Found problems: 6530
2007 Nicolae Coculescu, 3
Show that for any three numbers $ a,b,c\in (1,\infty ) , $ the following inequality is true:
$$ \log_{ab} c +\log_{bc} a +\log_{ca} b\ge log_{a^2bc} bc +log_{b^2ca} ca +log_{c^2ab} ab$$
[i]Costel Anghel[/i]
1996 Turkey MO (2nd round), 2
Let $ABCD$ be a square of side length 2, and let $M$ and $N$ be points on the sides $AB$ and $CD$ respectively. The lines $CM$ and $BN$ meet at $P$, while the lines $AN$ and $DM$ meet at $Q$. Prove that $\left| PQ \right|\ge 1$.
2018 Sharygin Geometry Olympiad, 3
Let $ABC$ be a triangle with $\angle A = 60^\circ$, and $AA', BB', CC'$ be its internal angle bisectors. Prove that $\angle B'A'C' \le 60^\circ$.
2016 Belarus Team Selection Test, 1
Given real numbers $a,b,c,d$ such that $\sin{a}+b >\sin{c}+d, a+\sin{b}>c+\sin{d}$, prove that $a+b>c+d$
2016 Estonia Team Selection Test, 8
Let $x, y$ and $z$ be positive real numbers such that $x + y + z = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$ . Prove that $xy + yz + zx \ge 3$.
2022 IFYM, Sozopol, 6
For the function $f : Z^2_{\ge0} \to Z_{\ge 0}$ it is known that
$$f(0, j) = f(i, 0) = 1, \,\,\,\,\, \forall i, j \in N_0$$
$$f(i, j) = if (i, j - 1) + jf(i - 1, j),\,\,\,\,\, \forall i, j \in N$$
Prove that for every natural number $n$ the following inequality holds:
$$\sum_{0\le i+j\le n+1} f(i, j) \le 2 \left(\sum^n_{k=0}\frac{1}{k!}\right)\left(\sum^n_{p=1}p!\right)+ 3$$
2010 Kazakhstan National Olympiad, 4
For $x;y \geq 0$ prove the inequality:
$\sqrt{x^2-x+1} \sqrt{y^2-y+1}+ \sqrt{x^2+x+1} \sqrt{y^2+y+1} \geq 2(x+y)$
2007 Bulgaria Team Selection Test, 4
Let $G$ is a graph and $x$ is a vertex of $G$. Define the transformation $\varphi_{x}$ over $G$ as deleting all incident edges with respect of $x$ and drawing the edges $xy$ such that $y\in G$ and $y$ is not connected with $x$ with edge in the beginning of the transformation. A graph $H$ is called $G-$[i]attainable[/i] if there exists a sequece of such transformations which transforms $G$ in $H.$ Let $n\in\mathbb{N}$ and $4|n.$ Prove that for each graph $G$ with $4n$ vertices and $n$ edges there exists $G-$[i]attainable[/i] graph with at least $9n^{2}/4$ triangles.
2005 Abels Math Contest (Norwegian MO), 4a
Show that for all positive real numbers $a, b$ and $c$, the inequality $(a+b)(a+c)\ge 2\sqrt{abc(a+b+c)}$ is true.
2024 Malaysia IMONST 2, 1
Suppose $a, b, c, d$ are positive reals such that $a \geq b \geq c \geq d$ and $ab^2c^3d^4 = 1$.
Help Janson prove that $a+b+c+d \geq 4$.
2000 All-Russian Olympiad, 4
Let $a_1, a_2, \cdots, a_n$ be a sequence of nonnegative integers. For $k=1,2,\cdots,n$ denote \[ m_k = \max_{1 \le l \le k} \frac{a_{k-l+1} + a_{k-l+2} + \cdots + a_k}{l}. \] Prove that for every $\alpha > 0$ the number of values of $k$ for which $m_k > \alpha$ is less than $\frac{a_1+a_2+ \cdots +a_n}{\alpha}.$
2011 Hanoi Open Mathematics Competitions, 5
Let $a, b, c$ be positive integers such that $a + 2b +3c = 100$.
Find the greatest value of $M = abc$
1998 Turkey MO (2nd round), 2
If $0\le a\le b\le c$ real numbers, prove that $(a+3b)(b+4c)(c+2a)\ge 60abc$.
2013 Dutch IMO TST, 5
Let $a, b$, and $c$ be positive real numbers satisfying $abc = 1$.
