Found problems: 6530
2009 Canada National Olympiad, 3
Define $f(x,y,z)=\frac{(xy+yz+zx)(x+y+z)}{(x+y)(y+z)(z+x)}$.
Determine the set of real numbers $r$ for which there exists a triplet of positive real numbers satisfying $f(x,y,z)=r$.
2023 India National Olympiad, 2
Suppose $a_0,\ldots, a_{100}$ are positive reals. Consider the following polynomial for each $k$ in $\{0,1,\ldots, 100\}$:
$$a_{100+k}x^{100}+100a_{99+k}x^{99}+a_{98+k}x^{98}+a_{97+k}x^{97}+\dots+a_{2+k}x^2+a_{1+k}x+a_k,$$where indices are taken modulo $101$, [i]i.e.[/i], $a_{100+i}=a_{i-1}$ for any $i$ in $\{1,2,\dots, 100\}$. Show that it is impossible that each of these $101$ polynomials has all its roots real.
[i]Proposed by Prithwijit De[/i]
2014 Contests, 4
Let $a,b,c$ be real numbers such that $a+b+c = 4$ and $a,b,c > 1$. Prove that:
\[\frac 1{a-1} + \frac 1{b-1} + \frac 1{c-1} \ge \frac 8{a+b} + \frac 8{b+c} + \frac 8{c+a}\]
2018 Saint Petersburg Mathematical Olympiad, 6
Let $a,b,c,d>0$ . Prove that $a^4+b^4+c^4+d^4 \geq 4abcd+4(a-b)^2 \sqrt{abcd}$
2021 Thailand Mathematical Olympiad, 3
Let $a$, $b$, and $c$ be positive real numbers satisfying $ab+bc+ca=abc$. Determine the minimum value of
$$a^abc + b^bca + c^cab.$$
2021 JHMT HS, 3
Let $(x,y)$ be the coordinates of a point chosen uniformly at random within the unit square with vertices at $(0,0), (0,1), (1,0),$ and $(1,1).$ The probability that $|x - \tfrac{1}{2}| + |y - \tfrac{1}{2}| < \tfrac{1}{2}$ is $\tfrac{p}{q},$ where $p$ and $q$ are relatively prime integers. Find $p + q.$
2011 Bogdan Stan, 3
Prove that
$$ a+b+c>\left( \sqrt\alpha +\sqrt\beta +\sqrt\gamma \right)^2, $$
for all positive real numbers $ a,b,c,\alpha ,\beta ,\gamma $ that are under the condition
$$ abc>\alpha bc+\beta ac+\gamma ab. $$
[i]Țuțescu Lucian[/i] and [i]Chiriță Aurel[/i]
2021 Flanders Math Olympiad, 4
(a) Prove that for every $x \in R$ holds that
$$-1 \le \frac{x}{x^2 + x + 1} \le \frac 13$$
(b) Determine all functions $f : R \to R$ for which for every $x \in R$ holds that
$$f \left( \frac{x}{x^2 + x + 1} \right) = \frac{x^2}{x^4 + x^2 + 1}$$
2016 Thailand TSTST, 2
Find the number of sequences $a_1,a_2,\dots,a_{100}$ such that
$\text{(i)}$ There exists $i\in\{1,2,\dots,100\}$ such that $a_i=3$, and
$\text{(ii)}$ $|a_i-a_{i+1}|\leq 1$ for all $1\leq i<100$.
2006 Baltic Way, 2
Suppose that the real numbers $a_i\in [-2,17],\ i=1,2,\ldots,59,$ satisfy $a_1+a_2+\ldots+a_{59}=0.$
Prove that
\[a_1^2+a_2^2+\ldots+a_{59}^2\le 2006\]
2003 Indonesia MO, 6
The hall in a castle is a regular hexagon where its sides' length is 6 meters. The floor of the hall is to be tiled with equilateral triangular tiles where its sides' length is 50 centimeters. Each tile is divided into three congruent triangles by their altitudes up to its orthocenter (see below). Each of these small triangles are colored such that each tile has different colors and no two tiles have identical colorings. How many colors at least are required?
A tile's pattern is:
[asy]
draw((0,0.000)--(2,0.000));
draw((2,0.000)--(1,1.732));
draw((1,1.732)--(0,0.000));
draw((1,0.577)--(0,0.000));
draw((1,0.577)--(2,0.000));
draw((1,0.577)--(1,1.732));
[/asy]
VI Soros Olympiad 1999 - 2000 (Russia), 10.4
Prove that the inequality $ r^2+r_a^2+r_b^2+ r_c^2 \ge 2S$ holds for an arbitrary triangle, where $r$ is the radius of the circle inscribed in the triangle, $r_a$, $r_b$, $r_c$ are the radii of its three excribed circles, $S$ is the area of the triangle.
