Found problems: 6530
2010 Iran MO (3rd Round), 3
prove that for each natural number $n$ there exist a polynomial with degree $2n+1$ with coefficients in $\mathbb{Q}[x]$ such that it has exactly $2$ complex zeros and it's irreducible in $\mathbb{Q}[x]$.(20 points)
2003 India National Olympiad, 2
Find all primes $p,q$ and even $n>2$ such that $p^n+p^{n-1}+...+1=q^2+q+1$.
2008 Indonesia TST, 4
Let $a, b, c$ be positive reals. Prove that $$\left(\frac{a}{a+b}\right)^2+\left(\frac{b}{b+c}\right)^2+\left(\frac{c}{c+a}\right)^2\ge \frac34$$
2013 Swedish Mathematical Competition, 6
Let $a, b, c$, be real numbers such that $$a^2b^2 + 18 abc > 4b^3+4a^3c+27c^2 .$$
Prove that $a^2>3b$.
2025 Poland - First Round, 11
Positive integer $l$ and positive real numbers $a_1, a_2, ..., a_l$ are given. For every positive integer $n$ we define
$$c_n=\sum_{k_1+k_2+...+k_l=n}\frac{(2n)!}{(2k_1)!(2k_2)!...(2k_l)!}a_1^{k_1}a_2^{k_2}...a_l^{k_l}.$$
Prove that for every positive integer $n$ the inequality $\sqrt[n]{c_n}\leq \sqrt[n+1]{c_{n+1}}$ holds.
2007 Grigore Moisil Intercounty, 4
Let $ \left( x_n \right)_{n\ge 1} $ be a sequence of positive real numbers, verifying the inequality $ x_n\le \frac{x_{n-1}+x_{n-2}}{2} , $ for any natural number $ n\ge 3. $
Show that $ \left( x_n \right)_{n\ge 1} $ is convergent.
2001 Vietnam National Olympiad, 1
Find the maximum value of $\frac{1}{x^{2}}+\frac{2}{y^{2}}+\frac{3}{z^{2}}$, where $x, y, z$ are positive reals satisfying $\frac{1}{\sqrt{2}}\leq z <\frac{ \min(x\sqrt{2}, y\sqrt{3})}{2}, x+z\sqrt{3}\geq\sqrt{6}, y\sqrt{3}+z\sqrt{10}\geq 2\sqrt{5}.$
2009 Estonia Team Selection Test, 4
Points $A', B', C'$ are chosen on the sides $BC, CA, AB$ of triangle $ABC$, respectively, so that $\frac{|BA'|}{|A'C|}=\frac{|CB'|}{|B'A|}=\frac{|AC'|}{|C'B|}$. The line which is parallel to line $B'C'$ and goes through point $A$ intersects the lines $AC$ and $AB$ at $P$ and $Q$, respectively. Prove that $\frac{|PQ|}{|B'C'|} \ge 2$
1983 National High School Mathematics League, 8
For any $\triangle ABC$, its girth is$l$, its circumradius is$R$, its inscribed radius is $r$.Which one is true?
$\text{(A)}l>R+r\qquad\text{(B)}l\leq R+r\qquad\text{(C)}\frac{l}{6}<R+r<6l\qquad\text{(D)}$None above
2000 IMO Shortlist, 1
Let $ a, b, c$ be positive real numbers so that $ abc \equal{} 1$. Prove that
\[ \left( a \minus{} 1 \plus{} \frac 1b \right) \left( b \minus{} 1 \plus{} \frac 1c \right) \left( c \minus{} 1 \plus{} \frac 1a \right) \leq 1.
\]
1969 All Soviet Union Mathematical Olympiad, 118
Given positive numbers $a,b,c,d$. Prove that the set of inequalities
$$a+b<c+d$$
$$(a+b)(c+d)<ab+cd$$
$$(a+b)cd<ab(c+d)$$
contain at least one wrong.
2010 Tournament Of Towns, 3
Each of $999$ numbers placed in a circular way is either $1$ or $-1$. (Both values appear). Consider the total sum of the products of every $10$ consecutive numbers.
$(a)$ Find the minimal possible value of this sum.
$(b)$ Find the maximal possible value of this sum.
1996 Romania Team Selection Test, 14
Let $ x,y,z $ be real numbers. Prove that the following conditions are equivalent:
(i) $ x,y,z $ are positive numbers and $ \dfrac 1x + \dfrac 1y + \dfrac 1z \leq 1 $;
(ii) $ a^2x+b^2y+c^2z>d^2 $ holds for every quadrilateral with sides $ a,b,c,d $.
2014 CentroAmerican, 3
Let $a$, $b$, $c$ and $d$ be real numbers such that no two of them are equal,
\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}=4\] and $ac=bd$. Find the maximum possible value of
\[\frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}.\]
1999 National Olympiad First Round, 4
If inequality $ \frac {\sin ^{3} x}{\cos x} \plus{} \frac {\cos ^{3} x}{\sin x} \ge k$ is hold for every $ x\in \left(0,\frac {\pi }{2} \right)$, what is the largest possible value of $ k$?
