Found problems: 6530
2014 Contests, 3
Let $n$ be an even positive integer, and let $G$ be an $n$-vertex graph with exactly $\tfrac{n^2}{4}$ edges, where there are no loops or multiple edges (each unordered pair of distinct vertices is joined by either 0 or 1 edge). An unordered pair of distinct vertices $\{x,y\}$ is said to be [i]amicable[/i] if they have a common neighbor (there is a vertex $z$ such that $xz$ and $yz$ are both edges). Prove that $G$ has at least $2\textstyle\binom{n/2}{2}$ pairs of vertices which are amicable.
[i]Zoltán Füredi (suggested by Po-Shen Loh)[/i]
2013 China Girls Math Olympiad, 3
In a group of $m$ girls and $n$ boys, any two persons either know each other or do not know each other. For any two boys and any two girls, there are at least one boy and one girl among them,who do not know each other. Prove that the number of unordered pairs of (boy, girl) who know each other does not exceed $m+\frac{n(n-1)}{2}$.
2012 Olympic Revenge, 1
Let $a$ and $b$ real numbers. Let $f:[a,b] \rightarrow \mathbb{R}$ a continuous function. We say that f is "smp" if $[a,b]=[c_0,c_1]\cup[c_1,c_2]...\cup[c_{n-1},c_n]$ satisfying $c_0<c_1...<c_n$ and for each $i\in\{0,1,2...n-1\}$:
$c_i<x<c_{i+1} \Rightarrow f(c_i)<f(x)<f(c_{i+1})$
or
$c_i>x>c_{i+1} \Rightarrow f(c_i)>f(x)>f(c_{i+1})$
Prove that if $f:[a,b] \rightarrow \mathbb{R}$ is continuous such that for each $v\in\mathbb{R}$ there are only finitely many $x$ satisfying $f(x)=v$, then $f$ is "smp".
2011 Estonia Team Selection Test, 4
Let $a,b,c$ be positive real numbers such that $2a^2 +b^2=9c^2$.Prove that $\displaystyle \frac{2c}{a}+\frac cb \ge\sqrt 3$.
2006 Iran MO (3rd Round), 3
In triangle $ABC$, if $L,M,N$ are midpoints of $AB,AC,BC$. And $H$ is orthogonal center of triangle $ABC$, then prove that \[LH^{2}+MH^{2}+NH^{2}\leq\frac14(AB^{2}+AC^{2}+BC^{2})\]
1989 China National Olympiad, 2
Let $x_1, x_2, \dots ,x_n$ ($n\ge 2$) be positive real numbers satisfying $\sum^{n}_{i=1}x_i=1$. Prove that:\[\sum^{n}_{i=1}\dfrac{x_i}{\sqrt{1-x_i}}\ge \dfrac{\sum_{i=1}^{n}\sqrt{x_i}}{\sqrt{n-1}}.\]
2017 Singapore MO Open, 2
Let $a_1,a_2,...,a_n,b_1,b_2,...,b_n,p$ be real numbers with $p >- 1$. Prove that
$$\sum_{i=1}^{n}(a_i-b_i)\left(a_i (a_1^2+a_2^2+...+a_n^2)^{p/2}-b_i (b_1^2+b_2^2+...+b_n^2)^{p/2}\right)\ge 0$$
1984 Dutch Mathematical Olympiad, 3
For $n = 1,2,3,...$. $a_n$ is defined by:
$$a_n =\frac{1 \cdot 4 \cdot 7 \cdot ... (3n-2)}{2 \cdot 5 \cdot 8 \cdot ... (3n-1)}$$
Prove that for every $n$ holds that
$$\frac{1}{\sqrt{3n+1}}\le a_n \le \frac{1}{\sqrt[3]{3n+1}}$$
2007 Germany Team Selection Test, 1
Prove the inequality:
\[\sum_{i < j}{\frac {a_{i}a_{j}}{a_{i} \plus{} a_{j}}}\leq \frac {n}{2(a_{1} \plus{} a_{2} \plus{}\cdots \plus{} a_{n})}\cdot \sum_{i < j}{a_{i}a_{j}}\]
for positive reals $ a_{1},a_{2},\ldots,a_{n}$.
