Found problems: 85335
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.
2012 JHMT, 2
A circle with radius $1$ has diameter $AB$. $C$ lies on this circle such that ratio of lengths of arcs $AC /BC= 4$. $\overline{AC}$ divides the circle into two parts, and we will label the smaller part Region I. Similarly, $\overline{BC}$ also divides the circle into two parts, and we will denote the smaller one as Region II. Find the positive difference between the areas of Regions I and II.
2013 Hanoi Open Mathematics Competitions, 5
The number of integer solutions $x$ of the equation below
$(12x -1)(6x - 1)(4x -1)(3x - 1) = 330$ is
(A): $0$, (B): $1$, (C): $2$, (D): $3$, (E): None of the above.
2011 Junior Balkan Team Selection Tests - Romania, 4
Let $k$ and $n$ be integer numbers with $2 \le k \le n - 1$. Consider a set $A$ of $n$ real numbers such that the sum of any $k$ distinct elements of $A$ is a rational number. Prove that all elements of the set $A$ are rational numbers.
2018 PUMaC Live Round, 5.2
Find $x^2$ given that $\tan^{-1}(x)+\tan^{-1}(3x)=\frac{\pi}{6}$ and $0<x<\frac{\pi}{6}$.
1988 IMO Shortlist, 20
Find the least natural number $ n$ such that, if the set $ \{1,2, \ldots, n\}$ is arbitrarily divided into two non-intersecting subsets, then one of the subsets contains 3 distinct numbers such that the product of two of them equals the third.
the 15th XMO, 3
$k$ is an integer, there exists a triangulation for a regular polygon with $2024$ sides and $2024$ colored dots with $k$ different colors meeting
$(1)$ each color will be used at least once
$(2)$ every small triangle will have at least $2$ dots that will be in the same color.
Try to find the maximum value of$k$
1986 China Team Selection Test, 3
Let $x_i,$ $1 \leq i \leq n$ be real numbers with $n \geq 3.$ Let $p$ and $q$ be their symmetric sum of degree $1$ and $2$ respectively. Prove that:
i) $p^2 \cdot \frac{n-1}{n}-2q \geq 0$
ii) $\left|x_i - \frac{p}{n}\right| \leq \sqrt{p^2 - \frac{2nq}{n-1}} \cdot \frac{n-1}{n}$ for every meaningful $i$.
1968 Putnam, A4
Let $S^{2}\subset \mathbb{R}^{3}$ be the unit sphere. Show that for any $n$ points on $ S^{2}$, the sum of the squares of the $\frac{n(n-1)}{2}$ distances between them is at most $n^{2}$.
2024 Thailand October Camp, 5
Find the maximal number of points, such that there exist a configuration of $2023$ lines on the plane, with each lines pass at least $2$ points.
2018 239 Open Mathematical Olympiad, 8-9.1
Given a prime number $p$. A positive integer $x$ is divided by $p$ with a remainder, and the number $p^2$ is divided by $x$ with a remainder. The remainders turned out to be equal. Find them
[i]Proposed by Sergey Berlov[/i]