Found problems: 6530
1978 Bulgaria National Olympiad, Problem 6
The base of the pyramid with vertex $S$ is a pentagon $ABCDE$ for which $BC>DE$ and $AB>CD$. If $AS$ is the longest edge of the pyramid prove that $BS>CS$.
[i]Jordan Tabov[/i]
2009 Indonesia TST, 4
Let $ a$, $ b$, and $ c$ be positive real numbers such that $ ab + bc + ca = 3$. Prove the inequality
\[ 3 + \sum_{\mathrm{\cyc}} (a - b)^2 \ge \frac {a + b^2c^2}{b + c} + \frac {b + c^2a^2}{c + a} + \frac {c + a^2b^2}{a + b} \ge 3.
\]
2007 Pre-Preparation Course Examination, 1
Let $a\geq 2$ be a natural number. Prove that $\sum_{n=0}^\infty\frac1{a^{n^{2}}}$ is irrational.
2020 Dutch IMO TST, 1
Given are real numbers $a_1, a_2,..., a_{2020}$, not necessarily different.
For every $n \ge 2020$, define $a_{n + 1}$ as the smallest real zero of the polynomial $$P_n (x) = x^{2n} + a_1x^{2n - 2} + a_2x^{2n - 4} +... + a_{n -1}x^2 + a_n$$, if it exists. Assume that $a_{n + 1}$ exists for all $n \ge 2020$.
Prove that $a_{n + 1} \le a_n$ for all $n \ge 2021$.
2018 Hanoi Open Mathematics Competitions, 1
Let $a, b$, and $c$ be distinct positive integers such that $a + 2b + 3c < 12$.
Which of the following inequalities must be true?
A. $a + b + c < 7$
B. $a- b + c < 4$
C. $b + c- a < 3$
D. $a + b- c <5 $
E. $5a + 3b + c < 27$
1968 Poland - Second Round, 4
Prove that if the numbers $ a, b, c $, are the lengths of the sides of a triangle and the sum of the numbers $x,y,z$ is zero, then $$a^2yz + b^2zx + c^2xy \leq 0.$$
1974 Czech and Slovak Olympiad III A, 1
Let $\left(a_k\right)_{k=1}^\infty$ be a sequence of positive numbers such that \[a_{k-1}a_{k+1}\ge a_k^2\] for all $k>1.$ For $n\ge1$ denote \[b_n=\left(a_1a_2\cdots a_n\right)^{1/n}.\] Show that also the inequality \[b_{n-1}b_{n+1}\ge b_n^2\] holds for every $n>1.$
1972 IMO Longlists, 32
If $n_1, n_2, \cdots, n_k$ are natural numbers and $n_1+n_2+\cdots+n_k = n$,
show that
\[max(n_1n_2\cdots n_k)=(t + 1)^rt^{k-r},\]
where $t =\left[\frac{n}{k}\right]$ and $r$ is the remainder of $n$ upon division by $k$; i.e., $n = tk + r, 0 \le r \le k- 1$.
1990 India Regional Mathematical Olympiad, 2
For all positive real numbers $ a,b,c$, prove that
\[ \frac {a}{b \plus{} c} \plus{} \frac {b}{c \plus{} a} \plus{} \frac {c}{a \plus{} b} \geq \frac {3}{2}.\]
1993 Chile National Olympiad, 5
Let $a,b,c$ three positive numbers less than $ 1$. Prove that cannot occur simultaneously these three inequalities:
$$a (1- b)>\frac14$$
$$b (1-c)>\frac14 $$
$$c (1-a)>\frac14$$
1991 Polish MO Finals, 3
If $x, y, z$ are real numbers satisfying $x^2 +y^2 +z^2 = 2$, prove the inequality
\[ x + y + z \leq 2 + xyz \]
When does equality occur?