Show that $a + b + c \ge \sqrt{\frac13 (a + 2)(b + 2)(c + 2)}$
2024 Turkey Team Selection Test, 6
For a positive integer $n$ and real numbers $a_1, a_2, \dots ,a_n$ we'll define $b_1, b_2, \dots ,b_{n+1}$ such that $b_k=a_k+\max({a_{k+1},a_{k+2}})$ for all $1\leq k \leq n$ and $b_{n+1}=b_1$. (Also $a_{n+1}=a_1$ and $a_{n+2}=a_2$) Find the least possible value of $\lambda$ such that for all $n, a_1, \dots, a_n$ the inequality
$$\lambda \Biggl[ \sum_{i=1}^n(a_i-a_{i+1})^{2024} \Biggr] \geq \sum_{i=1}^n(b_i-b_{i+1})^{2024}$$
holds.
1989 Romania Team Selection Test, 3
(a) Find the point $M$ in the plane of triangle $ABC$ for which the sum $MA + MB+ MC$ is minimal.
(b) Given a parallelogram $ABCD$ whose angles do not exceed $120^o$, determine $min \{MA+ MB+NC+ND+ MN | M,N$ are in the plane $ABCD\}$ in terms of the sides and angles of the parallelogram.
2023 China Second Round, 11
Find all real numbers $ t $ not less than $1 $ that satisfy the following requirements: for any $a,b\in [-1,t]$ , there always exists $c,d \in [-1,t ]$ such that $ (a+c)(b+d)=1.$
2015 Chile TST Ibero, 4
Let $x, y \in \mathbb{R}^+$. Prove that:
\[
\left( 1 + \frac{1}{x} \right) \left( 1 + \frac{1}{y} \right) \geq \left( 1 + \frac{2}{x + y} \right)^2.
\]
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
1993 Hungary-Israel Binational, 3
Distinct points $A, B , C, D, E$ are given in this order on a semicircle with radius $1$. Prove that
\[AB^{2}+BC^{2}+CD^{2}+DE^{2}+AB \cdot BC \cdot CD+BC \cdot CD \cdot DE < 4.\]
2010 Today's Calculation Of Integral, 562
(1) Show the following inequality for every natural number $ k$.
\[ \frac {1}{2(k \plus{} 1)} < \int_0^1 \frac {1 \minus{} x}{k \plus{} x}dx < \frac {1}{2k}\]
(2) Show the following inequality for every natural number $ m,\ n$ such that $ m > n$.
\[ \frac {m \minus{} n}{2(m \plus{} 1)(n \plus{} 1)} < \log \frac {m}{n} \minus{} \sum_{k \equal{} n \plus{} 1}^{m} \frac {1}{k} < \frac {m \minus{} n}{2mn}\]
2005 Miklós Schweitzer, 12
Let $x_1,x_2,\cdots,x_n$ be iid rv. $S_n=\sum x_k$
(a) let $P(|x_1|\leq 1)=1$ , $E[x_1]=0$ , $E[x_1^2]=\sigma^2>0$
Prove that $\exists C>0$ , $\forall u\geq 2n\sigma^2$
$P(S_n\geq u)\leq e^{-C u \log(u/n\sigma^2)}$
(b) let $P(x_1=1)=P(x_1=-1)=\sigma^2/2$ , $P(x_1=0)=1-\sigma^2$
Prove that $\exists B_1<1,B_2>1,B_3>0$ , $\forall u\geq1, B_1 n\geq u\geq B_2 n\sigma^2$
$P(S_n\geq u)>e^{-B_3 u \log(u/n\sigma^2)}$
1988 Iran MO (2nd round), 2
In tetrahedron $ABCD$ let $h_a, h_b, h_c$ and $h_d$ be the lengths of the altitudes from each vertex to the opposite side of that vertex. Prove that
\[\frac{1}{h_a} <\frac{1}{h_b}+\frac{1}{h_c}+\frac{1}{h_d}.\]
2014 BMO TST, 1
Prove that for $n\ge 2$ the following inequality holds:
$$\frac{1}{n+1}\left(1+\frac{1}{3}+\ldots +\frac{1}{2n-1}\right) >\frac{1}{n}\left(\frac{1}{2}+\ldots+\frac{1}{2n}\right).$$
2015 Peru IMO TST, 14
Let $ n$ be a positive integer and let $ a_1,a_2,\ldots,a_n$ be positive real numbers such that:
\[ \sum^n_{i \equal{} 1} a_i \equal{} \sum^n_{i \equal{} 1} \frac {1}{a_i^2}.
\]
Prove that for every $ i \equal{} 1,2,\ldots,n$ we can find $ i$ numbers with sum at least $ i$.