Found problems: 6530
2006 Poland - Second Round, 1
Positive integers $a,b,c,x,y,z$ satisfy:
$a^2+b^2=c^2$, $x^2+y^2=z^2$
and
$|x-a| \leq 1$ , $|y-b| \leq 1$.
Prove that sets $\{a,b\}$ and $\{x,y\}$ are equal.
2005 Serbia Team Selection Test, 5
Let $a,b,c$ be positive reals such that $abc=1$ .Prove the inequality $\frac{a}{a^2+2}+\frac{b}{b^2+2}+\frac{c}{c^2+2}\leq 1$
2008 District Olympiad, 1
Let $ \{a_n\}_{n\geq 1}$ be a sequence of real numbers such that $ |a_{n\plus{}1}\minus{}a_n|\leq 1$, for all positive integers $ n$. Let $ \{b_n\}_{n\geq 1}$ be the sequence defined by \[ b_n \equal{} \frac { a_1\plus{} a_2 \plus{} \cdots \plus{}a_n} {n}.\] Prove that $ |b_{n\plus{}1}\minus{}b_n | \leq \frac 12$, for all positive integers $ n$.
2008 Stars Of Mathematics, 2
Let $\sqrt{23}>\frac{m}{n}$ where $ m,n$ are positive integers.
i) Prove that $ \sqrt{23}>\frac{m}{n}\plus{}\frac{3}{mn}.$
ii) Prove that $ \sqrt{23}<\frac{m}{n}\plus{}\frac{4}{mn}$ occurs infinitely often, and give at least three such examples.
[i]Dan Schwarz[/i]
2017 Singapore Senior Math Olympiad, 4
Find all functions $f : Z^+ \to Z^+$ such that $f(k + 1) >f(f(k))$ for $k > 1$, where $Z^+$ is the set of positive integers.
1980 Czech And Slovak Olympiad IIIA, 5
Solve a set of inequalities in the domain of integer numbers:
$$3x^2 +2yz \le 1+y^2$$
$$3y^2 +2zx \le 1+z^2$$
$$3z^2 +2xy \le 1+x^2$$
2009 Estonia Team Selection Test, 1
For arbitrary pairwise distinct positive real numbers $a, b, c$, prove the inequality
$$\frac{(a^2- b^2)^3 + (b^2-c^2)^3+(c^2-a^2)^3}{(a- b)^3 + (b-c)^3+(c-a)^3}> 8abc$$
2019 Bundeswettbewerb Mathematik, 2
Determine the smallest possible value of the sum $S (a, b, c) = \frac{ab}{c}+\frac{bc}{a}+\frac{ca}{b}$ where $a, b, c$ are three positive real numbers with $a^2 + b^2 + c^2 = 1$
2014 Austria Beginners' Competition, 3
Let $a, b, c$ and $d$ be real numbers with $a < b < c < d$.
Sort the numbers $x = a \cdot b + c \cdot d, y = b \cdot c + a \cdot d$ and $z = c \cdot a + b \cdot d$ in ascending\order and prove the correctness of your result.
(R. Henner, Vienna)
1991 Baltic Way, 7
If $\alpha,\beta,\gamma$ are the angles of an acute-angled triangle, prove that
\[\sin \alpha + \sin \beta > \cos \alpha + \cos\beta + \cos\gamma.\]
2007 Romania Team Selection Test, 2
Let $f: \mathbb{Q}\rightarrow \mathbb{R}$ be a function such that \[|f(x)-f(y)|\leq (x-y)^{2}\] for all $x,y \in\mathbb{Q}$. Prove that $f$ is constant.
2022 CHMMC Winter (2022-23), 3
Suppose that $a,b,c$ are complex numbers with $a+b+c = 0$, $|abc| = 1$, $|b| = |c|$, and $$\frac{9-\sqrt{33}}{48}
\le \cos^2 \left( arg \left( \frac{b}{a} \right) \right)\le \frac{9+\sqrt{33}}{48} .$$
Find the maximum possible value of $|-a^6+b^6+c^6|$.
2005 MOP Homework, 4
Let $ABCD$ be a convex quadrilateral and let $K$, $L$, $M$, $N$ be the midpoints of sides $AB$, $BC$, $CD$, $DA$ respectively. Let $NL$ and $KM$ meet at point $T$. Show that $8[DNTM] < [ABCD] < 8[DNTM]$, where $[P]$ denotes area of $P$.