[i]Proposed by Dusan Dukic, Serbia[/i]
2012 German National Olympiad, 5
Let $a,b$ be the lengths of two nonadjacent edges of a tetrahedron with inradius $r$. Prove that \[r<\frac{ab}{2(a+b)}.\]
2008 Serbia National Math Olympiad, 3
Let $ a$, $ b$, $ c$ be positive real numbers such that $ a \plus{} b \plus{} c \equal{} 1$. Prove inequality:
\[ \frac{1}{bc \plus{} a \plus{} \frac{1}{a}} \plus{} \frac{1}{ac \plus{} b \plus{} \frac{1}{b}} \plus{} \frac{1}{ab \plus{} c \plus{} \frac{1}{c}} \leqslant \frac{27}{31}.\]
2009 Germany Team Selection Test, 3
Let $ a$, $ b$, $ c$, $ d$ be positive real numbers such that $ abcd \equal{} 1$ and $ a \plus{} b \plus{} c \plus{} d > \dfrac{a}{b} \plus{} \dfrac{b}{c} \plus{} \dfrac{c}{d} \plus{} \dfrac{d}{a}$. Prove that
\[ a \plus{} b \plus{} c \plus{} d < \dfrac{b}{a} \plus{} \dfrac{c}{b} \plus{} \dfrac{d}{c} \plus{} \dfrac{a}{d}\]
[i]Proposed by Pavel Novotný, Slovakia[/i]
2012 Balkan MO Shortlist, A6
Let $k$ be a positive integer. Find the maximum value of \[a^{3k-1}b+b^{3k-1}c+c^{3k-1}a+k^2a^kb^kc^k,\] where $a$, $b$, $c$ are non-negative reals such that $a+b+c=3k$.
2010 Junior Balkan MO, 2
Find all integers $n$, $n \ge 1$, such that $n \cdot 2^{n+1}+1$ is a perfect square.
2013 IMO Shortlist, A4
Let $n$ be a positive integer, and consider a sequence $a_1 , a_2 , \dotsc , a_n $ of positive integers. Extend it periodically to an infinite sequence $a_1 , a_2 , \dotsc $ by defining $a_{n+i} = a_i $ for all $i \ge 1$. If \[a_1 \le a_2 \le \dots \le a_n \le a_1 +n \] and \[a_{a_i } \le n+i-1 \quad\text{for}\quad i=1,2,\dotsc, n, \] prove that \[a_1 + \dots +a_n \le n^2. \]
2008 Indonesia TST, 4
Let $a, b, c$ be positive reals. Prove that $$\left(\frac{a}{a+b}\right)^2+\left(\frac{b}{b+c}\right)^2+\left(\frac{c}{c+a}\right)^2\ge \frac34$$
2021 Austrian MO Regional Competition, 1
Let $a$ and $b$ be positive integers and $c$ be a positive real number satisfying
$$\frac{a + 1}{b + c}=\frac{b}{a}.$$ Prove that $c \ge 1$ holds.
(Karl Czakler)
1998 APMO, 3
Let $a$, $b$, $c$ be positive real numbers. Prove that
\[ \biggl(1+\frac{a}{b}\biggr) \biggl(1+\frac{b}{c}\biggr) \biggl(1+\frac{c}{a}\biggr) \ge 2 \biggl(1+\frac{a+b+c}{\sqrt[3]{abc}}\biggr). \]
1970 IMO Longlists, 44
If $a, b, c$ are side lengths of a triangle, prove that
\[(a + b)(b + c)(c + a) \geq 8(a + b - c)(b + c - a)(c + a - b).\]
2016 Thailand TSTST, 1
Let $a_1, a_2, a_3, \dots$ be a sequence of integers such that
$\text{(i)}$ $a_1=0$
$\text{(ii)}$ for all $i\geq 1$, $a_{i+1}=a_i+1$ or $-a_i-1$.
Prove that $\frac{a_1+a_2+\cdots+a_n}{n}\geq-\frac{1}{2}$ for all $n\geq 1$.
2013 Saudi Arabia IMO TST, 2
Let $S = f\{0.1. 2.3,...\}$ be the set of the non-negative integers. Find all strictly increasing functions $f : S \to S$ such that $n + f(f(n)) \le 2f(n)$ for every $n$ in $S$
2008 Mathcenter Contest, 5
Let $a,b,c$ be positive real numbers where $ab+bc+ca = 3$. Prove that $$\dfrac{1}{a^2+1}+\dfrac{1}{b^2+1}+\dfrac{1}{c^2+1}\geq\dfrac{3} {2}.$$
[i](dektep)[/i]
1992 AMC 8, 17
The sides of a triangle have lengths $6.5$, $10$, and $s$, where $s$ is a whole number. What is the smallest possible value of $s$?
[asy]
pair A,B,C;
A=origin; B=(10,0); C=6.5*dir(15);
dot(A); dot(B); dot(C);
draw(B--A--C);
draw(B--C,dashed);
label("$6.5$",3.25*dir(15),NNW);
label("$10$",(5,0),S);
label("$s$",(8,1),NE);
[/asy]
$\text{(A)}\ 3 \qquad \text{(B)}\ 4 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 7$
2008 Hanoi Open Mathematics Competitions, 10
Let $a,b,c \in [1, 3]$ and satisfy the following conditions:
$ max \{a, b, c\}\ge 2$ and $ a + b + c = 5$
What is the smallest possible value of $a^2 + b^2 + c^2$?
2012 Bosnia and Herzegovina Junior BMO TST, 4
If $a$, $b$ and $c$ are sides of triangle which perimeter equals $1$, prove that:
$a^2+b^2+c^2+4abc<\frac{1}{2}$