Found problems: 85335
2014 BAMO, 3
Suppose that for two real numbers $x$ and $y$ the following equality is true:
$$(x+ \sqrt{1+ x^2})(y+\sqrt{1+y^2})=1$$
Find (with proof) the value of $x+y$.
2006 Mathematics for Its Sake, 1
[b]a)[/b] Show that there are $ 4 $ equidistant parallel planes that passes through the vertices of the same tetrahedron.
[b]b)[/b] How many such $ \text{4-tuplets} $ of planes does exist, in function of the tetrahedron?
2010 Purple Comet Problems, 16
Half the volume of a 12 foot high cone-shaped pile is grade A ore while the other half is grade B ore. The pile is worth \$62. One-third of the volume of a similarly shaped 18 foot pile is grade A ore while the other two-thirds is grade B ore. The second pile is worth \$162. Two-thirds of the volume of a similarly shaped 24 foot pile is grade A ore while the other one-third is grade B ore. What is the value in dollars (\$) of the 24 foot pile?
1998 Tournament Of Towns, 5
The sum of the length, width, and height of a rectangular parallelepiped will be called its size. Can it happen that one rectangular parallelepiped contains another one of greater size?
(A Shen)
2012 Pre - Vietnam Mathematical Olympiad, 1
Let $n \geq 2$ be a positive integer. Suppose there exist non-negative integers ${n_1},{n_2},\ldots,{n_k}$ such that $2^n - 1 \mid \sum_{i = 1}^k {{2^{{n_i}}}}$. Prove that $k \ge n$.
2018 Estonia Team Selection Test, 12
We call the polynomial $P (x)$ simple if the coefficient of each of its members belongs to the set $\{-1, 0, 1\}$.
Let $n$ be a positive integer, $n> 1$. Find the smallest possible number of terms with a non-zero coefficient in a simple $n$-th degree polynomial with all values at integer places are divisible by $n$.
2000 All-Russian Olympiad Regional Round, 9.5
In a $99\times 101$ table , cubes of natural numbers, as shown in figure . Prove that the sum of all numbers in the table are divisible by $200$.
[img]https://cdn.artofproblemsolving.com/attachments/3/e/dd3d38ca00a36037055acaaa0c2812ae635dcb.png[/img]
2011 Tournament of Towns, 4
Each diagonal of a convex quadrilateral divides it into two isosceles triangles. The two diagonals of the same quadrilateral divide it into four isosceles triangles. Must this quadrilateral be a square?
2014 NIMO Problems, 5
Let $r$, $s$, $t$ be the roots of the polynomial $x^3+2x^2+x-7$. Then \[ \left(1+\frac{1}{(r+2)^2}\right)\left(1+\frac{1}{(s+2)^2}\right)\left(1+\frac{1}{(t+2)^2}\right)=\frac{m}{n} \] for relatively prime positive integers $m$ and $n$. Compute $100m+n$.
[i]Proposed by Justin Stevens[/i]
2006 Baltic Way, 17
Determine all positive integers $n$ such that $3^{n}+1$ is divisible by $n^{2}$.
2006 Sharygin Geometry Olympiad, 9.1
Given a circle of radius $K$. Two other circles, the sum of the radii of which are also equal to $K$, tangent to the circle from the inside. Prove that the line connecting the points of tangency passes through one of the common points of these circles.
2019 EGMO, 1
Find all triples $(a, b, c)$ of real numbers such that $ab + bc + ca = 1$ and
$$a^2b + c = b^2c + a = c^2a + b.$$
1987 IMO Shortlist, 23
Prove that for every natural number $k$ ($k \geq 2$) there exists an irrational number $r$ such that for every natural number $m$,
\[[r^m] \equiv -1 \pmod k .\]
[i]Remark.[/i] An easier variant: Find $r$ as a root of a polynomial of second degree with integer coefficients.
[i]Proposed by Yugoslavia.[/i]
2018 Hong Kong TST, 3
On a rectangular board with $m$ rows and $n$ columns, where $m\leq n$, some squares are coloured black in a way that no two rows are alike. Find the biggest integer $k$ such that for every possible colouring to start with one can always color $k$ columns entirely red in such a way that no two rows are still alike.
2003 Cuba MO, 4
Let $f : N \to N$ such that $f(p) = 1$ for all p prime and $f(ab) =bf(a) + af(b)$ for all $a, b \in N$. Prove that if $n = p^{a_1}_1 p^{a_1}_2... p^{a_1}_k$ is the canonical distribution of $n$ and $p_i$ does not divide $a_i$ ($i = 1, 2, ..., k$) then $\frac{n}{gcd(n,f(n))}$ is square free (not divisible by a square greater than $1$).
