Found problems: 6530
2009 Ukraine National Mathematical Olympiad, 4
Let $x \leq y \leq z \leq t$ be real numbers such that $xy + xz + xt + yz + yt + zt = 1.$
[b]a)[/b] Prove that $xt < \frac 13,$
b) Find the least constant $C$ for which inequality $xt < C$ holds for all possible values $x$ and $t.$
2006 Cono Sur Olympiad, 3
Let $n$ be a natural number. The finite sequence $\alpha$ of positive integer terms, there are $n$ different numbers ($\alpha$ can have repeated terms). Moreover, if from one from its terms any we subtract 1, we obtain a sequence which has, between its terms, at least $n$ different positive numbers. What's the minimum value of the sum of all the terms of $\alpha$?
2007 Peru IMO TST, 4
Let $a,b$ and $c$ be sides of a triangle. Prove that:
$\frac{\sqrt{b+c-a}}{\sqrt{b}+\sqrt{c}-\sqrt{a}}+\frac{\sqrt{c+a-b}}{\sqrt{c}+\sqrt{a}-\sqrt{b}}+\frac{\sqrt{a+b-c}}{\sqrt{a}+\sqrt{b}-\sqrt{c}}\leq 3$
2016-2017 SDML (Middle School), 10
For how many positive integer values of $a$ is it true that $x = 2$ is the only positive integer solution of the system of inequalities $$\begin{cases} 2x > 3x - 3 \\ 3x - a > -6 \end{cases}$$
$\text{(A) }1\qquad\text{(B) }2\qquad\text{(C) }3\qquad\text{(D) }4\qquad\text{(E) }5$
2011 SEEMOUS, Problem 1
Let $f:[0,1]\rightarrow R$ be a continuous function and n be an integer number,n>0.Prove that $\int_0^1f(x)dx \le (n+1)*\int_0^1 x^n*f(x)dx $
2014 Romania National Olympiad, 4
Let $n \in \mathbb{N} , n \ge 2$ and $ a_0,a_1,a_2,\cdots,a_n \in \mathbb{C} ; a_n \not = 0 $. Then:
[b][size=100][i]P.[/i][/size][/b] $|a_nz^n + a_{n-1}z^z{n-1} + \cdots + a_1z + a_0 | \le |a_n+a_0|$ for any $z \in \mathbb{C}, |z|=1$
[b][size=100][i]Q[/i][/size][/b]. $a_1=a_2=\cdots=a_{n-1}=0$ and $a_0/a_n \in [0,\infty)$
Prove that $ P \Longleftrightarrow Q$
1992 Taiwan National Olympiad, 6
Find the greatest positive integer $A$ with the following property: For every permutation of $\{1001,1002,...,2000\}$ , the sum of some ten consecutive terms is great than or equal to $A$.
2013 NIMO Problems, 6
Let $n$ and $k$ be integers satisfying $\binom{2k}{2} + n = 60$. It is known that $n$ days before Evan's 16th birthday, something happened. Compute $60-n$.
[i]Proposed by Evan Chen[/i]
1997 Czech And Slovak Olympiad IIIA, 5
For a given integer $n \ge 2$, find the maximum possible value of $V_n = \sin x_1 \cos x_2 +\sin x_2 \cos x_3 +...+\sin x_n \cos x_1$, where $x_1,x_2,...,x_n$ are real numbers.
2011 South East Mathematical Olympiad, 2
Let $P_i$ $i=1,2,......n$ be $n$ points on the plane , $M$ is a point on segment $AB$ in the same plane , prove : $\sum_{i=1}^{n} |P_iM| \le \max( \sum_{i=1}^{n} |P_iA| , \sum_{i=1}^{n} |P_iB| )$. (Here $|AB|$ means the length of segment $AB$) .
2007 Nicolae Păun, 2
Consider a sequence of positive real numbers $ \left( x_n \right)_{n\ge 1} $ and a primitivable function $ f:\mathbb{R}\longrightarrow\mathbb{R} . $
[b]a)[/b] Prove that $ f $ is monotonic and continuous if for any natural numbers $ n $ and real numbers $ x, $ the inequality
$$ f\left( x+x_n \right)\geqslant f(x) $$
is true.
[b]b)[/b] Show that $ f $ is convex if for any natural numbers $ n $ and real numbers $ x, $ the inequality
$$ f\left( x+2x_n \right) +f(x)\geqslant 2f\left( x+x_n \right) $$
is true.
[i]Sorin Rădulescu[/i] and [i]Ion Savu[/i]
2025 Austrian MO National Competition, 1
Let $a$, $b$ and $c$ be pairwise distinct nonnegative real numbers. Prove that
\[
(a + b + c) \left( \frac{a}{(b - c)^2} + \frac{b}{(c - a)^2} + \frac{c}{(a - b)^2} \right) > 4.
