Found problems: 85335
2009 Tournament Of Towns, 3
Find all positive integers $a$ and $b$ such that $(a + b^2)(b + a^2) = 2^m$ for some integer $m.$
[i](6 points)[/i]
2010 Kazakhstan National Olympiad, 6
Let $ABCD$ be convex quadrilateral, such that exist $M,N$ inside $ABCD$ for which $\angle NAD= \angle MAB; \angle NBC= \angle MBA; \angle MCB=\angle NCD; \angle NDA=\angle MDC$
Prove, that $S_{ABM}+S_{ABN}+S_{CDM}+S_{CDN}=S_{BCM}+S_{BCN}+S_{ADM}+S_{ADN}$, where $S_{XYZ}$-area of triangle $XYZ$
2024 Sharygin Geometry Olympiad, 9.7
Let $P$ and $Q$ be arbitrary points on the side $BC$ of triangle ABC such that $BP = CQ$. The common points of segments $AP$ and $AQ$ with the incircle form a quadrilateral $XYZT$. Find the locus of common points of diagonals of such quadrilaterals.
2012 Grigore Moisil Intercounty, 2
Let $ \left( x_n \right)_{n\ge 0} $ be a sequence of positive real numbers with $ x_0=1 $ and defined recursively:
$$ x_{n+1}=x_n+\frac{x_0}{x_1+x_2+\cdots +x_n} $$
[b]a)[/b] Show that $ \lim_{n\to\infty } x_n=\infty . $
[b]b)[/b] Calculate $ \lim_{n\to\infty }\frac{x_n}{\sqrt{\ln n}} . $
[i]Ovidiu Furdui[/i]
1997 Croatia National Olympiad, Problem 4
On the sides of a triangle $ABC$ are constructed similar triangles $ABD,BCE,CAF$ with $k=AD/DB=BE/EC=CF/FA$ and $\alpha=\angle ADB=\angle BEC=\angle CFA$. Prove that the midpoints of the segments $AC,BC,CD$ and $EF$ form a parallelogram with an angle $\alpha$ and two sides whose ratio is $k$.
1985 Traian Lălescu, 1.2
For the triangles of fixed perimeter, find the maximum value of the product of the radius of the incircle with the radius of the excircle.
1985 Dutch Mathematical Olympiad, 4
A convex hexagon $ ABCDEF$ is such that each of the diagonals $ AD,BE,CF$ divides the hexagon into two parts of equal area. Prove that these three diagonals are concurrent.
2016 Canadian Mathematical Olympiad Qualification, 6
Determine all ordered triples of positive integers $(x, y, z)$ such that $\gcd(x+y, y+z, z+x) > \gcd(x, y, z)$.
1987 Kurschak Competition, 1
Find all quadruples of positive integers $(a,b,c,d)$ such that $a+b=cd$ and $c+d=ab$.
1955 AMC 12/AHSME, 9
A circle is inscribed in a triangle with sides $ 8$, $ 15$, and $ 17$. The radius of the circle is:
$ \textbf{(A)}\ 6 \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ 5 \qquad
\textbf{(D)}\ 3 \qquad
\textbf{(E)}\ 7$
2013 Junior Balkan Team Selection Tests - Moldova, 1
Given are positive integers $a, b, c$ such that $a$ is odd, $b>c$, $a, b, c$ are coprime and $a(b-c) =2bc$. Prove that $abc$ is square
1992 Kurschak Competition, 1
Define for $n$ given positive reals the [i]strange mean[/i] as the sum of the squares of these numbers divided by their sum. Decide which of the following statements hold for $n=2$:
a) The strange mean is never smaller than the third power mean.
b) The strange mean is never larger than the third power mean.
c) The strange mean, depending on the given numbers, can be larger or smaller than the third power mean.
Which statement is valid for $n=3$?
Gheorghe Țițeica 2025, P3
Let $\mathcal{P}_n$ be the set of all real monic polynomial functions of degree $n$. Prove that for any $a<b$, $$\inf_{P\in\mathcal{P}_n}\int_a^b |P(x)|\, dx >0.$$
[i]Cristi Săvescu[/i]
1949 Putnam, A1
Answer either (i) or (ii):
(i) Let $a>0.$ Three straight lines pass through the three points $(0,-a,a), (a,0,-a)$ and $(-a,a,0),$ parallel to the $x-,y-$ and $z-$axis, respectively. A variable straight line moves so that it has one point in common with each of the three given lines. Find the equation of the surface described by the variable line.
