Found problems: 1311
2007 Moldova National Olympiad, 12.2
For $p\in (0;\infty)$ find the area of the region bounded by the curves $y^{2}=4px$ and $16py^{2}=5(x-p)^{3}$
1975 Vietnam National Olympiad, 2
Solve this equation
$\frac{y^{3}+m^{3}}{\left ( y+m \right )^{3}}+\frac{y^{3}+n^{3}}{\left ( y+n \right )^{3}}+\frac{y^{3}+p^{3}}{\left ( y+p \right )^{3}}-\frac{3}{2}+\frac{3}{2}.\frac{y-m}{y+m}.\frac{y-n}{y+n}.\frac{y-p}{y+p}=0$
2008 Polish MO Finals, 2
A function $ f: R^3\rightarrow R$ for all reals $ a,b,c,d,e$ satisfies a condition:
\[ f(a,b,c)\plus{}f(b,c,d)\plus{}f(c,d,e)\plus{}f(d,e,a)\plus{}f(e,a,b)\equal{}a\plus{}b\plus{}c\plus{}d\plus{}e\]
Show that for all reals $ x_1,x_2,\ldots,x_n$ ($ n\geq 5$) equality holds:
\[ f(x_1,x_2,x_3)\plus{}f(x_2,x_3,x_4)\plus{}\ldots \plus{}f(x_{n\minus{}1},x_n,x_1)\plus{}f(x_n,x_1,x_2)\equal{}x_1\plus{}x_2\plus{}\ldots\plus{}x_n\]
2005 CHKMO, 1
Let $a_{1},a_{2},...,a_{n+1}(n>1)$ are positive real numbers such that $a_{2}-a_{1}=a_{3}-a_{2}=...=a_{n+1}-a_{n}$. Prove that $\sum_{k=2}^{n}\frac{1}{a_{k}^{2}}\leq\frac{n-1}{2}.\frac{a_{1}a_{n}+a_{2}a_{n+1}}{a_{1}a_{2}a_{n}a_{n+1}}$
2009 Iran Team Selection Test, 8
Find all polynomials $ P(x,y)$ such that for all reals $ x$ and $y$,
\[P(x^{2},y^{2}) =P\left(\frac {(x + y)^{2}}{2},\frac {(x - y)^{2}}{2}\right).\]
2005 China Northern MO, 6
Let $0 \leq \alpha , \beta , \gamma \leq \frac{\pi}{2}$, such that $\cos ^{2} \alpha + \cos ^{2} \beta + \cos ^{2} \gamma = 1$. Prove that
$2 \leq (1 + \cos ^{2} \alpha ) ^{2} \sin^{4} \alpha + (1 + \cos ^{2} \beta ) ^{2} \sin ^{4} \beta + (1 + \cos ^{2} \gamma ) ^{2} \sin ^{4} \gamma \leq (1 + \cos ^{2} \alpha )(1 + \cos ^{2} \beta)(1 + \cos ^{2} \gamma ).$
2014 Contests, 2
Does there exist a function $f: \mathbb R \to \mathbb R $ satisfying the following conditions:
(i) for each real $y$ there is a real $x$ such that $f(x)=y$ , and
(ii) $f(f(x)) = (x - 1)f(x) + 2$ for all real $x$ ?
[i]Proposed by Igor I. Voronovich, Belarus[/i]
2002 Romania National Olympiad, 4
Let $f:[0,1]\rightarrow [0,1]$ be a continuous and bijective function.
Describe the set:
\[A=\{f(x)-f(y)\mid x,y\in[0,1]\backslash\mathbb{Q}\}\]
[hide="Note"]
You are given the result that [i]there is no one-to-one function between the irrational numbers and $\mathbb{Q}$.[/i][/hide]
2008 Junior Balkan Team Selection Tests - Moldova, 10
Solve in prime numbers:
$ \{\begin{array}{c}\ \ 2a - b + 7c = 1826 \
3a + 5b + 7c = 2007 \end{array}$
2001 Brazil Team Selection Test, Problem 1
Find all functions $ f $ defined on real numbers and taking values in the set of real numbers such that $ f(x+y)+f(y+z)+f(z+x) \geq f(x+2y+3z) $ for all real numbers $ x,y,z $.
[hide]There is an infinity of such functions. Every function with the property that $ 3 \inf f \geq \sup f $ is a good one. I wonder if there is a way to find all the solutions. It seems very strange.[/hide]
2010 Indonesia TST, 2
Consider a polynomial with coefficients of real numbers $ \phi(x)\equal{}ax^3\plus{}bx^2\plus{}cx\plus{}d$ with three positive real roots. Assume that $ \phi(0)<0$, prove that \[ 2b^3\plus{}9a^2d\minus{}7abc \le 0.\]
[i]Hery Susanto, Malang[/i]