Found problems: 6530
1988 Romania Team Selection Test, 2
Let $OABC$ be a trihedral angle such that \[ \angle BOC = \alpha, \quad \angle COA = \beta, \quad \angle AOB = \gamma , \quad \alpha + \beta + \gamma = \pi . \] For any interior point $P$ of the trihedral angle let $P_1$, $P_2$ and $P_3$ be the projections of $P$ on the three faces. Prove that $OP \geq PP_1+PP_2+PP_3$.
[i]Constantin Cocea[/i]
2016 Hanoi Open Mathematics Competitions, 13
Find all triples $(a,b,c)$ of real numbers such that $|2a + b| \ge 4$ and $|ax^2 + bx + c| \le 1$ $ \forall x \in [-1, 1]$.
2000 Tuymaada Olympiad, 4
Prove for real $x_1$, $x_2$, ....., $x_n$,
$0 < x_k \leq {1\over 2}$, the inequality
\[ \left( {n \over x_1 + \dots + x_n} - 1 \right)^n \leq \left( {1 \over x_1} - 1 \right) \dots \left( {1 \over x_n} - 1 \right). \]
2000 Moldova National Olympiad, Problem 6
Let $(a_n)_{n\ge0}$ be a sequence of positive numbers that satisfy the relations $a_{i-1}a_{i+1}\le a_i^2$ for all $i\in\mathbb N$. For any integer $n>1$, prove the inequality
$$\frac{a_0+\ldots+a_n}{n+1}\cdot\frac{a_1+\ldots+a_{n-1}}{n-1}\ge\frac{a_0+\ldots+a_{n-1}}n\cdot\frac{a_1+\ldots+a_n}n.$$
2019 India IMO Training Camp, P2
Let $ABC$ be a triangle with $\angle A=\angle C=30^{\circ}.$ Points $D,E,F$ are chosen on the sides $AB,BC,CA$ respectively so that $\angle BFD=\angle BFE=60^{\circ}.$ Let $p$ and $p_1$ be the perimeters of the triangles $ABC$ and $DEF$, respectively. Prove that $p\le 2p_1.$
1981 Czech and Slovak Olympiad III A, 1
Determine all $a\in\mathbb R$ such that the inequality \[x^4+x^3-2(a+1)x^2-ax+a^2<0\] has at least one real solution $x.$
2008 China Girls Math Olympiad, 4
Equilateral triangles $ ABQ$, $ BCR$, $ CDS$, $ DAP$ are erected outside of the convex quadrilateral $ ABCD$. Let $ X$, $ Y$, $ Z$, $ W$ be the midpoints of the segments $ PQ$, $ QR$, $ RS$, $ SP$, respectively. Determine the maximum value of
\[ \frac {XZ\plus{}YW}{AC \plus{} BD}.
\]
2003 Manhattan Mathematical Olympiad, 3
Assume $a,b,c$ are positive numbers, such that
\[ a(1-b) = b(1-c) = c(1-a) = \dfrac14 \]
Prove that $a=b=c$.
2019 239 Open Mathematical Olympiad, 7
Given positive numbers $a_1, \ldots , a_n$, $b_1, \ldots , b_n$, $c_1, \ldots , c_n$. Let $m_k$ be the maximum of the products $a_ib_jc_l$ over the sets $(i, j, l)$ for which $max(i, j, l) = k$. Prove that
$$(a_1 + \ldots + a_n) (b_1 +\ldots + b_n) (c_1 +\ldots + c_n) \leq n^2 (m_1 + \ldots + m_n).$$
2009 Kyrgyzstan National Olympiad, 7
Does $ a^2 \plus{} b^2 \plus{} c^2 \leqslant 2(ab \plus{} bc \plus{} ca)$ hold for every $ a,b,c$ if it is known that $ a^4 \plus{} b^4 \plus{} c^4 \leqslant 2(a^2 b^2 \plus{} b^2 c^2 \plus{} c^2 a^2 )$.
2016 Azerbaijan Junior Mathematical Olympiad, 6
For all reals $x,y,z$ prove that $$\sqrt {x^2+\frac {1}{y^2}}+ \sqrt {y^2+\frac {1}{z^2}}+ \sqrt {z^2+\frac {1}{x^2}}\geq 3\sqrt {2}. $$
2004 Romania National Olympiad, 2
Prove that the equation $x^2+y^2+z^2+t^2=2^{2004}$, where $0 \leq x \leq y \leq z \leq t$, has exactly $2$ solutions in $\mathbb Z$.
[i]Mihai Baluna[/i]
1979 Romania Team Selection Tests, 4.
Give an example of a second degree polynomial $P\in \mathbb{R}[x]$ such that
\[\forall x\in \mathbb{R}\text{ with } |x|\leqslant 1: \;
\left|P(x)+\frac{1}{x-4}\right|
\leqslant 0.01.\]
Are there linear polynomials with this property?
[i]Octavian Stănășilă[/i]
2000 Junior Balkan Team Selection Tests - Moldova, 5
Let the real numbers $a, b, c$ be such that $a \ge b \ge c > 0$. Show that $$\frac{a^2-b^2}{c}+ \frac{c^2-b^2}{a}+ \frac{a^2-c^2}{b}\ge 3a - 4b + c.$$
When does equality hold?
