Found problems: 6530
2012 China Team Selection Test, 2
Given an integer $k\ge 2$. Prove that there exist $k$ pairwise distinct positive integers $a_1,a_2,\ldots,a_k$ such that for any non-negative integers $b_1,b_2,\ldots,b_k,c_1,c_2,\ldots,c_k$ satisfying $a_1\le b_i\le 2a_i, i=1,2,\ldots,k$ and $\prod_{i=1}^{k}b_i^{c_i}<\prod_{i=1}^{k}b_i$, we have
\[k\prod_{i=1}^{k}b_i^{c_i}<\prod_{i=1}^{k}b_i.\]
2012 ELMO Shortlist, 6
Let $a,b,c\ge0$. Show that $(a^2+2bc)^{2012}+(b^2+2ca)^{2012}+(c^2+2ab)^{2012}\le (a^2+b^2+c^2)^{2012}+2(ab+bc+ca)^{2012}$.
[i]Calvin Deng.[/i]
2012 Serbia Team Selection Test, 2
Let $\sigma(x)$ denote the sum of divisors of natural number $x$, including $1$ and $x$. For every $n\in \mathbb{N}$ define $f(n)$ as number of natural numbers $m, m\leq n$, for which $\sigma(m)$ is odd number. Prove that there are infinitely many natural numbers $n$, such that $f(n)|n$.
2017, SRMC, 3
Prove that among any $42$ numbers from the interval $[1,10^6]$, you can choose four numbers so that for any permutation $(a, b, c, d)$ of these numbers, the inequality
$$25 (ab + cd) (ad + bc) \ge 16 (ac + bd)^ 2$$
holds.
2011 Today's Calculation Of Integral, 737
Let $a,\ b$ real numbers such that $a>1,\ b>1.$
Prove the following inequality.
\[\int_{-1}^1 \left(\frac{1+b^{|x|}}{1+a^{x}}+\frac{1+a^{|x|}}{1+b^{x}}\right)\ dx<a+b+2\]
2007 Romania Team Selection Test, 1
If $a_{1}$, $a_{2}$, $\ldots$, $a_{n}\geq 0$ are such that \[a_{1}^{2}+\cdots+a_{n}^{2}=1,\]
then find the maximum value of the product $(1-a_{1})\cdots (1-a_{n})$.
2001 Austrian-Polish Competition, 5
The fields of the $8\times 8$ chessboard are numbered from $1$ to $64$ in the following manner: For $i=1,2,\cdots,63$ the field numbered by $i+1$ can be reached from the field numbered by $i$ by one move of the knight. Let us choose positive real numbers $x_{1},x_{2},\cdots,x_{64}$. For each white field numbered by $i$ define the number $y_{i}=1+x_{i}^{2}-\sqrt[3]{x_{i-1}^{2}x_{i+1}}$ and for each black field numbered by $j$ define the number $y_{j}=1+x_{j}^{2}-\sqrt[3]{x_{j-1}x_{j+1}^{2}}$ where $x_{0}=x_{64}$ and $x_{1}=x_{65}$. Prove that \[\sum_{i=1}^{64}y_{i}\geq 48\]
2025 Junior Macedonian Mathematical Olympiad, 4
Let $x, y$, and $z$ be positive real numbers, such that $x^2 + y^2 + z^2 = 3$. Prove the inequality
\[\frac{x^3}{2 + x} + \frac{y^3}{2 + y} + \frac{z^3}{2 + z} \ge 1.\]
When does the equality hold?
1970 IMO Longlists, 44
If $a, b, c$ are side lengths of a triangle, prove that
\[(a + b)(b + c)(c + a) \geq 8(a + b - c)(b + c - a)(c + a - b).\]
Ukrainian TYM Qualifying - geometry, XI.15
Let $I$ be the point of intersection of the angle bisectors of the $\vartriangle ABC$, $W_1,W_2,W_3$ be point of intersection of lines $AI, BI, CI$ with the circle circumscribed around the triangle, $r$ and $R$ be the radii of inscribed and circumscribed circles respectively. Prove the inequality $$IW_1+ IW_2 + IW_3\ge 2R + \sqrt{2Rr.}$$
2019 Jozsef Wildt International Math Competition, W. 33
Let $0 < \frac{1}{q} \leq \frac{1}{p} < 1$ and $\frac{1}{p}+\frac{1}{q}=1$. Let $u_k$, $v_k$, $a_k$ and $b_k$ be non-negative real sequences such as $u^2_k > a^p_k$ and $v_k > b^q_k$, where $k = 1, 2,\cdots , n$. If $0 < m_1\leq u_k \leq M_1$ and $0 < m_2 \leq v_k \leq M_2$ , then $$\left(\sum \limits_{k=1}^n\left(l^p\left(u_k+v_k\right)^2-\left(a_k+b_k\right)^p\right)\right)^{\frac{1}{p}}\geq \left(\sum \limits_{k=1}^n\left(u_k^2-a_k^p\right)\right)^{\frac{1}{p}}\left(\sum \limits_{k=1}^n\left(v_k^2-b_k^p\right)\right)^{\frac{1}{p}}$$where $$l=\frac{M_1M_2+m_1m_2}{2\sqrt{m_1M_1m_2M_2}}$$
2018-IMOC, A6
Given $ a, b, c > 0$. Prove that: $ (1\plus{}a\plus{}b\plus{}c)(1\plus{}ab\plus{}bc\plus{}ca) \ge 4\sqrt{2}\sqrt{(a\plus{}bc)(b\plus{}ca)(c\plus{}ab)}$
:)
2012 Spain Mathematical Olympiad, 1
Find all positive integers $n$ and $k$ such that $(n+1)^n=2n^k+3n+1$.