2023 Brazil Undergrad MO, 3
Prove that there exists a constant $C > 0$ such that, for any integers $m, n$ with $n \geq m > 1$ and any real number $x > 1$, $$\sum_{k=m}^{n}\sqrt[k]{x} \leq C\bigg(\frac{m^2 \cdot \sqrt[m-1]{x}}{\log{x}} + n\bigg)$$
1991 Vietnam National Olympiad, 2
Let $G$ be centroid and $R$ the circunradius of a triangle $ABC$. The extensions of $GA,GB,GC$ meet the circuncircle again at $D,E,F$. Prove that:
$\frac{3}{R} \leq \frac{1}{GD} + \frac{1}{GE} + \frac{1}{GF} \leq \sqrt{3} \leq \frac{1}{AB} + \frac{1}{BC} + \frac{1}{CA}$
2019 Latvia Baltic Way TST, 1
Prove that for all positive real numbers $a, b, c$ with $\frac{1}{a}+\frac{1}{b}+\frac{1}{c} =1$ the following inequality holds:
$$3(ab+bc+ca)+\frac{9}{a+b+c} \le \frac{9abc}{a+b+c} + 2(a^2+b^2+c^2)+1$$
2017 Polish MO Finals, 6
Three sequences $(a_0, a_1, \ldots, a_n)$, $(b_0, b_1, \ldots, b_{n})$, $(c_0, c_1, \ldots, c_{2n})$ of non-negative real numbers are given such that for all $0\leq i,j\leq n$ we have $a_ib_j\leq (c_{i+j})^2$. Prove that
$$\sum_{i=0}^n a_i\cdot\sum_{j=0}^n b_j\leq \left( \sum_{k=0}^{2n} c_k\right)^2.$$
2008 Germany Team Selection Test, 1
Determine $ Q \in \mathbb{R}$ which is so big that a sequence with non-negative reals elements $ a_1 ,a_2, \ldots$ which satisfies the following two conditions:
[b](i)[/b] $ \forall m,n \geq 1$ we have $ a_{m \plus{} n} \leq 2 \left(a_m \plus{} a_n \right)$
[b](ii)[/b] $ \forall k \geq 0$ we have $ a_{2^k} \leq \frac {1}{(k \plus{} 1)^{2008}}$
such that for each sequence element we have the inequality $ a_n \leq Q.$
2017 Moldova Team Selection Test, 6
Let $a,b,c$ be positive real numbers that satisfy $a+b+c=abc$. Prove that
$$\sqrt{(1+a^2)(1+b^2)}+\sqrt{(1+b^2)(1+c^2)}+\sqrt{(1+a^2)(1+c^2)}-\sqrt{(1+a^2)(1+b^2)(1+c^2)} \ge 4.$$
2015 Silk Road, 1 (original)
Given positive real numbers $a,b,c,d$ such that
$ \frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}=6 \quad \text{and} \quad \frac{b}{a}+\frac{c}{b}+\frac{d}{c}+\frac{a}{d}=36.$
Prove the inequality
${{a}^{2}}+{{b}^{2}}+{{c}^{2}}+{{d}^{2}}>ab+ac+ad+bc+bd+cd.$
2024 Moldova Team Selection Test, 4
Let $a_1,a_2,\dots,a_{2023}$ be positive integers such that
[list=disc]
[*] $a_1,a_2,\dots,a_{2023}$ is a permutation of $1,2,\dots,2023$, and
[*] $|a_1-a_2|,|a_2-a_3|,\dots,|a_{2022}-a_{2023}|$ is a permutation of $1,2,\dots,2022$.
[/list]
Prove that $\max(a_1,a_{2023})\ge 507$.
2002 AMC 10, 19
If $a,b,c$ are real numbers such that $a^2+2b=7$, $b^2+4c=-7$, and $c^2+6a=-14$, find $a^2+b^2+c^2$.
$\textbf{(A) }14\qquad\textbf{(B) }21\qquad\textbf{(C) }28\qquad\textbf{(D) }35\qquad\textbf{(E) }49$
II Soros Olympiad 1995 - 96 (Russia), 11.1
Find $a$ and $b$ for which the largest and smallest is values of the function $y=\frac{x^2+ax+b}{x^2-x+1}$ are equal to the $2$ and $-3$ respectively.
1992 IberoAmerican, 3
Let $ABC$ be an equilateral triangle of sidelength 2 and let $\omega$ be its incircle.
a) Show that for every point $P$ on $\omega$ the sum of the squares of its distances to $A$, $B$, $C$ is 5.
b) Show that for every point $P$ on $\omega$ it is possible to construct a triangle of sidelengths $AP$, $BP$, $CP$. Also, the area of such triangle is $\frac{\sqrt{3}}{4}$.
2006 China Team Selection Test, 3
Given $n$ real numbers $a_1$, $a_2$ $\ldots$ $a_n$. ($n\geq 1$). Prove that there exists real numbers $b_1$, $b_2$ $\ldots$ $b_n$ satisfying:
(a) For any $1 \leq i \leq n$, $a_i - b_i$ is a positive integer.
(b)$\sum_{1 \leq i < j \leq n} (b_i - b_j)^2 \leq \frac{n^2-1}{12}$