Found problems: 85335
2008 Singapore Junior Math Olympiad, 4
Six distinct positive integers $a,b,c.d,e, f$ are given. Jack and Jill calculated the sums of each pair of these numbers. Jack claims that he has $10$ prime numbers while Jill claims that she has $9$ prime numbers among the sums. Who has the correct claim?
2015 Kyiv Math Festival, P3
Is it true that every positive integer greater than $100$ is a sum of $4$ positive integers such that each two of them have a common divisor greater than $1$?
2002 France Team Selection Test, 2
Let $ ABC$ be a non-equilateral triangle. Denote by $ I$ the incenter and by $ O$ the circumcenter of the triangle $ ABC$. Prove that $ \angle AIO\leq\frac{\pi}{2}$ holds if and only if $ 2\cdot BC\leq AB\plus{}AC$.
2008 Tournament Of Towns, 3
In his triangle $ABC$ Serge made some measurements and informed Ilias about the lengths of median $AD$ and side $AC$. Based on these data Ilias proved the assertion: angle $CAB$ is obtuse, while angle $DAB$ is acute. Determine a ratio $AD/AC$ and prove Ilias' assertion (for any triangle with such a ratio).
2006 Sharygin Geometry Olympiad, 2
Points $A, B$ move with equal speeds along two equal circles.
Prove that the perpendicular bisector of $AB$ passes through a fixed point.
2010 Postal Coaching, 4
Five distinct points $A, B, C, D$ and $E$ lie in this order on a circle of radius $r$ and satisfy $AC = BD = CE = r$. Prove that the orthocentres of the triangles $ACD, BCD$ and $BCE$ are the vertices of a right-angled triangle.
2023 Oral Moscow Geometry Olympiad, 1
In trapezoid $ABCD$ with bases $AD, BC$, $AD = 2BC$ and $M$ is midpoint of $AB$. Prove that line $BD$ passes through the midpoint of segment $CM$.
2005 CentroAmerican, 4
Two players, Red and Blue, play in alternating turns on a 10x10 board. Blue goes first. In his turn, a player picks a row or column (not chosen by any player yet) and color all its squares with his own color. If any of these squares was already colored, the new color substitutes the old one.
The game ends after 20 turns, when all rows and column were chosen. Red wins if the number of red squares in the board exceeds at least by 10 the number of blue squares; otherwise Blue wins.
Determine which player has a winning strategy and describe this strategy.
2014 Math Hour Olympiad, 8-10.5
An infinite number of lilypads grow in a line, numbered $\dots$, $-2$, $-1$, $0$, $1$, $2$, $\dots$ Thumbelina and her pet frog start on one of the lilypads. She wants to make a sequence of jumps that will end on either pad $0$ or pad $96$. On each jump, Thumbelina tells her frog the distance (number of pads) to leap, but the frog chooses whether to jump left or right. From which starting pads can she always get to pad $0$ or pad $96$, regardless of her frog's decisions?
2018 Saudi Arabia BMO TST, 3
The partition of $2n$ positive integers into $n$ pairs is called [i]square-free[/i] if the product of numbers in each pair is not a perfect square.Prove that if for $2n$ distinct positive integers, there exists one square-free partition, then there exists at least $n!$ square-free partitions.
2000 Nordic, 4
The real-valued function $f$ is defined for $0 \le x \le 1, f(0) = 0, f(1) = 1$, and $\frac{1}{2} \le \frac{ f(z) - f(y)}{f(y) - f(x)} \le 2$ for all $0 \le x < y < z \le 1$ with $z - y = y -x$. Prove that $\frac{1}{7} \le f (\frac{1}{3} ) \le \frac{4}{7}$.
2009 Croatia Team Selection Test, 3
On sides $ AB$ and $ AC$ of triangle $ ABC$ there are given points $ D,E$ such that $ DE$ is tangent of circle inscribed in triangle $ ABC$ and $ DE \parallel BC$. Prove
$ AB\plus{}BC\plus{}CA\geq 8DE$
2016-2017 SDML (Middle School), 15
A regular hexagon $ABCDEF$ has area $36$. Find the area of the region which lies in the overlap of the triangles $ACE$ and $BDF$.
