is over when there is only one student left. What position was the remaining student sitting at the beginning of the game?
2014 Danube Mathematical Competition, 3
Let $ABC$ be a triangle with $\angle A<90^o, AB \ne AC$. Denote $H$ the orthocenter of triangle $ABC$, $N$ the midpoint of segment $[AH]$, $M$ the midpoint of segment $[BC]$ and $D$ the intersection point of the angle bisector of $\angle BAC$ with the segment $[MN]$. Prove that $<ADH=90^o$
2021/2022 Tournament of Towns, P7
A checkered square of size $2\times2$ is covered by two triangles. Is it necessarily true that:
[list=a]
[*]at least one of its four cells is fully covered by one of the triangles;
[*]some square of size $1\times1$ can be placed into one of these triangles?
[/list]
[i]Alexandr Shapovalov[/i]
2002 Chile National Olympiad, 5
Given a right triangle $T$, where the coordinates of its vertices are integers, let $E$ be the number of points of integer coordinates that belong to the edge of the triangle $T$, $I$ the number of points of integer coordinates that belong to the interior of the triangle $T$. Show that the area $A(T)$ of triangle $T$ is given by: $A(T) = \frac{E}{2}+I -1$.
2017 APMO, 5
Let $n$ be a positive integer. A pair of $n$-tuples $(a_1,\cdots{}, a_n)$ and $(b_1,\cdots{}, b_n)$ with integer entries is called an [i]exquisite pair[/i] if
$$|a_1b_1+\cdots{}+a_nb_n|\le 1.$$
Determine the maximum number of distinct $n$-tuples with integer entries such that any two of them form an exquisite pair.
[i]Pakawut Jiradilok and Warut Suksompong, Thailand[/i]
2020 Princeton University Math Competition, A1/B3
Let $f(x) =\frac{x+a}{x+b}$ satisfy $f(f(f(x))) = x$ for real numbers $a, b$. If the maximum value of a is $p/q$, where $p, q$ are relatively prime integers, what is $|p| + |q|$?
1988 Mexico National Olympiad, 5
If $a$ and $b$ are coprime positive integers and $n$ an integer, prove that the greatest common divisor of $a^2+b^2-nab$ and $a+b$ divides $n+2$.
V Soros Olympiad 1998 - 99 (Russia), 11.9
It is known that unequal numbers $a$,$b$ and $c$ are successive members of an arithmetic progression, all of them are greater than $1000$ and all are squares of natural numbers. Find the smallest possible value of $b$.
2010 Pan African, 1
a) Show that it is possible to pair off the numbers $1,2,3,\ldots ,10$ so that the sums of each of the five pairs are five different prime numbers.
b) Is it possible to pair off the numbers $1,2,3,\ldots ,20$ so that the sums of each of the ten pairs are ten different prime numbers?
1992 National High School Mathematics League, 11
For real numbers $a_1,a_2,\cdots,a_{100}$, $a_1=a_2=1,a_3=2$. For any positive integer $n$, $a_na_{n+1}a_{n+2}\neq1,a_na_{n+1}a_{n+2}a_{n+3}=a_n+a_{n+1}+a_{n+2}+a_{n+3}$, then $a_1+a_2+\cdots+a_{100}=$________.
2010 Purple Comet Problems, 5
If $a$ and $b$ are positive integers such that $a \cdot b = 2400,$ find the least possible value of $a + b.$
2009 AMC 12/AHSME, 3
Twenty percent less than $ 60$ is one-third more than what number?
$ \textbf{(A)}\ 16\qquad
\textbf{(B)}\ 30\qquad
\textbf{(C)}\ 32\qquad
\textbf{(D)}\ 36\qquad
\textbf{(E)}\ 48$
2002 Junior Balkan Team Selection Tests - Romania, 2
We are given $n$ circles which have the same center. Two lines $D_1,D_2$ are concurent in $P$, a point inside all circles. The rays determined by $P$ on the line $D_i$ meet the circles in points $A_1,A_2,...,A_n$ and $A'_1, A'_2,..., A'_n$ respectively and the rays on $D_2$ meet the circles at points $B_1,B_2, ... ,B_n$ and $B'_2, B'_2 ..., B'_n$ (points with the same indices lie on the same circle). Prove that if the arcs $A_1B_1$ and $A_2B_2$ are equal then the arcs $A_iB_i$ and $A'_iB'_i$ are equal, for all $i = 1,2,... n$.
2015 239 Open Mathematical Olympiad, 4
On a circle $4$ points are chosen and for each point we wrote the multiple of its distances to the rest. Could the written numbers be $1,2,3, 4$ in some order?
1953 AMC 12/AHSME, 35
If $ f(x)\equal{}\frac{x(x\minus{}1)}{2}$, then $ f(x\plus{}2)$ equals:
$ \textbf{(A)}\ f(x)\plus{}f(2) \qquad\textbf{(B)}\ (x\plus{}2)f(x) \qquad\textbf{(C)}\ x(x\plus{}2)f(x) \qquad\textbf{(D)}\ \frac{xf(x)}{x\plus{}2}\\
\textbf{(E)}\ \frac{(x\plus{}2)f(x\plus{}1)}{x}$