$\textbf{(A)}\ \frac {1}{2} \qquad\textbf{(B)}\ \frac {3}{4} \qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\ \frac {3}{2} \qquad\textbf{(E)}\ \text{None}$
2003 AIME Problems, 11
An angle $x$ is chosen at random from the interval $0^\circ < x < 90^\circ$. Let $p$ be the probability that the numbers $\sin^2 x$, $\cos^2 x$, and $\sin x \cos x$ are not the lengths of the sides of a triangle. Given that $p = d/n$, where $d$ is the number of degrees in $\arctan m$ and $m$ and $n$ are positive integers with $m + n < 1000$, find $m + n$.
1965 German National Olympiad, 6
Let $\alpha,\beta, \gamma$ be the angles of a triangle. Prove that $\cos\alpha, + \cos\beta + \cos\gamma \le \frac{3}{2} $ and find the cases of equality.
2004 India IMO Training Camp, 3
Suppose the polynomial $P(x) \equiv x^3 + ax^2 + bx +c$ has only real zeroes and let $Q(x) \equiv 5x^2 - 16x + 2004$. Assume that $P(Q(x)) = 0$ has no real roots. Prove that $P(2004) > 2004$
2009 Ukraine Team Selection Test, 2
Let $ a$, $ b$, $ c$ are sides of a triangle. Find the least possible value $ k$ such that the following inequality always holds:
$ \left|\frac{a\minus{}b}{a\plus{}b}\plus{}\frac{b\minus{}c}{b\plus{}c}\plus{}\frac{c\minus{}a}{c\plus{}a}\right|<k$
[i](Vitaly Lishunov)[/i]
2005 Mediterranean Mathematics Olympiad, 3
Let $A_1,A_2,\ldots , A_n$ $(n\geq 3)$ be finite sets of positive integers. Prove, that
\[ \displaystyle \frac{1}{n} \left( \sum_{i=1}^n |A_i|\right) + \frac{1}{{{n}\choose{3}}}\sum_{1\leq i < j < k \leq n} |A_i \cap A_j \cap A_k| \geq \frac{2}{{{n}\choose{2}}}\sum_{1\leq i < j \leq n}|A_i \cap A_j| \]
holds, where $|E|$ is the cardinality of the set $E$
2011 Saudi Arabia BMO TST, 2
Let $a_1,a_2,..., a_n$ be real numbers such that $a_1 + a_2 + ... + a_n = 0$ and $|a_1| + |a_2 | + ... + |a_n | = 1$. Prove that $$ |a_1 + 2a_2 + ... + na_n | \le \frac{n-1}{2} $$
2010 Brazil Team Selection Test, 3
Let $a$, $b$, $c$ be positive real numbers such that $\dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c} = a+b+c$. Prove that:
\[\frac{1}{(2a+b+c)^2}+\frac{1}{(a+2b+c)^2}+\frac{1}{(a+b+2c)^2}\leq \frac{3}{16}.\]
[i]Proposed by Juhan Aru, Estonia[/i]
2003 IMO Shortlist, 3
Consider pairs of the sequences of positive real numbers \[a_1\geq a_2\geq a_3\geq\cdots,\qquad b_1\geq b_2\geq b_3\geq\cdots\] and the sums \[A_n = a_1 + \cdots + a_n,\quad B_n = b_1 + \cdots + b_n;\qquad n = 1,2,\ldots.\] For any pair define $c_n = \min\{a_i,b_i\}$ and $C_n = c_1 + \cdots + c_n$, $n=1,2,\ldots$.
(1) Does there exist a pair $(a_i)_{i\geq 1}$, $(b_i)_{i\geq 1}$ such that the sequences $(A_n)_{n\geq 1}$ and $(B_n)_{n\geq 1}$ are unbounded while the sequence $(C_n)_{n\geq 1}$ is bounded?
(2) Does the answer to question (1) change by assuming additionally that $b_i = 1/i$, $i=1,2,\ldots$?
Justify your answer.
2002 Vietnam National Olympiad, 1
Let $ a$, $ b$, $ c$ be real numbers for which the polynomial $ x^3 \plus{} ax^2 \plus{} bx \plus{} c$ has three real roots. Prove that \[ 12ab \plus{} 27c \le 6a^3 \plus{} 10\left(a^2 \minus{} 2b\right)^{\frac {3}{2}}\] When does equality occur?
2007 All-Russian Olympiad Regional Round, 11.8
Prove that $ \prod_{i\equal{}1}^{n}(1\plus{}x_{1}\plus{}x_{2}\plus{}...\plus{}x_{i})\geq\sqrt{(n\plus{}1)^{n\plus{}1}x_{1}x_{2}...x_{n}}\forall x_{1},...,x_{n}> 0$.