Found problems: 85335
2019 Moldova EGMO TST, 4
Let $x,y>0$ be real numbers.Prove that: $$\frac{1}{x^2+y^2} +\frac{1}{x^2}+\frac{1}{y^2}\ge\frac{10}{(x+y)^2}$$
I tried CBS, but it doesn't work... Can you give an idea, please?
1997 ITAMO, 4
Let $ABCD$ be a tetrahedron. Let $a$ be the length of $AB$ and let $S$ be the area of the projection of the tetrahedron onto a plane perpendicular to $AB$. Determine the volume of the tetrahedron in terms of $a$ and $S$.
2009 International Zhautykov Olympiad, 1
Find all pairs of integers $ (x,y)$, such that
\[ x^2 \minus{} 2009y \plus{} 2y^2 \equal{} 0
\]
2018 Polish Junior MO Second Round, 5
Each integer has been colored in one of three colors. Prove that exist two different numbers of the same color, whose difference is a perfect square.
2001 AMC 8, 22
On a twenty-question test, each correct answer is worth 5 points, each unanswered question is worth 1 point and each incorrect answer is worth 0 points. Which of the following scores is NOT possible?
$ \text{(A)}\ 90\qquad\text{(B)}\ 91\qquad\text{(C)}\ 92\qquad\text{(D)}\ 95\qquad\text{(E)}\ 97 $
2007 Grigore Moisil Intercounty, 1
In a triangle $ ABC $ with $ AB\neq AC, $ let $ D $ be the midpoint of the side $ BC $ and denote with $ E $ the feet of the bisector of $ \angle BAC. $ Also, let $ M,N $ be two points situated in the exterior of $ ABC $ such that $ AMB\sim ANC. $ Prove the following propositions:
$ \text{(a)} MN\perp AD\iff MA\perp AB $
$ \text{(b)} MN\perp AE \iff\angle MAN=180^{\circ } $
2011 AIME Problems, 6
Suppose that a parabola has vertex $\left(\tfrac{1}{4},-\tfrac{9}{8}\right)$, and equation $y=ax^2+bx+c$, where $a>0$ and $a+b+c$ is an integer. The minimum possible value of $a$ can be written as $\tfrac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
2005 Brazil Undergrad MO, 2
Let $f$ and $g$ be two continuous, distinct functions from $[0,1] \rightarrow (0,+\infty)$ such that
$\int_{0}^{1}f(x)dx = \int_{0}^{1}g(x)dx$
Let
$y_n=\int_{0}^{1}{\frac{f^{n+1}(x)}{g^{n}(x)}dx}$, for $n\geq 0$, natural.
Prove that $(y_n)$ is an increasing and divergent sequence.
2013 Puerto Rico Team Selection Test, 4
If $x_0=x_1=1$, and for $n\geq1$
$x_{n+1}=\frac{x_n^2}{x_{n-1}+2x_n}$,
find a formula for $x_n$ as a function of $n$.
2015 HMMT Geometry, 8
Let $S$ be the set of [b]discs[/b] $D$ contained completely in the set $\{ (x,y) : y<0\}$ (the region below the $x$-axis) and centered (at some point) on the curve $y=x^2-\frac{3}{4}$. What is the area of the union of the elements of $S$?
2019 ELMO Shortlist, C5
Given a permutation of $1,2,3,\dots,n$, with consecutive elements $a,b,c$ (in that order), we may perform either of the [i]moves[/i]:
[list]
[*] If $a$ is the median of $a$, $b$, and $c$, we may replace $a,b,c$ with $b,c,a$ (in that order)
[*] If $c$ is the median of $a$, $b$, and $c$, we may replace $a,b,c$ with $c,a,b$ (in that order)
[/list]
What is the least number of sets in a partition of all $n!$ permutations, such that any two permutations in the same set are obtainable from each other by a sequence of moves?
[i]Proposed by Milan Haiman[/i]
2014 Stars Of Mathematics, 3
i) Show there exist (not necessarily distinct) non-negative real numbers $a_1,a_2,\ldots,a_{10}$; $b_1,b_2,\ldots,b_{10}$, with $a_k+b_k \leq 4$ for all $1\leq k \leq 10$, such that $\max\{|a_i-a_j|, |b_i-b_j|\} \geq \dfrac{4}{3} > 1$ for all $1\leq i < j \leq 10$.
ii) Prove for any (not necessarily distinct) non-negative real numbers $a_1,a_2,\ldots,a_{11}$; $b_1,b_2,\ldots,b_{11}$, with $a_k+b_k \leq 4$ for all $1\leq k \leq 11$, there exist $1\leq i < j \leq 11$ such that $\max\{|a_i-a_j|, |b_i-b_j|\} \leq 1$.