(II) Which planes cut the surface $xy+yz+xz=0$ in (1) circles, (2) parabolas?
2017 China Team Selection Test, 4
An integer $n>1$ is given . Find the smallest positive number $m$ satisfying the following conditions： for any set $\{a,b\}$ $\subset \{1,2,\cdots,2n-1\}$ ,there are non-negative integers $ x, y$ ( not all zero) such that $2n|ax+by$ and $x+y\leq m.$
2010 China Team Selection Test, 2
Let $A=\{a_1,a_2,\cdots,a_{2010}\}$ and $B=\{b_1,b_2,\cdots,b_{2010}\}$ be two sets of complex numbers. Suppose
\[\sum_{1\leq i<j\leq 2010} (a_i+a_j)^k=\sum_{1\leq i<j\leq 2010}(b_i+b_j)^k\]
holds for every $k=1,2,\cdots, 2010$. Prove that $A=B$.
1997 All-Russian Olympiad, 1
Find all integer solutions of the equation $(x^2 - y^2)^2 = 1 + 16y$.
[i]M. Sonkin[/i]
2022 Novosibirsk Oral Olympiad in Geometry, 4
Fold the next seven corners into a rectangle.
[img]https://cdn.artofproblemsolving.com/attachments/b/b/2b8b9d6d4b72024996a66d41f865afb91bb9b7.png[/img]
2015 India PRMO, 14
$14.$ If $3^x+2^y=985.$ and $3^x-2^y=473.$ What is the value of $xy ?$
2020 Baltic Way, 13
Let $ABC$ be an acute triangle with circumcircle $\omega$. Let $\ell$ be the tangent line to $\omega$ at $A$. Let $X$ and $Y$ be the projections of $B$ onto lines $\ell$ and $AC$, respectively. Let $H$ be the orthocenter of $BXY$. Let $CH$ intersect $\ell$ at $D$. Prove that $BA$ bisects angle $CBD$.
2006 Federal Math Competition of S&M, Problem 1
Let $x,y,z$ be positive numbers with the sum $1$. Prove that
$$\frac x{y^2+z}+\frac y{z^2+x}+\frac z{x^2+y}\ge\frac94.$$
1979 IMO Longlists, 51
Let $ABC$ be an arbitrary triangle and let $S_1, S_2,\cdots, S_7$ be circles satisfying the following conditions:
$S_1$ is tangent to $CA$ and $AB$,
$S_2$ is tangent to $S_1, AB$, and $BC,$
$S_3$ is tangent to $S_2, BC$, and $CA,$
..............................
$S_7$ is tangent to $S_6, CA$ and $AB.$
Prove that the circles $S_1$ and $S_7$ coincide.
2009 Hong Kong TST, 5
Let $ a,b,c$ be the three sides of a triangle. Determine all possible values of $ \frac {a^2 \plus{} b^2 \plus{} c^2}{ab \plus{} bc \plus{} ca}$
1996 AMC 8, 22
The horizontal and vertical distances between adjacent points equal $1$ unit. The area of triangle $ABC$ is
[asy]
for (int a = 0; a < 5; ++a)
{
for (int b = 0; b < 4; ++b)
{
dot((a,b));
}
}
draw((0,0)--(3,2)--(4,3)--cycle);
label("$A$",(0,0),SW);
label("$B$",(3,2),SE);
label("$C$",(4,3),NE);
[/asy]
$\text{(A)}\ 1/4 \qquad \text{(B)}\ 1/2 \qquad \text{(C)}\ 3/4 \qquad \text{(D)}\ 1 \qquad \text{(E)}\ 5/4$
2020-2021 OMMC, 11
In equilateral triangle $XYZ$ with side length $10$, define points $A, B$ on $XY,$ points $C, D$ on $YZ,$ and points $E, F$ on $ZX$ such that $ABDE$ and $ACEF$ are rectangles, $XA<XB,$ $YC<YD,$ and $ZE<ZF$. The area of hexagon $ABCDEF$ can be written as $\sqrt{x}$ for some positive integer $x$. Find $x$.