Found problems: 85335
2020 Polish Junior MO First Round, 7.
Consider the right prism with the rhombus with side $a$ and acute angle $60^{\circ}$ as a base. This prism was intersected by some plane intersecting its side edges, such that the cross-section of the prism and the plane is a square. Determine all possible lengths of the side of this square.
1988 AMC 8, 11
$ \sqrt{164} $ is
$ \text{(A)}\ 42\qquad\text{(B)}\ \text{less than }10\qquad\text{(C)}\ \text{between }10\text{ and }11\qquad\text{(D)}\ \text{between }11\text{ and }12\qquad\text{(E)}\ \text{between }12\text{ and }13 $
2001 Moldova National Olympiad, Problem 5
Consider all quadratic trinomials $x^2+px+q$ with $p,q\in\{1,\ldots,2001\}$. Which of them has more elements: those having integer roots, or those having no real roots?
2016 IOM, 1
Find all positive integers $n$ such that there exist $n$ consecutive positive integers whose sum is a perfect square.
2022 Thailand TSTST, 3
Let $S$ be the set of the positive integers greater than $1$, and let $n$ be from $S$. Does there exist a function $f$ from $S$ to itself such that for all pairwise distinct positive integers $a_1, a_2,...,a_n$ from $S$, we have $f(a_1)f(a_2)...f(a_n)=f(a_1^na_2^n...a_n^n)$?
1985 AMC 12/AHSME, 6
One student in a class of boys and girls is chosen to represent the class. Each student is equally likely to be chosen and the probability that a boy is chosen is $ \frac23$ of the probability that a girl is chosen. The ratio of the number of boys to the total number of boys and girls is
$ \textbf{(A)}\ \frac13 \qquad \textbf{(B)}\ \frac25 \qquad \textbf{(C)}\ \frac12 \qquad \textbf{(D)}\ \frac35 \qquad \textbf{(E)}\ \frac23$
2019 Purple Comet Problems, 22
Let $a$ and $b$ positive real numbers such that $(65a^2 + 2ab + b^2)(a^2 + 8ab + 65b^2) = (8a^2 + 39ab + 7b^2)^2$. Then one possible value of $\frac{a}{b}$ satis es $2 \left(\frac{a}{b}\right) = m +\sqrt{n}$, where $m$ and $n$ are positive integers. Find $m + n$.
2006 China Team Selection Test, 3
Given positive integers $m$ and $n$ so there is a chessboard with $mn$ $1 \times 1$ grids. Colour the grids into red and blue (Grids that have a common side are not the same colour and the grid in the left corner at the bottom is red). Now the diagnol that goes from the left corner at the bottom to the top right corner is coloured into red and blue segments (Every segment has the same colour with the grid that contains it). Find the sum of the length of all the red segments.
2014 China Team Selection Test, 3
$A$ is the set of points of a convex $n$-gon on a plane. The distinct pairwise distances between any $2$ points in $A$ arranged in descending order is $d_1>d_2>...>d_m>0$. Let the number of unordered pairs of points in $A$ such that their distance is $d_i$ be exactly $\mu _i$, for $i=1, 2,..., m$.
Prove: For any positive integer $k\leq m$, $\mu _1+\mu _2+...+\mu _k\leq (3k-1)n$.
2007 Iran Team Selection Test, 1
In triangle $ABC$, $M$ is midpoint of $AC$, and $D$ is a point on $BC$ such that $DB=DM$. We know that $2BC^{2}-AC^{2}=AB.AC$. Prove that \[BD.DC=\frac{AC^{2}.AB}{2(AB+AC)}\]
1997 Iran MO (3rd Round), 6
Let $\mathcal P$ be the set of all points in $\mathbb R^n$ with rational coordinates. For the points $A,B \in \mathcal l{P}$, one can move from $A$ to $B$ if the distance $AB$ is $1$. Prove that every point in $\mathcal l{ P}$ can be reached from any other point in $\mathcal{P}$ by a finite sequence of moves if and only if $n \geq 5$.
1913 Eotvos Mathematical Competition, 3
Let $d$ denote the greatest common divisor of the natural numbers $a$ and $b$, and let $d'$ denote the greatest common divisor of the natural numbers $a'$ and $b'$. Prove that $dd'$ is the greatest common divisor of the four numbers $$ aa' , \ \ ab' , \ \ ba' , \ \ bb' .$$
1989 IMO Longlists, 85
Let a regular $ (2n \plus{}1)\minus{}$gon be inscribed in a circle of radius $ r.$ We consider all the triangles whose vertices are from those of the regular $ (2n \plus{} 1)\minus{}$gon.
[b](a)[/b] How many triangles among them contain the center of the circle in their interior?
[b](b)[/b] Find the sum of the areas of all those triangles that contain the center of the circle in their interior.
