Found problems: 592
1996 French Mathematical Olympiad, Problem 4
(a) A function $f$ is defined by $f(x)=x^x$ for all $x>0$. Find the minimum value of $f$.
(b) If $x$ and $y$ are two positive real numbers, show that $x^y+y^x>1$.
1985 IMO, 3
For any polynomial $P(x)=a_0+a_1x+\ldots+a_kx^k$ with integer coefficients, the number of odd coefficients is denoted by $o(P)$. For $i-0,1,2,\ldots$ let $Q_i(x)=(1+x)^i$. Prove that if $i_1,i_2,\ldots,i_n$ are integers satisfying $0\le i_1<i_2<\ldots<i_n$, then: \[ o(Q_{i_1}+Q_{i_2}+\ldots+Q_{i_n})\ge o(Q_{i_1}). \]
2017 Azerbaijan Junior National Olympiad, P3
Show that $\frac{(x + y + z)^2}{3} \ge x\sqrt{yz} + y\sqrt{zx} + z\sqrt{xy}$ for all non-negative reals $x, y, z$.
2009 Bosnia and Herzegovina Junior BMO TST, 2
Let $a$ , $b$, $c$ and $d$ be positive real numbers such that $a+b+c+d=8$. Prove that $\frac{1}{a}+\frac{1}{b}+\frac{4}{c}+\frac{16}{d}\geq8$
2010 IMO Shortlist, 3
Let $A_1A_2 \ldots A_n$ be a convex polygon. Point $P$ inside this polygon is chosen so that its projections $P_1, \ldots , P_n$ onto lines $A_1A_2, \ldots , A_nA_1$ respectively lie on the sides of the polygon. Prove that for arbitrary points $X_1, \ldots , X_n$ on sides $A_1A_2, \ldots , A_nA_1$ respectively,
\[\max \left\{ \frac{X_1X_2}{P_1P_2}, \ldots, \frac{X_nX_1}{P_nP_1} \right\} \geq 1.\]
[i]Proposed by Nairi Sedrakyan, Armenia[/i]
2013 Balkan MO Shortlist, A1
Positive real numbers $a, b,c$ satisfy $ab + bc+ ca = 3$. Prove the inequality $$\frac{1}{4+(a+b)^2}+\frac{1}{4+(b+c)^2}+\frac{1}{4+(c+a)^2}\le \frac{3}{8}$$
2016 Macedonia JBMO TST, 4
Let $x$, $y$, and $z$ be positive real numbers. Prove that
$\sqrt {\frac {xy}{x^2 + y^2 + 2z^2}} + \sqrt {\frac {yz}{y^2 + z^2 + 2x^2}}+\sqrt {\frac {zx}{z^2 + x^2 + 2y^2}} \le \frac{3}{2}$.
When does equality hold?
2022 Macedonian Mathematical Olympiad, Problem 1
Let $(x_n)_{n=1}^\infty$ be a sequence defined recursively with: $x_1=2$ and $x_{n+1}=\frac{x_n(x_n+n)}{n+1}$ for all $n \ge 1$. Prove that $$n(n+1) >\frac{(x_1+x_2+ \ldots +x_n)^2}{x_{n+1}}.$$
[i]Proposed by Nikola Velov[/i]
1985 IMO Shortlist, 18
Let $x_1, x_2, \cdots , x_n$ be positive numbers. Prove that
\[\frac{x_1^2}{x_1^2+x_2x_3} + \frac{x_2^2}{x_2^2+x_3x_4} + \cdots +\frac{x_{n-1}^2}{x_{n-1}^2+x_nx_1} +\frac{x_n^2}{x_n^2+x_1x_2} \leq n-1\]
2021 Alibaba Global Math Competition, 6
Let $M(t)$ be measurable and locally bounded function, that is,
\[M(t) \le C_{a,b}, \quad \forall 0 \le a \le t \le b<\infty\]
with some constant $C_{a,b}$, from $[0,\infty)$ to $[0,\infty)$ such that
\[M(t) \le 1+\int_0^t M(t-s)(1+t)^{-1}s^{-1/2} ds, \quad \forall t \ge 0.\]
Show that
\[M(t) \le 10+2\sqrt{5}, \quad \forall t \ge 0.\]
2016 Bosnia And Herzegovina - Regional Olympiad, 1
Find minimal value of $A=\frac{\left(x+\frac{1}{x}\right)^6-\left(x^6+\frac{1}{x^6}\right)-2}{\left(x+\frac{1}{x}\right)^3+\left(x^3+\frac{1}{x^3}\right)}$
2016 Junior Balkan Team Selection Test, 4
Let $a,b,c\in \mathbb{R}^+$, prove that: $$\frac{2a}{\sqrt{3a+b}}+\frac{2b}{\sqrt{3b+c}}+\frac{2c}{\sqrt{3c+a}}\leq \sqrt{3(a+b+c)}$$
2024 HMIC, 2
Suppose that $a$, $b$, $c$, and $d$ are real numbers such that $a+b+c+d=8$. Compute the minimum possible value of \[20(a^2+b^2+c^2+d^2)-\sum_{\text{sym}}a^3b,\] where the sum is over all $12$ symmetric terms.
