Found problems: 6530
1994 Irish Math Olympiad, 5
Let $ f(n)$ be defined for $ n \in \mathbb{N}$ by $ f(1)\equal{}2$ and $ f(n\plus{}1)\equal{}f(n)^2\minus{}f(n)\plus{}1$ for $ n \ge 1$. Prove that for all $ n >1:$
$ 1\minus{}\frac{1}{2^{2^{n\minus{}1}}}<\frac{1}{f(1)}\plus{}\frac{1}{f(2)}\plus{}...\plus{}\frac{1}{f(n)}<1\minus{}\frac{1}{2^{2^n}}$
2015 Czech-Polish-Slovak Match, 3
Real numbers $x,y,z$ satisfy $$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+x+y+z=0$$ and none of them lies in the open interval $(-1,1)$. Find the maximum value of $x+y+z$.
[i]Proposed by Jaromír Šimša[/i]
2017 Kosovo National Mathematical Olympiad, 2
2 .Solve the inequality $|x-1|-2|x-4|>3+2x$
2023 Ukraine National Mathematical Olympiad, 8.3
Positive integers $x, y$ satisfy the following conditions:
$$\{\sqrt{x^2 + 2y}\}> \frac{2}{3}; \hspace{10mm} \{\sqrt{y^2 + 2x}\}> \frac{2}{3}$$
Show that $x = y$.
Here $\{x\}$ denotes the fractional part of $x$. For example, $\{3.14\} = 0.14$.
[i]Proposed by Anton Trygub[/i]
2010 Putnam, B2
Given that $A,B,$ and $C$ are noncollinear points in the plane with integer coordinates such that the distances $AB,AC,$ and $BC$ are integers, what is the smallest possible value of $AB?$
VI Soros Olympiad 1999 - 2000 (Russia), 9.4
Is there a function $f(x)$, which satisfies both of the following conditions:
a) if $x \ne y$, then $f(x)\ne f(y)$
b) for all real $x$, holds the inequality $f(x^2-1998x)-f^2(2x-1999)\ge \frac14$?
1977 IMO Longlists, 7
Prove the following assertion: If $c_1,c_2,\ldots ,c_n\ (n\ge 2)$ are real numbers such that
\[ (n-1)(c_1^2+c_2^2+\cdots +c_n^2)=(c_1+c_2+\cdots + c_n)^2,\]
then either all these numbers are nonnegative or all these numbers are nonpositive.
2010 South africa National Olympiad, 4
Given $n$ positive real numbers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n \ge 0$ and $x_1^2+x_2^2+\cdots+x_n^2=1$, prove that
\[\frac{x_1}{\sqrt{1}}+\frac{x_2}{\sqrt{2}}+\cdots+\frac{x_n}{\sqrt{n}}\ge 1.\]
2008 Moldova National Olympiad, 12.1
Consider the equation $ x^4 \minus{} 4x^3 \plus{} 4x^2 \plus{} ax \plus{} b \equal{} 0$, where $ a,b\in\mathbb{R}$. Determine the largest value $ a \plus{} b$ can take, so that the given equation has two distinct positive roots $ x_1,x_2$ so that $ x_1 \plus{} x_2 \equal{} 2x_1x_2$.
1990 China Team Selection Test, 1
Given a triangle $ ABC$ with angle $ C \geq 60^{\circ}$. Prove that:
$ \left(a \plus{} b\right) \cdot \left(\frac {1}{a} \plus{} \frac {1}{b} \plus{} \frac {1}{c} \right) \geq 4 \plus{} \frac {1}{\sin\left(\frac {C}{2}\right)}.$
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 Italy TST, 3
Let $p \geq 5$ be a prime.
(a) Show that exists a prime $q \neq p$ such that $q| (p-1)^{p}+1$
(b) Factoring in prime numbers $(p-1)^{p}+1 = \prod_{i=1}^{n}p_{i}^{a_{i}}$ show that:
\[\sum_{i=1}^{n}p_{i}a_{i}\geq \frac{p^{2}}2 \]
Azerbaijan Al-Khwarizmi IJMO TST 2025, 2
For $a,b,c$ positive real numbers satisfying $a^2+b^2+c^2 \geq 3$,show that:
$\sqrt[3]{\frac{a^3+b^3+c^3}{3}}+\frac{a+b+c}{9} \geq \frac{4}{3}$.