\]
[i](Karl Czakler)[/i]
1999 Rioplatense Mathematical Olympiad, Level 3, 4
Prove the following inequality:
$$ \frac{1}{\sqrt[3]{1^2}+\sqrt[3]{1 \cdot 2}+\sqrt[3]{2^2} }+\frac{1}{\sqrt[3]{3^2}+\sqrt[3]{3 \cdot 4}+\sqrt[3]{4^2} }+...+ \frac{1}{\sqrt[3]{999^2}+\sqrt[3]{999 \cdot 1000}+\sqrt[3]{1000^2} }> \frac{9}{2}$$
(The member on the left has 500 fractions.)
2007 China Girls Math Olympiad, 2
Let $ ABC$ be an acute triangle. Points $ D$, $ E$, and $ F$ lie on segments $ BC$, $ CA$, and $ AB$, respectively, and each of the three segments $ AD$, $ BE$, and $ CF$ contains the circumcenter of $ ABC$. Prove that if any two of the ratios $ \frac{BD}{DC}$, $ \frac{CE}{EA}$, $ \frac{AF}{FB}$, $ \frac{BF}{FA}$, $ \frac{AE}{EC}$, $ \frac{CD}{DB}$ are integers, then triangle $ ABC$ is isosceles.
1979 Romania Team Selection Tests, 4.
Give an example of a second degree polynomial $P\in \mathbb{R}[x]$ such that
\[\forall x\in \mathbb{R}\text{ with } |x|\leqslant 1: \;
\left|P(x)+\frac{1}{x-4}\right|
\leqslant 0.01.\]
Are there linear polynomials with this property?
[i]Octavian Stănășilă[/i]
2024 China Team Selection Test, 18
Let $m,n\in\mathbb Z_{\ge 0},$ $a_0,a_1,\ldots ,a_m,b_0,b_1,\ldots ,b_n\in\mathbb R_{\ge 0}$ For any integer $0\le k\le m+n,$ define $c_k:=\max_{i+j=k}a_ib_j.$ Proof
$$\frac 1{m+n+1}\sum_{k=0}^{m+n}c_k\ge\frac 1{(m+1)(n+1)}\sum_{i=0}^{m}a_i\sum_{j=0}^{n}b_j.$$
[i]Created by Yinghua Ai[/i]
2009 South East Mathematical Olympiad, 3
Let $x,y,z $ be positive reals such that $\sqrt{a}=x(y-z)^2$, $\sqrt{b}=y(z-x)^2$ and $\sqrt{c}=z(x-y)^2$. Prove that
\[a^2+b^2+c^2 \geq 2(ab+bc+ca)\]
2021 Indonesia TST, A
A positive real $M$ is $strong$ if for any positive reals $a$, $b$, $c$ satisfying
$$ \text{max}\left\{ \frac{a}{b+c} , \frac{b}{c+a} , \frac{c}{a+b} \right\} \geqslant M $$
then the following inequality holds:
$$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b} > 20.$$
(a) Prove that $M=20-\frac{1}{20}$ is not $strong$.
(b) Prove that $M=20-\frac{1}{21}$ is $strong$.
2022 Malaysia IMONST 2, 2
Without using a calculator, determine which number is greater: $17^{24}$ or $31^{19}$
2024 Indonesia MO, 7
Suppose $P(x) = x^n + a_{n-1} x^{n-1} + \cdots + a_1x + a_0$ where $a_0, a_1, \ldots, a_{n-1}$ are reals for $n\geq 1$ (monic $n$th-degree polynomial with real coefficients). If the inequality
\[ 3(P(x)+P(y)) \geq P(x+y) \] holds for all reals $x,y$, determine the minimum possible value of $P(2024)$.
2008 Cuba MO, 1
Given a polynomial of degree $2$, $p(x) = ax^2 +bx+c$ define the function $$S(p) = (a -b)^2 + (b - c)^2 + (c - a)^2.$$ Determine the real number$ r$such that, for any polynomial $p(x)$ of degree $2$ with real roots, holds $S(p) \ge ra^2$
2012 Turkey Team Selection Test, 2
In a plane, the six different points $A, B, C, A', B', C'$ are given such that triangles $ABC$ and $A'B'C'$ are congruent, i.e. $AB=A'B', BC=B'C', CA=C'A'.$ Let $G$ be the centroid of $ABC$ and $A_1$ be an intersection point of the circle with diameter $AA'$ and the circle with center $A'$ and passing through $G.$ Define $B_1$ and $C_1$ similarly. Prove that
\[ AA_1^2+BB_1^2+CC_1^2 \leq AB^2+BC^2+CA^2 \]
2023 China Northern MO, 4
Given the sequence $(a_n) $ satisfies $1=a_1< a_2 < a_3< \cdots<a_n $ and there exist real number $m$ such that
$$\displaystyle\sum_{i=1}^{n-1} \sqrt[3]{\frac{a_{i+1}-a_i}{(2+a_i)^4}}\leq m $$
for any positive integer $ n $ not less than 2 . Find the minimum of $m.$
2011 JBMO Shortlist, 3
$\boxed{\text{A3}}$If $a,b$ be positive real numbers, show that:$$ \displaystyle{\sqrt{\dfrac{a^2+ab+b^2}{3}}+\sqrt{ab}\leq a+b}$$
1973 Chisinau City MO, 67
The product of $10$ natural numbers is equal to $10^{10}$. What is the largest possible sum of these numbers?