2019 India IMO Training Camp, P1
Let $a_1,a_2,\ldots, a_m$ be a set of $m$ distinct positive even numbers and $b_1,b_2,\ldots,b_n$ be a set of $n$ distinct positive odd numbers such that
\[a_1+a_2+\cdots+a_m+b_1+b_2+\cdots+b_n=2019\]
Prove that
\[5m+12n\le 581.\]
2024 pOMA, 3
Let $ABC$ be a triangle with circumcircle $\Omega$, and let $P$ be a point on the arc $BC$ of $\Omega$ not containing $A$. Let $\omega_B$ and $\omega_C$ be circles respectively passing through $B$ and $C$ and such that both of them are tangent to line $AP$ at point $P$. Let $R$, $R_B$, $R_C$ be the radii of $\Omega$, $\omega_B$, and $\omega_C$, respectively.
Prove that if $h$ is the distance from $A$ to line $BC$, then
\[
\frac{R_B+R_C}{R} \le \frac{BC}{h}.
\]
2009 Princeton University Math Competition, 5
Suppose that for some positive integer $n$, the first two digits of $5^n$ and $2^n$ are identical. Suppose the first two digits are $a$ and $b$ in this order. Find the two-digit number $\overline{ab}$.
2005 Taiwan TST Round 3, 1
Let ${a_1,a_2,\dots,a_n}$ be positive real numbers, ${n>1}$. Denote by $g_n$ their geometric mean, and by $A_1,A_2,\dots,A_n$ the sequence of arithmetic means defined by \[ A_k=\frac{a_1+a_2+\cdots+a_k}{k},\qquad k=1,2,\dots,n. \] Let $G_n$ be the geometric mean of $A_1,A_2,\dots,A_n$. Prove the inequality \[
n \root n\of{\frac{G_n}{A_n}}+ \frac{g_n}{G_n}\le n+1 \] and establish the cases of equality.
[i]Proposed by Finbarr Holland, Ireland[/i]
2010 Poland - Second Round, 3
Positive integer numbers $k$ and $n$ satisfy the inequality $k > n!$. Prove that there exist pairwisely different prime numbers $p_1, p_2, \ldots, p_n$ which are divisors of the numbers $k+1, k+2, \ldots, k+n$ respectively (i.e. $p_i|k+i$).
2015 Greece JBMO TST, 1
If $x,y,z>0$, prove that $(3x+y)(3y+z)(3z+x) \ge 64xyz$. When we have equality;
1989 Balkan MO, 3
Let $G$ be the centroid of a triangle $ABC$ and let $d$ be a line that intersects $AB$ and $AC$ at $B_{1}$ and $C_{1}$, respectively, such that the points $A$ and $G$ are not separated by $d$.
Prove that: $[BB_{1}GC_{1}]+[CC_{1}GB_{1}] \geq \frac{4}{9}[ABC]$.
1990 All Soviet Union Mathematical Olympiad, 532
If every altitude of a tetrahedron is at least $1$, show that the shortest distance between each pair of opposite edges is more than $2$.
1962 Czech and Slovak Olympiad III A, 2
Determine the set of all points $(x,y)$ in two-dimensional cartesian coordinate system such that \begin{align*}0\le &\,x\le\frac{\pi}{2}, \\ \sqrt{1-\sin 2x}-\sqrt{1+\sin 2x}\le &\,y\le\sqrt{1-\cos2x}-\sqrt{1+\cos2x}.\end{align*}
Draw a picture of the set.
1987 IMO Longlists, 14
Given $n$ real numbers $0 < t_1 \leq t_2 \leq \cdots \leq t_n < 1$, prove that
\[(1-t_n^2) \left( \frac{t_1}{(1-t_1^2)^2}+\frac{t_2}{(1-t_2^3)^2}+\cdots +\frac{t_n}{(1-t_n^{n+1})^2} \right) < 1.\]
2008 Croatia Team Selection Test, 1
Let $ x$, $ y$, $ z$ be positive numbers. Find the minimum value of:
$ (a)\quad \frac{x^2 \plus{} y^2 \plus{} z^2}{xy \plus{} yz}$
$ (b)\quad \frac{x^2 \plus{} y^2 \plus{} 2z^2}{xy \plus{} yz}$
2013 Serbia National Math Olympiad, 6
Find the largest constant $K\in \mathbb{R}$ with the following property:
if $a_1,a_2,a_3,a_4>0$ are numbers satisfying $a_i^2 + a_j^2 + a_k^2 \geq 2
(a_ia_j + a_ja_k + a_ka_i)$, for every $1\leq i<j<k\leq 4$, then \[a_1^2+a_2^2+a_3^2+a_4^2 \geq K
(a_1a_2+a_1a_3+a_1a_4+a_2a_3+a_2a_4+a_3a_4).\]
2011 Today's Calculation Of Integral, 737
Let $a,\ b$ real numbers such that $a>1,\ b>1.$
Prove the following inequality.
\[\int_{-1}^1 \left(\frac{1+b^{|x|}}{1+a^{x}}+\frac{1+a^{|x|}}{1+b^{x}}\right)\ dx<a+b+2\]