Kvant 2023, M2734
Real numbers are placed at the vertices of an $n{}$-gon. On each side, we write the sum of the numbers on its endpoints. For which $n{}$ is it possible that the numbers on the sides form a permutation of $1, 2, 3,\ldots , n$?
[i]From the folklore[/i]
1999 Dutch Mathematical Olympiad, 3
Let $ABCD$ be a square and let $\ell$ be a line. Let $M$ be the centre of the square. The diagonals of the square have length 2 and the distance from $M$ to $\ell$ exceeds 1. Let $A',B',C',D'$ be the orthogonal projections of $A,B,C,D$ onto $\ell$. Suppose that one rotates the square, such that $M$ is invariant. The positions of $A,B,C,D,A',B',C',D'$ change. Prove that the value of $AA'^2 + BB'^2 + CC'^2 + DD'^2$ does not change.
2009 Junior Balkan Team Selection Tests - Romania, 3
The plane is divided into a net of equilateral triangles of side length $1$, with disjoint interiors. A checker is placed initialy inside a triangle. The checker can be moved into another triangle sharing a common vertex (with the triangle hosting the checker) and having the opposite sides (with respect to this vertex) parallel. A path consists in a finite sequence of moves. Prove that there is no path between two triangles sharing a common side.
2005 Mexico National Olympiad, 6
Let $ABC$ be a triangle and $AD$ be the angle bisector of $<BAC$, with $D$ on $BC$. Let $E$ be a point on segment $BC$ such that $BD = EC$. Through $E$ draw $l$ a parallel line to $AD$ and let $P$ be a point in $l$ inside the triangle. Let $G$ be the point where $BP$ intersects $AC$ and $F$ be the point where $CP$ intersects $AB$. Show $BF = CG$.
2003 China Girls Math Olympiad, 4
(1) Prove that there exist five nonnegative real numbers $ a, b, c, d$ and $ e$ with their sum equal to 1 such that for any arrangement of these numbers around a circle, there are always two neighboring numbers with their product not less than $ \frac{1}{9}.$
(2) Prove that for any five nonnegative real numbers with their sum equal to 1 , it is always possible to arrange them around a circle such that there are two neighboring numbers with their product not greater than $ \frac{1}{9}.$
2018 USAMTS Problems, 2:
Given a set of positive integers $R$, we define the [i]friend set[/i] of $R$ to be all positive integers that are divisible by at least one number in $R$. The friend set of $R$ is denoted by $\mathcal{F}(S_1)=\mathcal{F}(S_2)$. Show that $S_1=S_2$.
1986 IMO Longlists, 27
In an urn there are n balls numbered $1, 2, \cdots, n$. They are drawn at random one by one without replacement and the numbers are recorded. What is the probability that the resulting random permutation has only one local maximum?
A term in a sequence is a local maximum if it is greater than all its neighbors.
2016 BMT Spring, 10
An $m \times n$ rectangle is tiled with $\frac{mn}{2}$ $1 \times 2$ dominoes. The tiling is such that whenever the rectangle is partitioned into two smaller rectangles, there exists a domino that is part of the interior of both rectangles. Given $mn > 2$, what is the minimum possible value of $mn$?
For instance, the following tiling of a $4 \times 3$ rectangle doesn’t work because we can partition along the line shown, but that doesn’t necessarily mean other $4 \times 3$ tilings don’t work.
[img]https://cdn.artofproblemsolving.com/attachments/d/3/cb1fed407e45463950542b3cc64185892afdc5.png[/img]
2008 Junior Balkan Team Selection Tests - Romania, 4
Determine the maximum possible real value of the number $ k$, such that
\[ (a \plus{} b \plus{} c)\left (\frac {1}{a \plus{} b} \plus{} \frac {1}{c \plus{} b} \plus{} \frac {1}{a \plus{} c} \minus{} k \right )\ge k\]
for all real numbers $ a,b,c\ge 0$ with $ a \plus{} b \plus{} c \equal{} ab \plus{} bc \plus{} ca$.
1982 IMO Longlists, 57
Let $K$ be a convex polygon in the plane and suppose that $K$ is positioned in the coordinate system in such a way that
\[\text{area } (K \cap Q_i) =\frac 14 \text{area } K \ (i = 1, 2, 3, 4, ),\]
where the $Q_i$ denote the quadrants of the plane. Prove that if $K$ contains no nonzero lattice point, then the area of $K$ is less than $4.$