2003 China Girls Math Olympiad, 6
Let $ n \geq 2$ be an integer. Find the largest real number $ \lambda$ such that the inequality \[ a^2_n \geq \lambda \sum^{n\minus{}1}_{i\equal{}1} a_i \plus{} 2 \cdot a_n.\] holds for any positive integers $ a_1, a_2, \ldots a_n$ satisfying $ a_1 < a_2 < \ldots < a_n.$
2011 Romania National Olympiad, 2
The numbers $x, y, z, t, a$ and $b$ are positive integers, so that $xt-yz = 1$ and $$\frac{x}{y} \ge \frac{a}{b} \ge \frac{z}{t} .$$Prove that $$ab \le (x + z) (y +t)$$
2023 All-Russian Olympiad Regional Round, 11.9
If $a, b, c$ are non-zero reals, prove that $|\frac{b} {a}-\frac{b} {c}|+|\frac{c} {a}-\frac{c}{b}|+|bc+1|>1$.
2017 Kosovo National Mathematical Olympiad, 2
2 .Solve the inequality $|x-1|-2|x-4|>3+2x$
1986 Federal Competition For Advanced Students, P2, 5
Show that for every convex $ n$-gon $ ( n \ge 4)$, the arithmetic mean of the lengths of its sides is less than the arithmetic mean of the lengths of all its diagonals.
1989 APMO, 4
Let $S$ be a set consisting of $m$ pairs $(a,b)$ of positive integers with the property that $1 \leq a < b \leq n$. Show that there are at least
\[ 4m \cdot \dfrac{(m - \dfrac{n^2}{4})}{3n} \]
triples $(a,b,c)$ such that $(a,b)$, $(a,c)$, and $(b,c)$ belong to $S$.
2014 Benelux, 4
Let $ABCD$ be a square. Consider a variable point $P$ inside the square for which $\angle BAP \ge 60^\circ.$ Let $Q$ be the intersection of the line $AD$ and the perpendicular to $BP$ in $P$. Let $R$ be the intersection of the line $BQ$ and the perpendicular to $BP$ from $C$.
[list]
[*] [b](a)[/b] Prove that $|BP|\ge |BR|$
[*] [b](b)[/b] For which point(s) $P$ does the inequality in [b](a)[/b] become an equality?[/list]
1955 Polish MO Finals, 4
Prove that $$
\sin^2 \alpha + \sin^2 \beta \geq
\sin \alpha \sin \beta + \sin \alpha + \sin \beta - 1.$$
1999 Turkey MO (2nd round), 5
In an acute triangle $\vartriangle ABC$ with circumradius $R$, altitudes $\overline{AD},\overline{BE},\overline{CF}$ have lengths ${{h}_{1}},{{h}_{2}},{{h}_{3}}$, respectively. If ${{t}_{1}},{{t}_{2}},{{t}_{3}}$ are lengths of the tangents from $A,B,C$, respectively, to the circumcircle of triangle $\vartriangle DEF$, prove that
$\sum\limits_{i=1}^{3}{{{\left( \frac{t{}_{i}}{\sqrt{h{}_{i}}} \right)}^{2}}\le }\frac{3}{2}R$.
2010 BAMO, 5
Let $a$, $b$, $c$, $d$ be positive real numbers such that $abcd=1$. Prove that
$1/[(1/2 +a+ab+abc)^{1/2}]+ 1/[(1/2+b+bc+bcd)^{1/2}] + 1/[(1/2+c+cd+cda)^{1/2}] + 1/[1(1/2+d+da+dab)^{1/2}]$ is greater than or equal to $2^{1/2}$.
2001 Romania Team Selection Test, 2
Prove that there is no function $f:(0,\infty )\rightarrow (0,\infty)$ such that
\[f(x+y)\ge f(x)+yf(f(x)) \]
for every $x,y\in (0,\infty )$.
2013 Federal Competition For Advanced Students, Part 2, 4
For a positive integer $n$, let $a_1, a_2, \ldots a_n$ be nonnegative real numbers such that for all real numbers $x_1>x_2>\ldots>x_n>0$ with $x_1+x_2+\ldots+x_n<1$, the inequality $\sum_{k=1}^na_kx_k^3<1$ holds. Show that \[na_1+(n-1)a_2+\ldots+(n-j+1)a_j+\ldots+a_n\leqslant\frac{n^2(n+1)^2}{4}.\]