Found problems: 6530
1962 Putnam, A4
Assume that $|f(x)|\leq 1$ and $|f''(x)|\leq 1$ for all $x$ on an interval of length at least $2.$ Show that $|f'(x)|\leq 2 $ on the interval.
1986 French Mathematical Olympiad, Problem 1
Let $ABCD$ be a tetrahedron.
(a) Prove that the midpoints of the edges $AB,AC,BD$, and $CD$ lie in a plane.
(b) Find the point in that plane, whose sum of distances from the lines $AD$ and $BC$ is minimal.
2020 Bundeswettbewerb Mathematik, 4
In each cell of a table with $m$ rows and $n$ columns, where $m<n$, we put a non-negative real number such that each column contains at least one positive number.
Show that there is a cell with a positive number such that the sum of the numbers in its row is larger than the sum of the numbers in its column.
2022 Switzerland Team Selection Test, 5
Let $a, b, c, \lambda$ be positive real numbers with $\lambda \geq 1/4$. Show that $$\frac{a}{\sqrt{b^2+\lambda bc+c^2}}+\frac{b}{\sqrt{c^2+\lambda ca+a^2}}+\frac{c}{\sqrt{a^2+\lambda ab+b^2}} \geq \frac{3}{\sqrt{\lambda +2}}.$$
2010 Purple Comet Problems, 27
Let $a$ and $b$ be real numbers satisfying $2(\sin a + \cos a) \sin b = 3 - \cos b$. Find $3 \tan^2a+4\tan^2 b$.
2013 BMT Spring, P2
If $f(x)=x^n-7x^{n-1}+17x^{n-2}+a_{n-3}x^{n-3}+\ldots+a_0$ is a real-valued function of degree $n>2$ with all real roots, prove that no root has value greater than $4$ and at least one root has value less than $0$ or greater than $2$.
2020 Federal Competition For Advanced Students, P1, 1
Let $x, y$ and $z$ be positive real numbers such that $x \geq y+z$.
Proof that
$$\frac{x+y}{z} + \frac{y+z}{x} +\frac{z+x}{y} \geq 7$$
When does equality occur?
(Walther Janous)
2014 Junior Balkan Team Selection Tests - Romania, 3
Consider two integers $n \ge m \ge 4$ and $A = \{a_1, a_2, ..., a_m\}$ a subset of the set $\{1, 2, ..., n\}$ such that:
[i]for all $a, b \in A, a \ne b$, if $a + b \le n$, then $a + b \in A$.[/i]
Prove that $\frac{a_1 + a_2 + ... + a_m}{m} \ge \frac{n + 1}{2}$ .
2011 USAMO, 6
Let $A$ be a set with $|A|=225$, meaning that $A$ has 225 elements. Suppose further that there are eleven subsets $A_1, \ldots, A_{11}$ of $A$ such that $|A_i|=45$ for $1\leq i\leq11$ and $|A_i\cap A_j|=9$ for $1\leq i<j\leq11$. Prove that $|A_1\cup A_2\cup\ldots\cup A_{11}|\geq 165$, and give an example for which equality holds.
2021 Indonesia TST, A
Let $a$ and $b$ be real numbers. It is known that the graph of the parabola $y =ax^2 +b$ cuts the graph of the curve $y = x+1/x$ in exactly three points. Prove that $3ab < 1$.
2010 Korea National Olympiad, 1
$ x, y, z $ are positive real numbers such that $ x+y+z=1 $. Prove that
\[ \sqrt{ \frac{x}{1-x} } + \sqrt{ \frac{y}{1-y} } + \sqrt{ \frac{z}{1-z} } > 2 \]
1990 Czech and Slovak Olympiad III A, 2
Determine all values $\alpha\in\mathbb R$ with the following property: if positive numbers $(x,y,z)$ satisfy the inequality \[x^2+y^2+z^2\le\alpha(xy+yz+zx),\] then $x,y,z$ are sides of a triangle.
1998 Belarus Team Selection Test, 1
Let $S(n)$ be the sum of all different natural divisors of odd natural number $n> 1$ (including $n$ and $1$).
Prove that $(S(n))^3 <n^4$.
