Found problems: 6530
2015 China Western Mathematical Olympiad, 5
Let $a,b,c,d$ are lengths of the sides of a convex quadrangle with the area equal to $S$, set $S =\{x_1, x_2,x_3,x_4\}$ consists of permutations $x_i$ of $(a, b, c, d)$. Prove that \[S \leq \frac{1}{2}(x_1x_2+x_3x_4).\]
1982 Tournament Of Towns, (017) 3
a) Prove that in an infinite sequence ${a_k}$ of integers, pairwise distinct and each member greater than $1$, one can find $100$ members for which $a_k > k$.
b) Prove that in an infinite sequence ${a_k}$ of integers, pairwise distinct and each member greater than $1$ there are infinitely many such numbers $a_k$ such that $a_k > k$.
(A Andjans, Riga)
PS. (a) for juniors (b) for seniors
I Soros Olympiad 1994-95 (Rus + Ukr), 9.8
Let $f(x) =x^2-2x$. Find all $x$ for which $f(f(x))<3$.
2007 Iran Team Selection Test, 1
Find all polynomials of degree 3, such that for each $x,y\geq 0$: \[p(x+y)\geq p(x)+p(y)\]
2009 Princeton University Math Competition, 2
Tetrahedron $ABCD$ has sides of lengths, in increasing order, $7, 13, 18, 27, 36, 41$. If $AB=41$, then what is the length of $CD$?
1997 Croatia National Olympiad, Problem 2
Let $a,b,c$ be positive reals. Prove that $$a^ab^bc^c \geq a^bb^cc^a$$
1995 IMC, 2
Let $f$ be a continuous function on $[0,1]$ such that for every $x\in [0,1]$,
we have $\int_{x}^{1}f(t)dt \geq\frac{1-x^{2}}{2}$. Show that $\int_{0}^{1}f(t)^{2}dt \geq \frac{1}{3}$.
2010 ISI B.Math Entrance Exam, 4
If $a,b,c\in (0,1)$ satisfy $a+b+c=2$ , prove that
$\frac{abc}{(1-a)(1-b)(1-c)}\ge 8$
2023 Brazil National Olympiad, 4
Let $x, y, z$ be three real distinct numbers such that
$$\begin{cases} x^2-x=yz \\ y^2-y=zx \\ z^2-z=xy \end{cases}$$ Show that $-\frac{1}{3} < x,y,z < 1$.
2002 Taiwan National Olympiad, 4
Let $0<x_{1},x_{2},x_{3},x_{4}\leq\frac{1}{2}$ are real numbers. Prove that $\frac{x_{1}x_{2}x_{3}x_{4}}{(1-x_{1})(1-x_{2})(1-x_{3})(1-x_{4})}\leq\frac{x_{1}^{4}+x_{2}^{4}+x_{3}^{4}+x_{4}^{4}}{(1-x_{1})^{4}+(1-x_{2})^{4}+(1-x_{3})^{4}+(1-x_{4})^{4}}$.
2011 Macedonia National Olympiad, 1
Let $~$ $ a,\,b,\,c,\,d\, >\, 0$ $~$ and $~$ $a+b+c+d\, =\, 1\, .$ $~$ Prove the inequality
\[ \frac{1}{4a+3b+c}+\frac{1}{3a+b+4d}+\frac{1}{a+4c+3d}+\frac{1}{4b+3c+d}\; \ge\; 2\, . \]
2017 Iran Team Selection Test, 1
Let $a,b,c,d$ be positive real numbers with $a+b+c+d=2$. Prove the following inequality:
$$\frac{(a+c)^{2}}{ad+bc}+\frac{(b+d)^{2}}{ac+bd}+4\geq 4\left ( \frac{a+b+1}{c+d+1}+\frac{c+d+1}{a+b+1} \right).$$
[i]Proposed by Mohammad Jafari[/i]
1954 Moscow Mathematical Olympiad, 281
*. Positive numbers $x_1, x_2, ..., x_{100}$ satisfy the system $$\begin{cases} x^2_1+ x^2_2+ ... + x^2_{100} > 10 000 \\
x_1 + x_2 + ...+ x_{100} < 300 \end{cases}$$
Prove that among these numbers there are three whose sum is greater than $100$.