2014 Cezar Ivănescu, 2
[b]a)[/b] Let be two nonegative integers $ n\ge 1,k, $ and $ n $ real numbers $ a,b,\ldots ,c. $ Prove that
$$ (1/a+1/b+\cdots 1/c)\left( a^{1+k} +b^{1+k}+\cdots c^{1+k} \right)\ge n\left(a^k+b^k+\cdots +c^k\right) . $$
[b]b)[/b] If $ 1\le d\le e\le f\le g\le h\le i\le 1000 $ are six real numbers, determine the minimum value the expression
$$ d/e+f/g+h/i $$
can take.
2012 Romania National Olympiad, 3
[color=darkred]Let $\mathcal{C}$ be the set of integrable functions $f\colon [0,1]\to\mathbb{R}$ such that $0\le f(x)\le x$ for any $x\in [0,1]$ . Define the function $V\colon\mathcal{C}\to\mathbb{R}$ by
\[V(f)=\int_0^1f^2(x)\ \text{d}x-\left(\int_0^1f(x)\ \text{d}x\right)^2\ ,\ f\in\mathcal{C}\ .\]
Determine the following two sets:
[list][b]a)[/b] $\{V(f_a)\, |\, 0\le a\le 1\}$ , where $f_a(x)=0$ , if $0\le x\le a$ and $f_a(x)=x$ , if $a<x\le 1\, ;$
[b]b)[/b] $\{V(f)\, |\, f\in\mathcal{C}\}\ .$[/list] [/color]
2023 Korea Junior Math Olympiad, 6
Find the maximum value of real number $A$ such that $$3x^2 + y^2 + 1 \geq A(x^2 + xy + x)$$ for all positive integers $x, y.$
1980 Czech And Slovak Olympiad IIIA, 4
Let $a_1 < a_2< ...< a_n$ are real numbers, $$f(x) = \sum_{i=1}^n|x-a_i|,$$ for $n$ even. Find the minimum of this function.
1998 All-Russian Olympiad, 4
A connected graph has $1998$ points and each point has degree $3$. If $200$ points, no two of them joined by an edge, are deleted, show that the result is a connected graph.
2011 Graduate School Of Mathematical Sciences, The Master Cource, The University Of Tokyo, 2
Let $f(x,\ y)=\frac{x+y}{(x^2+1)(y^2+1)}.$
(1) Find the maximum value of $f(x,\ y)$ for $0\leq x\leq 1,\ 0\leq y\leq 1.$
(2) Find the maximum value of $f(x,\ y),\ \forall{x,\ y}\in{\mathbb{R}}.$
1997 Romania Team Selection Test, 1
We are given in the plane a line $\ell$ and three circles with centres $A,B,C$ such that they are all tangent to $\ell$ and pairwise externally tangent to each other. Prove that the triangle $ABC$ has an obtuse angle and find all possible values of this this angle.
[i]Mircea Becheanu[/i]
1999 North Macedonia National Olympiad, 5
If $a,b,c$ are positive numbers with $a^2 +b^2 +c^2 = 1$, prove that $a+b+c+\frac{1}{abc} \ge 4\sqrt3$
1982 IMO Longlists, 39
Let $S$ be the unit circle with center $O$ and let $P_1, P_2,\ldots, P_n$ be points of $S$ such that the sum of vectors $v_i=\stackrel{\longrightarrow}{OP_i}$ is the zero vector. Prove that the inequality $\sum_{i=1}^n XP_i \geq n$ holds for every point $X$.
2008 China Team Selection Test, 6
Find the maximal constant $ M$, such that for arbitrary integer $ n\geq 3,$ there exist two sequences of positive real number $ a_{1},a_{2},\cdots,a_{n},$ and $ b_{1},b_{2},\cdots,b_{n},$ satisfying
(1):$ \sum_{k \equal{} 1}^{n}b_{k} \equal{} 1,2b_{k}\geq b_{k \minus{} 1} \plus{} b_{k \plus{} 1},k \equal{} 2,3,\cdots,n \minus{} 1;$
(2):$ a_{k}^2\leq 1 \plus{} \sum_{i \equal{} 1}^{k}a_{i}b_{i},k \equal{} 1,2,3,\cdots,n, a_{n}\equiv M$.
2002 District Olympiad, 3
Let be two real numbers $ a,b, $ that satisfy $ 3^a+13^b=17^a $ and $ 5^a+7^b=11^b. $
Show that $ a<b. $
1983 National High School Mathematics League, 6
Let $a,b,c,d,m,n$ be positive real numbers. $P=\sqrt{ab}+\sqrt{cd},Q=\sqrt{ma+nc}\cdot\sqrt{\frac{b}{m}+\frac{d}{n}}$. Then
$\text{(A)}P\geq Q\qquad\text{(B)}P\leq Q\qquad\text{(C)}P<Q\qquad\text{(D)}$Not sure