2007 Today's Calculation Of Integral, 232
For $ f(x)\equal{}1\minus{}\sin x$, let $ g(x)\equal{}\int_0^x (x\minus{}t)f(t)\ dt.$
Show that $ g(x\plus{}y)\plus{}g(x\minus{}y)\geq 2g(x)$ for any real numbers $ x,\ y.$
2016 Germany Team Selection Test, 2
The positive integers $a_1,a_2, \dots, a_n$ are aligned clockwise in a circular line with $n \geq 5$. Let $a_0=a_n$ and $a_{n+1}=a_1$. For each $i \in \{1,2,\dots,n \}$ the quotient \[ q_i=\frac{a_{i-1}+a_{i+1}}{a_i} \] is an integer. Prove \[ 2n \leq q_1+q_2+\dots+q_n < 3n. \]
2018 Saint Petersburg Mathematical Olympiad, 6
Let $a,b,c,d>0$ . Prove that $a^4+b^4+c^4+d^4 \geq 4abcd+4(a-b)^2 \sqrt{abcd}$
2023 Peru MO (ONEM), 2
For each positive real number $x$, let $f(x)=\frac{x}{1+x}$ . Prove that if $a$, $b,$ $c$ are the sidelengths of a triangle, then $f(a)$, $f(b),$ $f(c)$ are sidelengths of a triangle.
1999 Miklós Schweitzer, 5
Let $\alpha>-2$ , $n\in \mathbb{N}$ and $y_1,\cdots,y_n$ be the solutions to the system of equations:
$\sum_{j=1}^n \frac{y_j}{j+k+\alpha}= \frac{1}{n+1+k+\alpha}$ , $k=1,\cdots,n$
Prove that $y_{j-1}y_{j+1}\leq y_j^2 \,\forall 1<j<n$
2005 Germany Team Selection Test, 2
If $a$, $b$, $c$ are positive reals such that $a+b+c=1$, prove that
\[\frac{1+a}{1-a}+\frac{1+b}{1-b}+\frac{1+c}{1-c}\leq 2\left(\frac{b}{a}+\frac{c}{b}+\frac{a}{c}\right).\]
1987 India National Olympiad, 4
If $ x$, $ y$, $ z$, and $ n$ are natural numbers, and $ n\geq z$ then prove that the relation $ x^n \plus{} y^n \equal{} z^n$ does not hold.
2012 Dutch IMO TST, 2
Let $a, b, c$ and $d$ be positive real numbers. Prove that
$$\frac{a - b}{b + c}+\frac{b - c}{c + d}+\frac{c - d}{d + a} +\frac{d - a}{a + b } \ge 0 $$
2022 China Second Round A2, 1
$a_1,a_2,...,a_9$ are nonnegative reals with sum $1$. Define $S$ and $T$ as below:
$$S=\min\{a_1,a_2\}+2\min\{a_2,a_3\}+...+9\min\{a_9,a_1\}$$
$$T=\max\{a_1,a_2\}+2\max\{a_2,a_3\}+...+9\max\{a_9,a_1\}$$
When $S$ reaches its maximum, find all possible values of $T$.
2015 Poland - Second Round, 2
Let $A$ be an integer and $A>1$. Let $a_{1}=A^{A}$, $a_{n+1}=A^{a_{n}}$ and $b_{1}=A^{A+1}$, $b_{n+1}=2^{b_{n}}$, $n=1, 2, 3, ...$. Prove that $a_{n}<b_{n}$ for each $n$.
1998 IMC, 6
Let $f: [0,1]\rightarrow\mathbb{R}$ be a continuous function satisfying $xf(y)+yf(x)\le 1$ for every $x,y\in[0,1]$.
(a) Show that $\int^1_0 f(x)dx \le \frac{\pi}4$.
(b) Find such a funtion for which equality occurs.
2022 Austrian MO National Competition, 1
Prove that for all positive real numbers $x, y$ and $z$, the double inequality $$0 < \frac{1}{x + y + z + 1} -\frac{1}{(x + 1)(y + 1)(z + 1)} \le \frac18$$ holds. When does equality hold in the right inequality?
[i](Walther Janous)[/i]
1982 Putnam, B6
Denote by $S(a,b,c)$ the area of a triangle whose lengthes of three sides are $a,b,c$
Prove that for any positive real numbers $a_{1},b_{1},c_{1}$ and $a_{2},b_{2},c_{2}$ which can serve as the lengthes of three sides of two triangles respectively ,we have
$ \sqrt{S(a_{1},b_{1},c_{1})}+\sqrt{S(a_{2},b_{2},c_{2})}\le\sqrt{S(a_{1}+a_{2},b_{1}+b_{2},c_{1}+c_{2})}$
2010 Contests, 3
Given $a_1\ge 1$ and $a_{k+1}\ge a_k+1$ for all $k\ge 1,2,\dots,n$, show that $a_1^3+a_2^3+\dots+a_n^3\ge (a_1+a_2+\dots+a_n)^2$