1992 Balkan MO, 2
Prove that for all positive integers $n$ the following inequality takes place \[ (2n^2+3n+1)^n \geq 6^n (n!)^2 . \]
[i]Cyprus[/i]
2019 Jozsef Wildt International Math Competition, W. 31
Let $a, b \in \Gamma$, $a < b$ and the differentiable function $f : [a, b] \to \Gamma$, such that $f (a) = a$ and $f (b) = b$. Prove that $$\int \limits_{a}^{b} \left(f'(x)\right)^2dx \geq b-a$$
2011 Czech-Polish-Slovak Match, 1
Let $a$, $b$, $c$ be positive real numbers satisfying $a^2<bc$. Prove that $b^3+ac^2>ab(a+c)$.
2022 Israel National Olympiad, P6
Let $x,y,z$ be non-negative real numbers. Prove that:
\[\sqrt{(2x+y)(2x+z)}+\sqrt{(2y+x)(2y+z)}+\sqrt{(2z+x)(2z+y)}\geq \]
\[\geq \sqrt{(x+2y)(x+2z)}+\sqrt{(y+2x)(y+2z)}+\sqrt{(z+2x)(z+2y)}.\]
2007 IMC, 5
For each positive integer $ k$, find the smallest number $ n_{k}$ for which there exist real $ n_{k}\times n_{k}$ matrices $ A_{1}, A_{2}, \ldots, A_{k}$ such that all of the following conditions hold:
(1) $ A_{1}^{2}= A_{2}^{2}= \ldots = A_{k}^{2}= 0$,
(2) $ A_{i}A_{j}= A_{j}A_{i}$ for all $ 1 \le i, j \le k$, and
(3) $ A_{1}A_{2}\ldots A_{k}\ne 0$.
1991 Poland - Second Round, 1
The numbers $ a_i $, $ b_i $, $ c_i $, $ d_i $ satisfy the conditions $ 0\leq c_i \leq a_i \leq b_i \leq d_i $ and $ a_i+b_i = c_i+d_i $ for $ i=1,2 ,\ldots,n$. Prove that
$$ \prod_{i=1}^n a_i + \prod_{i=1}^n b_i \leq \prod_{i=1}^n c_i + \prod_{i=1}^n d_i$$
1997 IMO, 3
Let $ x_1$, $ x_2$, $ \ldots$, $ x_n$ be real numbers satisfying the conditions:
\[ \left\{\begin{array}{cccc} |x_1 \plus{} x_2 \plus{} \cdots \plus{} x_n | & \equal{} & 1 & \ \\
|x_i| & \leq & \displaystyle \frac {n \plus{} 1}{2} & \ \textrm{ for }i \equal{} 1, 2, \ldots , n. \end{array} \right.
\]
Show that there exists a permutation $ y_1$, $ y_2$, $ \ldots$, $ y_n$ of $ x_1$, $ x_2$, $ \ldots$, $ x_n$ such that
\[ | y_1 \plus{} 2 y_2 \plus{} \cdots \plus{} n y_n | \leq \frac {n \plus{} 1}{2}.
\]
1992 AIME Problems, 13
Triangle $ABC$ has $AB=9$ and $BC: AC=40: 41$. What's the largest area that this triangle can have?
1988 India National Olympiad, 9
Show that for a triangle with radii of circumcircle and incircle equal to $ R$, $ r$ respectively, the inequality $ R \geq 2r$ holds.
2011 Indonesia TST, 1
For all positive integer $n$, define $f_n(x)$ such that $f_n(x) = \sum_{k=1}^n{|x - k|}$.
Determine all solution from the inequality $f_n(x) < 41$ for all positive $2$-digit integers $n$ (in decimal notation).
2014 Online Math Open Problems, 28
In the game of Nim, players are given several piles of stones. On each turn, a player picks a nonempty pile and removes any positive integer number of stones from that pile. The player who removes the last stone wins, while the first player who cannot move loses.
Alice, Bob, and Chebyshev play a 3-player version of Nim where each player wants to win but avoids losing at all costs (there is always a player who neither wins nor loses). Initially, the piles have sizes $43$, $99$, $x$, $y$, where $x$ and $y$ are positive integers. Assuming that the first player loses when all players play optimally, compute the maximum possible value of $xy$.
[i]Proposed by Sammy Luo[/i]
2011 Kyrgyzstan National Olympiad, 3
Given positive numbers ${a_1},{a_2},...,{a_n}$ with ${a_1} + {a_2} + ... + {a_n} = 1$. Prove that $\left( {\frac{1}{{a_1^2}} - 1} \right)\left( {\frac{1}{{a_2^2}} - 1} \right)...\left( {\frac{1}{{a_n^2}} - 1} \right) \geqslant {({n^2} - 1)^n}$.