([i]Dan Schwarz[/i])
2019 Korea USCM, 4
For any $n\times n$ unitary matrices $A,B$, prove that $|\det (A+2B)|\leq 3^n$.
2025 Bangladesh Mathematical Olympiad, P8
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that$$f(xf(y)-y) + f(xy-x) + f(x+y) = 2xy$$for all $x, y \in \mathbb{R}$.
2020 Junior Balkan Team Selection Tests-Serbia, 1#
Given is triangle $ABC$ with arbitrary point $D$ on $AB$ and central of inscribed circle $I$. The perpendicular bisector of $AB$ intersects $AI$ and $BI$ at $P$ and $Q$, respectively. The circle $(ADP)$ intersects $CA$ at $E$, and the circle $(BDQ)$ intersects $BC$ at $F$ and $(ADP)$ intersects $(BDQ)$ at $K$. Prove that $E, F, K, I$ lie on one circle.
2021 Regional Olympiad of Mexico West, 4
Some numbers from $1$ to $100$ are painted red so that the following two conditions are met:
$\bullet$ The number $1 $ is painted red.
$\bullet$ If the numbers other than $a$ and $b$ are painted red then no number between $a$ and $b$ divides the number $ab$.
What is the maximum number of numbers that can be painted red?
2001 Estonia Team Selection Test, 1
Consider on the coordinate plane all rectangles whose
(i) vertices have integer coordinates;
(ii) edges are parallel to coordinate axes;
(iii) area is $2^k$, where $k = 0,1,2....$
Is it possible to color all points with integer coordinates in two colors so that no such rectangle has all its vertices of the same color?
1998 IberoAmerican Olympiad For University Students, 4
Four circles of radius $1$ with centers $A,B,C,D$ are in the plane in such a way that each circle is tangent to two others. A fifth circle passes through the center of two of the circles and is tangent to the other two.
Find the possible values of the area of the quadrilateral $ABCD$.
2021 Ukraine National Mathematical Olympiad, 4
Find all the following functions $f:R\to R$ , which for arbitrary valid $x,y$ holds equality: $$f(xf(x+y))+f((x+y)f(y))=(x+y)^2$$
(Vadym Koval)
1967 Miklós Schweitzer, 9
Let $ F$ be a surface of nonzero curvature that can be represented around one of its points $ P$ by a power series and is symmetric around the normal planes parallel to the principal directions at $ P$. Show that the derivative with respect to the arc length of the curvature of an arbitrary normal section at $ P$ vanishes at $ P$. Is it possible to replace the above symmetry condition by a weaker one?
[i]A. Moor[/i]
2004 Purple Comet Problems, 20
A $70$ foot pole stands vertically in a plane supported by three $490$ foot wires, all attached to the top of the pole, pulled taut, and anchored to three equally spaced points in the plane. How many feet apart are any two of those anchor points?
2010 Peru IMO TST, 5
Let $\Bbb{N}$ be the set of positive integers. For each subset $\mathcal{X}$ of $\Bbb{N}$ we define the set $\Delta(\mathcal{X})$ as the set of all numbers $| m - n |,$ where $m$ and $n$ are elements of $\mathcal{X}$, ie: $$\Delta (\mathcal{X}) = \{ |m-n| \ | \ m, n \in \mathcal{X} \}$$ Let $\mathcal A$ and $\mathcal B$ be two infinite, disjoint sets whose union is $\Bbb{N.}$
a) Prove that the set $\Delta (\mathcal A) \cap \Delta (\mathcal B)$ has infinitely many elements.
b) Prove that there exists an infinite subset $\mathcal C$ of $\Bbb{N}$ such that $\Delta (\mathcal C)$ is a subset of $\Delta (\mathcal A) \cap \Delta (\mathcal B).$
2004 Germany Team Selection Test, 1
Let $a_{ij}$ $i=1,2,3$; $j=1,2,3$ be real numbers such that $a_{ij}$ is positive for $i=j$ and negative for $i\neq j$.
Prove the existence of positive real numbers $c_{1}$, $c_{2}$, $c_{3}$ such that the numbers \[a_{11}c_{1}+a_{12}c_{2}+a_{13}c_{3},\qquad a_{21}c_{1}+a_{22}c_{2}+a_{23}c_{3},\qquad a_{31}c_{1}+a_{32}c_{2}+a_{33}c_{3}\] are either all negative, all positive, or all zero.
[i]Proposed by Kiran Kedlaya, USA[/i]
2022-23 IOQM India, 5
Let $m$ be the smallest positive integer such that $m^2+(m+1)^2+\cdots+(m+10)^2$ is the square of a positive integer $n$. Find $m+n$
PEN H Problems, 10
Prove that there are unique positive integers $a$ and $n$ such that \[a^{n+1}-(a+1)^{n}= 2001.\]