2017 Tuymaada Olympiad, 2
$ABCD $ is a cyclic quadrilateral such that the diagonals $AC $ and $BD $ are perpendicular and their intersection is $P $. Point $Q $ on the segment $CP$ is such that $CQ=AP $. Prove that the perimeter of triangle $BDQ $ is at least $2AC $.
Tuymaada 2017 Q2 Juniors
1999 Greece JBMO TST, 2
For $a,b,c>0$, prove that
(i) $\frac{a+b+c}{2}-\frac{ab}{a+b}-\frac{bc}{b+c}-\frac{ca}{c+a}\ge 0$
(ii) $a(1+b)+b(1+c)+c(1+a)\ge 6\sqrt{abc}$
2008 ITest, 23
Find the number of positive integers $n$ that are solutions to the simultaneous system of inequalities \begin{align*}4n-18&<2008,\\7n+17&>2008.\end{align*}
1964 IMO Shortlist, 2
Suppose $a,b,c$ are the sides of a triangle. Prove that \[ a^2(b+c-a)+b^2(a+c-b)+c^2(a+b-c) \leq 3abc \]
2012 Mathcenter Contest + Longlist, 5 sl13
Define $f : \mathbb{R}^+ \rightarrow \mathbb{R}$ as the strictly increasing function such that
$$f(\sqrt{xy})=\frac{f(x)+f(y)}{2}$$ for all positive real numbers $x,y$. Prove that there are some positive real numbers $a$ where $f(a)<0$.
[i] (PP-nine) [/i]
Gheorghe Țițeica 2025, P2
Let $k\geq 2$ be a positive integer and $x_1,x_2,\dots ,x_k\in (0,1)$. Also, let $m_1,m_2,\dots ,m_k$ and $n_1,n_2,\dots ,n_k$ be integers. Define $$A=x_1^{m_1}x_2^{m_2}\dots x_k^{m_k},\quad B=x_1^{n_1}x_2^{n_2}\dots x_k^{n_k}.$$ Let $$C=x_1^{\min(m_1,n_1)}x_2^{\min(m_2,n_2)}\dots x_k^{\min(m_k,n_k)}$$ $$D=x_1^{\max(m_1,n_1)}x_2^{\max(m_2,n_2)}\dots x_k^{\max(m_k,n_k)}.$$ Prove that $A+B\leq C+D$. When does equality hold?
[i]Dorel Miheț[/i]
1997 Bosnia and Herzegovina Team Selection Test, 4
$a)$ In triangle $ABC$ let $A_1$, $B_1$ and $C_1$ be touching points of incircle $ABC$ with $BA$, $CA$ and $AB$, respectively. Let $l_1$, $l_2$ and $l_3$ be lenghts of arcs $ B_1C_1$, $A_1C_1$, $B_1A_1$ of incircle $ABC$, respectively, which does not contain points $A_1$, $B_1$ and $C_1$, respectively.
Does the following inequality hold: $$ \frac{a}{l_1}+\frac{b}{l_2}+\frac{c}{l_3} \geq \frac{9\sqrt{3}}{\pi}$$
$b)$ Tetrahedron $ABCD$ has three pairs of equal opposing sides. Find length of height of tetrahedron in function od lengths of sides
2013 District Olympiad, 2
Let $a,b\in \mathbb{C}$. Prove that $\left| az+b\bar{z} \right|\le 1$, for every $z\in \mathbb{C}$, with $\left| z \right|=1$, if and only if $\left| a \right|+\left| b \right|\le 1$.
2012 Estonia Team Selection Test, 5
Let $x, y, z$ be positive real numbers whose sum is $2012$. Find the maximum value of $$ \frac{(x^2 + y^2 + z^2)(x^3 + y^3 + z^3)}{(x^4 + y^4 + z^4)}$$
1992 Romania Team Selection Test, 4
Let $x_1,x_2,...,x_n$ be real numbers with $1 \ge x_1 \ge x_2\ge ... \ge x_n \ge 0$ and $x_1^2 +x_2^2+...+x_n^2= 1$.
If $[x_1 +x_2 +...+x_n] = m$, prove that $x_1 +x_2 +...+x_m \ge 1$.
2012 ELMO Shortlist, 9
Let $a,b,c$ be distinct positive real numbers, and let $k$ be a positive integer greater than $3$. Show that
\[\left\lvert\frac{a^{k+1}(b-c)+b^{k+1}(c-a)+c^{k+1}(a-b)}{a^k(b-c)+b^k(c-a)+c^k(a-b)}\right\rvert\ge \frac{k+1}{3(k-1)}(a+b+c)\]
and
\[\left\lvert\frac{a^{k+2}(b-c)+b^{k+2}(c-a)+c^{k+2}(a-b)}{a^k(b-c)+b^k(c-a)+c^k(a-b)}\right\rvert\ge \frac{(k+1)(k+2)}{3k(k-1)}(a^2+b^2+c^2).\]
[i]Calvin Deng.[/i]