2013 Junior Balkan Team Selection Tests - Romania, 1
Let $a, b, c, d > 0$ satisfying $abcd = 1$. Prove that $$\frac{1}{a + b + 2}+\frac{1}{b + c + 2}+\frac{1}{c + d + 2}+\frac{1}{d + a + 2} \le 1$$
1972 AMC 12/AHSME, 10
For $x$ real, the inequality $1\le |x-2|\le 7$ is equivalent to
$\textbf{(A) }x\le 1\text{ or }x\ge 3\qquad\textbf{(B) }1\le x\le 3\qquad\textbf{(C) }-5\le x\le 9\qquad$
$\textbf{(D) }-5\le x\le 1\text{ or }3\le x\le 9\qquad \textbf{(E) }-6\le x\le 1\text{ or }3\le x\le 10$
2010 China Team Selection Test, 2
Let $M=\{1,2,\cdots,n\}$, each element of $M$ is colored in either red, blue or yellow. Set
$A=\{(x,y,z)\in M\times M\times M|x+y+z\equiv 0\mod n$, $x,y,z$ are of same color$\},$
$B=\{(x,y,z)\in M\times M\times M|x+y+z\equiv 0\mod n,$ $x,y,z$ are of pairwise distinct color$\}.$
Prove that $2|A|\geq |B|$.
2001 Switzerland Team Selection Test, 2
If $a,b$, and $c$ are the sides of a triangle, prove the inequality $\sqrt{a+b-c}+\sqrt{c+a-b}+\sqrt{b+c-a } \le \sqrt{a}+\sqrt{b}+\sqrt{c}$.
When does equality occur?
2010 Indonesia TST, 1
Let $ a$, $ b$, and $ c$ be non-negative real numbers and let $ x$, $ y$, and $ z$ be positive real numbers such that $ a\plus{}b\plus{}c\equal{}x\plus{}y\plus{}z$. Prove that
\[ \dfrac{a^3}{x^2}\plus{}\dfrac{b^3}{y^2}\plus{}\dfrac{c^3}{z^2} \ge a\plus{}b\plus{}c.\]
[i]Hery Susanto, Malang[/i]
2018 Azerbaijan BMO TST, 2
Let $M = \{(a,b,c)\in R^3 :0 <a,b,c<\frac12$ with $a+b+c=1 \}$ and $f: M\to R$ given as $$f(a,b,c)=4\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)-\frac{1}{abc}$$
Find the best (real) bounds $\alpha$ and $\beta$ such that $f(M) = \{f(a,b,c): (a,b,c)\in M\}\subseteq [\alpha,\beta]$ and determine whether any of them is achievable.
1990 Baltic Way, 18
Numbers $1, 2,\dots , 101$ are written in the cells of a $101\times 101$ square board so that each number is repeated $101$ times. Prove that there exists either a column or a row containing at least $11$ different numbers.
1985 IMO Longlists, 63
Let $x_n = \sqrt[2]{2+\sqrt[3]{3+\cdots+\sqrt[n]{n}}}.$ Prove that
\[x_{n+1}-x_n <\frac{1}{n!} \quad n=2,3,\cdots\]
1982 National High School Mathematics League, 8
$a,b$ are two different positive real numbers, then which one is the largest?
$$A=(a+\frac{1}{a})(b+\frac{1}{b}), B=(\sqrt{ab}+\frac{1}{\sqrt{ab}})^2, C=(\frac{a+b}{2}+\frac{2}{a+b})^2.$$
$\text{(A)}A\qquad\text{(B)}B\qquad\text{(C)}C\qquad\text{(D)}$Not sure.
2012 Junior Balkan Team Selection Tests - Romania, 1
Prove that if the positive real numbers $p$ and $q$ satisfy $\frac{1}{p}+\frac{1}{q}= 1$, then
a) $\frac{1}{3} \le \frac{1}{p (p + 1)} +\frac{1}{q (q + 1)} <\frac{1}{2}$
b) $\frac{1}{p (p - 1)} + \frac{1}{q (q - 1)} \ge 1$
2001 Saint Petersburg Mathematical Olympiad, 11.4
For any two positive integers $n>m$ prove the following inequality:
$$[m,n]+[m+1,n+1]\geq \dfrac{2nm}{\sqrt{m-n}}$$
As always, $[x,y]$ means the least common multiply of $x,y$.
[I]Proposed by A. Golovanov[/i]
1985 Polish MO Finals, 3
The function $f : R \to R$ satisfies $f(3x) = 3f(x) - 4f(x)^3$ for all real $x$ and is continuous at $x = 0$. Show that $|f(x)| \le 1$ for all $x$.