$\text{(A) }3\qquad\text{(B) }9\qquad\text{(C) }12\qquad\text{(D) }18\qquad\text{(E) }24$
2018 Bosnia And Herzegovina - Regional Olympiad, 3
If numbers $x_1$, $x_2$,...,$x_n$ are from interval $\left( \frac{1}{4},1 \right)$ prove the inequality:
$\log _{x_1} {\left(x_2-\frac{1}{4} \right)} + \log _{x_2} {\left(x_3-\frac{1}{4} \right)}+ ... + \log _{x_{n-1}} {\left(x_n-\frac{1}{4} \right)} + \log _{x_n} {\left(x_1-\frac{1}{4} \right)} \geq 2n$
VI Soros Olympiad 1999 - 2000 (Russia), 11.3
A convex quadrilateral $ABCD$ has an inscribed circle touching its sides $AB$, $BC$, $CD$, $DA$ at the points $M$,$N$,$P$,$K$, respectively. Let $O$ be the center of the inscribed circle, the area of the quadrilateral $MNPK$ is equal to $8$. Prove the inequality $$2S \le OA \cdot OC+ OB \cdot OD.$$
1977 All Soviet Union Mathematical Olympiad, 243
Seven elves are sitting at a round table. Each elf has a cup. Some cups are filled with some milk. Each elf in turn and clockwise divides all his milk between six other cups. After the seventh has done this, every cup was containing the initial amount of milk. How much milk did every cup contain, if there was three litres of milk total?
2018 Purple Comet Problems, 20
Let $ABCD$ be a square with side length $6$. Circles $X, Y$ , and $Z$ are congruent circles with centers inside the square such that $X$ is tangent to both sides $\overline{AB}$ and $\overline{AD}$, $Y$ is tangent to both sides $\overline{AB}$ and $\overline{BC}$, and $Z$ is tangent to side $\overline{CD}$ and both circles $X$ and $Y$ . The radius of the circle $X$ can be written $m -\sqrt{n}$, where $m$ and $n$ are positive integers. Find $m + n$.
2011 Uzbekistan National Olympiad, 3
In acute triangle $ABC$ $AD$ is bisector. $O$ is circumcenter, $H$ is orthocenter. If $AD=AC$ and $AC\perp OH$ . Find all of the value of $\angle ABC$ and $\angle ACB$.
BIMO 2022, 3
Find all functions $ f : \mathbb{R} \rightarrow \mathbb{R} $ such that for all reals $ x, y $,$$ f(x^2+f(x+y))=y+xf(x+1) $$
MathLinks Contest 2nd, 6.1
Determine the parity of the positive integer $N$, where $$N = \lfloor \frac{2002!}{2001 \cdot2003} \rfloor.$$
2016 Irish Math Olympiad, 1
If the three-digit number $ABC$ is divisible by $27$, prove that the three-digit numbers $BCA$ and $CAB$ are also divisible by $27$.
2010 Mediterranean Mathematics Olympiad, 3
Let $A'\in(BC),$ $B'\in(CA),C'\in(AB)$ be the points of tangency of the excribed circles of triangle $\triangle ABC$ with the sides of $\triangle ABC.$ Let $R'$ be the circumradius of triangle $\triangle A'B'C'.$ Show that \[
R'=\frac{1}{2r}\sqrt{2R\left(2R-h_{a}\right)\left(2R-h_{b}\right)\left(2R-h_{c}\right)}\]
where as usual, $R$ is the circumradius of $\triangle ABC,$ r is the inradius of $\triangle ABC,$ and $h_{a},h_{b},h_{c}$ are the lengths of altitudes of $\triangle ABC.$
2011 Oral Moscow Geometry Olympiad, 6
One triangle lies inside another. Prove that at least one of the two smallest sides (out of six) is the side of the inner triangle.
2008 Pan African, 3
Let $a,b,c$ be three positive integers such that $a<b<c$. Consider the the sets $A,B,C$ and $X$, defined as follows: $A=\{ 1,2,\ldots ,a \}$, $B=\{a+1,a+2,\ldots,b\}$, $C=\{b+1,b+2,\ldots ,c\}$ and $X=A\cup B\cup C$.
Determine, in terms of $a,b$ and $c$, the number of ways of placing the elements of $X$ in three boxes such that there are $x,y$ and $z$ elements in the first, second and third box respectively, knowing that:
i) $x\le y\le z$;
ii) elements of $B$ cannot be put in the first box;
iii) elements of $C$ cannot be put in the third box.
1998 IMO Shortlist, 7
A solitaire game is played on an $m\times n$ rectangular board, using $mn$ markers which are white on one side and black on the other. Initially, each square of the board contains a marker with its white side up, except for one corner square, which contains a marker with its black side up. In each move, one may take away one marker with its black side up, but must then turn over all markers which are in squares having an edge in common with the square of the removed marker. Determine all pairs $(m,n)$ of positive integers such that all markers can be removed from the board.