Found problems: 85335
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}$
2021 Purple Comet Problems, 27
Let $ABCD$ be a cyclic quadrilateral with $AB = 5$, $BC = 10$, $CD = 11$, and $DA = 14$. The value of $AC + BD$ can be written as $\tfrac{n}{\sqrt{pq}}$, where $n$ is a positive integer and $p$ and $q$ are distinct primes. Find $n + p + q$.
2022 Germany Team Selection Test, 2
Let $r>1$ be a rational number. Alice plays a solitaire game on a number line. Initially there is a red bead at $0$ and a blue bead at $1$. In a move, Alice chooses one of the beads and an integer $k \in \mathbb{Z}$. If the chosen bead is at $x$, and the other bead is at $y$, then the bead at $x$ is moved to the point $x'$ satisfying $x'-y=r^k(x-y)$.
Find all $r$ for which Alice can move the red bead to $1$ in at most $2021$ moves.
2022 Taiwan TST Round 2, A
Let $n\geqslant 1$ be an integer, and let $x_0,x_1,\ldots,x_{n+1}$ be $n+2$ non-negative real numbers that satisfy $x_ix_{i+1}-x_{i-1}^2\geqslant 1$ for all $i=1,2,\ldots,n.$ Show that \[x_0+x_1+\cdots+x_n+x_{n+1}>\bigg(\frac{2n}{3}\bigg)^{3/2}.\][i]Pakawut Jiradilok and Wijit Yangjit, Thailand[/i]
2020 Iranian Geometry Olympiad, 4
Let $P$ be an arbitrary point in the interior of triangle $\triangle ABC$. Lines$\overline{BP}$ and $\overline{CP}$
intersect $\overline{AC}$ and $\overline{AB}$ at $E$ and $F$, respectively. Let $K$ and $L$ be the midpoints of the segments $BF$ and $CE$, respectively. Let the lines through $L$ and $K$ parallel to $\overline{CF}$ and $\overline{BE}$ intersect $\overline{BC}$ at $S$ and $T$, respectively; moreover, denote by $M$ and $N$ the reflection of $S$ and $T$ over the points $L$ and $K$, respectively. Prove that as $P$ moves in the interior of triangle $\triangle ABC$, line $\overline{MN}$ passes through a fixed point.
[i]Proposed by Ali Zamani[/i]
2011 Albania National Olympiad, 2
Find all the values that can take the last digit of a "perfect" even number. (The natural number $n$ is called "perfect" if the sum of all its natural divisors is equal twice the number itself.For example: the number $6$ is perfect ,because $1+2+3+6=2\cdot6$).
2002 Portugal MO, 6
On March $6$, $2002$, the celebrations of the $500$th anniversary of the birth of by mathematician Pedro Nunes. That morning, only ten people entered the Viva bookstore for science. Each of these people bought exactly $3$ different books. Furthermore, any two people bought at least one copy of the same book. The Adventures of Mathematics by Pedro Nunes was one of the books that achieved the highest number of sales in this morning. What is the smallest value this number could have taken?
1964 Kurschak Competition, 2
At a party every girl danced with at least one boy, but not with all of them. Similarly, every boy danced with at least one girl, but not with all of them. Show that there were two girls $G$ and $G'$ and two boys $B$ and $B'$, such that each of $B$ and $G$ danced, $B'$ and $G'$ danced, but $B$ and $G'$ did not dance, and $B'$ and $G$ did not dance.
1929 Eotvos Mathematical Competition, 2
Let $k \le n$ be positive integers and $x$ be a real number with $0 \le x < 1/n$. Prove that
$${n \choose 0} - {n \choose 1} x +{n \choose 2} x^2 - ... + (-1)^k {n \choose k} x^k > 0$$
2024 IMO, 6
Let $\mathbb{Q}$ be the set of rational numbers. A function $f: \mathbb{Q} \to \mathbb{Q}$ is called aquaesulian if the following property holds: for every $x,y \in \mathbb{Q}$,
\[ f(x+f(y)) = f(x) + y \quad \text{or} \quad f(f(x)+y) = x + f(y). \]
Show that there exists an integer $c$ such that for any aquaesulian function $f$ there are at most $c$ different rational numbers of the form $f(r) + f(-r)$ for some rational number $r$, and find the smallest possible value of $c$.
1990 China Team Selection Test, 2
Find all functions $f,g,h: \mathbb{R} \mapsto \mathbb{R}$ such that $f(x) - g(y) = (x-y) \cdot h(x+y)$ for $x,y \in \mathbb{R}.$