[i]Derek Liu[/i]
2016 Belarus Team Selection Test, 1
Prove for positive $a,b,c$ that
$$ (a^2+\frac{b^2}{c^2})(b^2+\frac{c^2}{a^2})(c^2+\frac{a^2}{b^2}) \geq abc (a+\frac{1}{a})(b+\frac{1}{b})(c+\frac{1}{c})$$
2022 Romania EGMO TST, P3
Let $ABCD$ be a convex quadrilateral and let $O$ be the intersection of its diagonals. Let $P,Q,R,$ and $S$ be the projections of $O$ on $AB,BC,CD,$ and $DA$ respectively. Prove that \[2(OP+OQ+OR+OS)\leq AB+BC+CD+DA.\]
2018 Azerbaijan Junior NMO, 3
$a;b\in\mathbb{R^+}$. Prove the following inequality: $$\sqrt[3]{\frac{a}{b}}+\sqrt[3]{\frac{b}{a}}\leq\sqrt[3]{2(a+b)(\frac1{a}+\frac1{b})}$$
1991 IMO Shortlist, 26
Let $ n \geq 2, n \in \mathbb{N}$ and let $ p, a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n \in \mathbb{R}$ satisfying $ \frac{1}{2} \leq p \leq 1,$ $ 0 \leq a_i,$ $ 0 \leq b_i \leq p,$ $ i \equal{} 1, \ldots, n,$ and \[ \sum^n_{i\equal{}1} a_i \equal{} \sum^n_{i\equal{}1} b_i.\] Prove the inequality: \[ \sum^n_{i\equal{}1} b_i \prod^n_{j \equal{} 1, j \neq i} a_j \leq \frac{p}{(n\minus{}1)^{n\minus{}1}}.\]
2019 ELMO Shortlist, A1
Let $a$, $b$, $c$ be positive reals such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$. Show that $$a^abc+b^bca+c^cab\ge 27bc+27ca+27ab.$$
[i]Proposed by Milan Haiman[/i]
2019 JBMO Shortlist, A1
Real numbers $a$ and $b$ satisfy $a^3+b^3-6ab=-11$. Prove that $-\frac{7}{3}<a+b<-2$.
[i]Proposed by Serbia[/i]
2019 South East Mathematical Olympiad, 1
Find the largest real number $k$, such that for any positive real numbers $a,b$,
$$(a+b)(ab+1)(b+1)\geq kab^2$$
2007 Korea Junior Math Olympiad, 5
For all positive real numbers $a, b,c.$ Prove the folllowing inequality$$\frac{a}{c+5b}+\frac{b}{a+5c}+\frac{c}{b+5a}\geq\frac{1}{2}.$$
BIMO 2022, 1
Let $a, b, c,$ be nonnegative reals with $ a+b+c=3 $, find the largest positive real $ k $ so that for all $a,b,c,$ we have $$ a^2+b^2+c^2+k(abc-1)\ge 3 $$
1970 IMO Shortlist, 10
The real numbers $a_0,a_1,a_2,\ldots$ satisfy $1=a_0\le a_1\le a_2\le\ldots. b_1,b_2,b_3,\ldots$ are defined by $b_n=\sum_{k=1}^n{1-{a_{k-1}\over a_k}\over\sqrt a_k}$.
[b]a.)[/b] Prove that $0\le b_n<2$.
[b]b.)[/b] Given $c$ satisfying $0\le c<2$, prove that we can find $a_n$ so that $b_n>c$ for all sufficiently large $n$.
2012 IFYM, Sozopol, 4
Prove that if $x$, $y$, and $z$ are non-negative numbers and $x^2+y^2+z^2=1$, then the following inequality is true:
$\frac{x}{1-x^2}+\frac{y}{1-y^2}+\frac{z}{1-z^2 }\geq \frac{3\sqrt{3}}{2}$
2001 China Team Selection Test, 2
Let ${a_n}$ be a non-increasing sequence of positive numbers. Prove that if for $n \ge 2001$, $na_{n} \le 1$, then for any positive integer $m \ge 2001$ and $x \in \mathbb{R}$, the following inequality holds:
$\left | \sum_{k=2001}^{m} a_{k} \sin kx \right | \le 1 + \pi$