1978 IMO Longlists, 19
We consider three distinct half-lines $Ox, Oy, Oz$ in a plane. Prove the existence and uniqueness of three points $A \in Ox, B \in Oy, C \in Oz$ such that the perimeters of the triangles $OAB,OBC,OCA$ are all equal to a given number $2p > 0.$
2024 Mathematical Talent Reward Programme, 3
The smallest positive integer which can be expressed as sum of positive perfect cubes (possibly with repetition and/or with a single element sum) in at least two different ways in
$$(A) 8$$
$$(B) 1729$$
$$(C) 2023$$
$$(D) 2024$$
2009 Dutch Mathematical Olympiad, 2
Consider the sequence of integers $0, 1, 2, 4, 6, 9, 12,...$ obtained by starting with zero, adding $1$, then adding $1$ again, then adding $2$, and adding $2$ again, then adding $3$, and adding $3$ again, and so on. If we call the subsequent terms of this sequence $a_0, a_1, a_2, ...$, then we have $a_0 = 0$, and $a_{2n-1} = a_{2n-2} + n$ , $a_{2n} = a_{2n-1} + n$ for all integers $n \ge 1$.
Find all integers $k \ge 0$ for which $a_k$ is the square of an integer.
1996 AIME Problems, 4
A wooden cube, whose edges are one centimeter long, rests on a horizontal surface. Illuminated by a point source of light that is $x$ centimeters directly above an upper vertex, the cube casts a shadow on the horizontal surface. The area of the shadow, which does not inclued the area beneath the cube is 48 square centimeters. Find the greatest integer that does not exceed $1000x.$
2017 Germany Team Selection Test, 1
The leader of an IMO team chooses positive integers $n$ and $k$ with $n > k$, and announces them to the deputy leader and a contestant. The leader then secretly tells the deputy leader an $n$-digit binary string, and the deputy leader writes down all $n$-digit binary strings which differ from the leader’s in exactly $k$ positions. (For example, if $n = 3$ and $k = 1$, and if the leader chooses $101$, the deputy leader would write down $001, 111$ and $100$.) The contestant is allowed to look at the strings written by the deputy leader and guess the leader’s string. What is the minimum number of guesses (in terms of $n$ and $k$) needed to guarantee the correct answer?
2003 Croatia National Olympiad, Problem 3
In an isosceles triangle with base $a$, lateral side $b$, and height to the base $v$, it holds that $\frac a2+v\ge b\sqrt2$. Find the angles of the triangle. Compute its area if $b=8\sqrt2$.
2014 Tournament of Towns., 5
There are several white and black points. Every white point is connected with every black point by a segment. Each segment is equipped with a positive integer. For any closed circuit the product of the integers on the segments passed in the direction from white to black point is equal to the product of the integers on the segments passed in the opposite direction. Can one always place the integer at each point so that the integer on each segment is the product of the integers at its ends?
1975 Czech and Slovak Olympiad III A, 6
Let $\mathbf M\subseteq\mathbb R^2$ be a set with the following properties:
1) there is a pair $(a,b)\in\mathbf M$ such that $ab(a-b)\neq0,$
2) if $\left(x_1,y_1\right),\left(x_2,y_2\right)\in\mathbf M$ and $c\in\mathbb R$ then also \[\left(cx_1,cy_1\right),\left(x_1+x_2,y_1+y_2\right),\left(x_1x_2,y_1y_2\right)\in\mathbf M.\]
Show that in fact \[\mathbf M=\mathbb R^2.\]
2019 Saint Petersburg Mathematical Olympiad, 4
Given a convex quadrilateral $ABCD$. The medians of the triangle $ABC$ intersect at point $M$, and the medians of the triangle $ACD$ at point$ N$. The circle, circumscibed around the triangle $ACM$, intersects the segment $BD$ at the point $K$ lying inside the triangle $AMB$ . It is known that $\angle MAN = \angle ANC = 90^o$. Prove that $\angle AKD = \angle MKC$.
2005 AIME Problems, 5
Robert has 4 indistinguishable gold coins and 4 indistinguishable silver coins. Each coin has an engraving of one face on one side, but not on the other. He wants to stack the eight coins on a table into a single stack so that no two adjacent coins are face to face. Find the number of possible distinguishable arrangements of the 8 coins.
1993 Tournament Of Towns, (382) 4
Three players Alexander, Beverley and Catherine participate in a tournament (all of them play the same number of games with each other). Is it possible that Alexander gets more points than the others, Catherine gets less points than the others, but Alexander has a smaller number of wins than the others and Catherine has a greater number of wins than the others? (A win scores $1$ point, a draw scores $\frac12$.)
(A Rubin,)
1987 IMO Longlists, 29
Is it possible to put $1987$ points in the Euclidean plane such that the distance between each pair of points is irrational and each three points determine a non-degenerate triangle with rational area? [i](IMO Problem 5)[/i]
[i]Proposed by Germany, DR[/i]