Found problems: 85335
2015 Junior Regional Olympiad - FBH, 3
Let $AD$ be a median of $ABC$ and $S$ its midpoint. Let $E$ be a intersection point of $AB$ and $CS$. Prove that $BE=2AE$
2021 Cyprus JBMO TST, 2
Find all pairs of natural numbers $(\alpha,\beta)$ for which, if $\delta$ is the greatest common divisor of $\alpha,\beta$, and $\varDelta$ is the least common multiple of $\alpha,\beta$, then
\[ \delta + \Delta = 4(\alpha + \beta) + 2021\]
2013 BMT Spring, 12
Triangle $ABC$ satisfies the property that $\angle A = a \log x$, $\angle B = a \log 2x$, and $\angle C = a \log 4x$ radians, for some real numbers $a$ and $x$. If the altitude to side $AB$ has length $8$ and the altitude to side $BC$ has length $9$, find the area of $\vartriangle ABC$.
2010 Argentina Team Selection Test, 6
Suppose $a_1, a_2, ..., a_r$ are integers with $a_i \geq 2$ for all $i$ such that $a_1 + a_2 + ... + a_r = 2010$.
Prove that the set $\{1,2,3,...,2010\}$ can be partitioned in $r$ subsets $A_1, A_2, ..., A_r$ each with $a_1, a_2, ..., a_r$ elements respectively, such that the sum of the numbers on each subset is divisible by $2011$.
Decide whether this property still holds if we replace $2010$ by $2011$ and $2011$ by $2012$ (that is, if the set to be partitioned is $\{1,2,3,...,2011\}$).
1941 Moscow Mathematical Olympiad, 087
On a plane, several points are chosen so that a disc of radius $1$ can cover every $3$ of them. Prove that a disc of radius $1$ can cover all the points.
2005 Cuba MO, 3
Determine all the quadruples of real numbers that satisfy the following:
[i]The product of any three of these numbers plus the fourth is constant.[/i]
2013 China Western Mathematical Olympiad, 7
Label sides of a regular $n$-gon in clockwise direction in order 1,2,..,n. Determine all integers n ($n\geq 4$) satisfying the following conditions:
(1) $n-3$ non-intersecting diagonals in the $n$-gon are selected, which subdivide the $n$-gon into $n-2$ non-overlapping triangles;
(2) each of the chosen $n-3$ diagonals are labeled with an integer, such that the sum of labeled numbers on three sides of each triangles in (1) is equal to the others;
1988 Tournament Of Towns, (173) 6
The first quadrant of the Cartesian $0-x-y$ plane can be considered to be divided into an infinite set of squares of unit side length, arranged in rows and columns , formed by the axes and lines $x = i$ and $y = j$ , where $i$ and $j$ are non-negative integers. Is it possible to write a natural number $(1,2, 3,...)$ in each square , so that each row and column contains each natural number exactly once?
(V . S . Shevelev)
2023 Saint Petersburg Mathematical Olympiad, 2
Given is a triangle $ABC$ with median $BM$. The point $D$ lies on the line $AC$ after $C$, such that $BD=2CD$. The circle $(BMC)$ meets the segment $BD$ at $N$. Show that $AC+BM>2MN$.
2003 Romania Team Selection Test, 9
Let $n\geq 3$ be a positive integer. Inside a $n\times n$ array there are placed $n^2$ positive numbers with sum $n^3$. Prove that we can find a square $2\times 2$ of 4 elements of the array, having the sides parallel with the sides of the array, and for which the sum of the elements in the square is greater than $3n$.
[i]Radu Gologan[/i]
2022 Dutch IMO TST, 4
Determine all positive integers $d,$ such that there exists an integer $k\geq 3,$ such that
One can arrange the numbers $d,2d,\ldots,kd$ in a row, such that the sum of every two consecutive of them is a perfect square.
1990 IMO Longlists, 38
Let $\alpha$ be the positive root of the quadratic equation $x^2 = 1990x + 1$. For any $m, n \in \mathbb N$, define the operation $m*n = mn + [\alpha m][ \alpha n]$, where $[x]$ is the largest integer no larger than $x$. Prove that $(p*q)*r = p*(q*r)$ holds for all $p, q, r \in \mathbb N.$
1969 IMO, 6
Given real numbers $x_1,x_2,y_1,y_2,z_1,z_2$ satisfying $x_1>0,x_2>0,x_1y_1>z_1^2$, and $x_2y_2>z_2^2$, prove that: \[ {8\over(x_1+x_2)(y_1+y_2)-(z_1+z_2)^2}\le{1\over x_1y_1-z_1^2}+{1\over x_2y_2-z_2^2}. \] Give necessary and sufficient conditions for equality.
2023 Purple Comet Problems, 15
A rectangle with integer side lengths has the property that its area minus $5$ times its perimeter equals $2023$. Find the minimum possible perimeter of this rectangle.
2004 India IMO Training Camp, 4
Let $f$ be a bijection of the set of all natural numbers on to itself. Prove that there exists positive integers $a < a+d < a+ 2d$ such that $f(a) < f(a+d) <f(a+2d)$
1963 Miklós Schweitzer, 6
Show that if $ f(x)$ is a real-valued, continuous function on the half-line $ 0\leq x < \infty$, and \[ \int_0^{\infty} f^2(x)dx
<\infty\] then the function \[ g(x)\equal{}f(x)\minus{}2e^{\minus{}x}\int_0^x e^tf(t)dt\] satisfies \[ \int _0^{\infty}g^2(x)dx\equal{}\int_0^{\infty}f^2(x)dx.\] [B. Szokefalvi-Nagy]
1967 IMO Longlists, 55
Find all $x$ for which, for all $n,$ \[\sum^n_{k=1} \sin {k x} \leq \frac{\sqrt{3}}{2}.\]
2023 Chile National Olympiad, 1
Let $n$ be a natural number such that $n!$ is a multiple of $2023$ and is not divisible by $37$. Find the largest power of $11$ that divides $n!$.
2005 Czech-Polish-Slovak Match, 6
Determine all pairs of integers $(x, y)$ satisfying the equation
\[y(x + y) = x^3- 7x^2 + 11x - 3.\]
2006 Germany Team Selection Test, 3
Suppose that $ a_1$, $ a_2$, $ \ldots$, $ a_n$ are integers such that $ n\mid a_1 \plus{} a_2 \plus{} \ldots \plus{} a_n$.
Prove that there exist two permutations $ \left(b_1,b_2,\ldots,b_n\right)$ and $ \left(c_1,c_2,\ldots,c_n\right)$ of $ \left(1,2,\ldots,n\right)$ such that for each integer $ i$ with $ 1\leq i\leq n$, we have
\[ n\mid a_i \minus{} b_i \minus{} c_i
\]
[i]Proposed by Ricky Liu & Zuming Feng, USA[/i]
1993 All-Russian Olympiad Regional Round, 9.3
Points $M$ and $N$ are chosen on the sides $AB$ and BC of a triangle $ABC$. The segments $AN$ and $CM$ meet at $O$ such that $AO =CO$. Is the triangle $ABC$ necessarily isosceles, if
(a) $AM = CN$?
(b) $BM = BN$?
1998 Croatia National Olympiad, Problem 1
Let there be a given parabola $y^2=4ax$ in the coordinate plane. Consider all chords of the parabola that are visible at a right angle from the origin of the coordinate system. Prove that all these chords pass through a fixed point.
2007 Vietnam National Olympiad, 2
Given a number $b>0$, find all functions $f: \mathbb{R}\rightarrow\mathbb{R}$ such that:
$f(x+y)=f(x).3^{b^{y}+f(y)-1}+b^{x}.\left(3^{b^{y}+f(y)-1}-b^{y}\right) \forall x,y\in\mathbb{R}$
Denmark (Mohr) - geometry, 2000.2
Three identical spheres fit into a glass with rectangular sides and bottom and top in the form of regular hexagons such that every sphere touches every side of the glass. The glass has volume $108$ cm$^3$. What is the sidelength of the bottom?
[img]https://1.bp.blogspot.com/-hBkYrORoBHk/XzcDt7B83AI/AAAAAAAAMXs/P5PGKTlNA7AvxkxMqG-qxqDVc9v9cU0VACLcBGAsYHQ/s0/2000%2BMohr%2Bp2.png[/img]
2005 All-Russian Olympiad Regional Round, 9.3
Two players take turns placing the numbers $1, 2, 3,. . . , 24$, in each of the $24$ squares on the surface of a $2 \times 2 \times 2$ cube (each number can be placed once). The second player wants the sum of the numbers in each cell the rings of $8$ cells encircling the cube were identical. Will he be able to the first player to stop him?