Found problems: 6530
2019 Thailand TSTST, 2
Let $a,b,c\in(0,\frac{4}{3})$ and $a + b + c = 3$. Prove that $$\frac{4abc}{(a+b)(a+c)}+\frac{(a+b)^2+(a+c)^2}{(a+b)+(a+c)}\leq\sum_{cyc}\frac{1}{a^2(3b+3c-5)}.$$
2009 Mediterranean Mathematics Olympiad, 4
Let $x,y,z$ be positive real numbers. Prove that
\[ \sum_{cyclic} \frac{xy}{xy+x^2+y^2} ~\le~ \sum_{cyclic} \frac{x}{2x+z} \]
[i](Proposed by Šefket Arslanagić, Bosnia and Herzegovina)[/i]
2015 Estonia Team Selection Test, 10
Let $n$ be an integer and $a, b$ real numbers such that $n > 1$ and $a > b > 0$. Prove that $$(a^n - b^n) \left ( \frac{1}{b^{n- 1}} - \frac{1}{a^{n -1}}\right) > 4n(n -1)(\sqrt{a} - \sqrt{b})^2$$
1980 AMC 12/AHSME, 6
A positive number $x$ satisfies the inequality $\sqrt{x} < 2x$ if and only if
$\text{(A)} \ x > \frac{1}{4} \qquad \text{(B)} \ x > 2 \qquad \text{(C)} x > 4 \qquad \text{(D)} \ x < \frac{1}{4}\qquad \text{(E)} x < 4$
2012 JBMO ShortLists, 2
Let $a$ , $b$ , $c$ be positive real numbers such that $abc=1$ . Show that :
\[\frac{1}{a^3+bc}+\frac{1}{b^3+ca}+\frac{1}{c^3+ab} \leq \frac{ \left (ab+bc+ca \right )^2 }{6}\]
2006 Putnam, B2
Prove that, for every set $X=\{x_{1},x_{2},\dots,x_{n}\}$ of $n$ real numbers, there exists a non-empty subset $S$ of $X$ and an integer $m$ such that
\[\left|m+\sum_{s\in S}s\right|\le\frac1{n+1}\]
2006 Hong Kong TST., 4
Let x,y,z be positive real numbers such that $x+y+z=1$.
For positive integer n, define $S_n = x^n+y^n+z^n$
Furthermore, let $P=S_2 S_{2005}$ and $Q=S_3 S_{2004}$.
(a) Find the smallest possible value of Q.
(b) If $x,y,z$ are pairwise distinct, determine whether P or Q is larger.
2024 Benelux, 1
Let $a_0,a_1,\dots,a_{2024}$ be real numbers such that $\left|a_{i+1}-a_i\right| \le 1$ for $i=0,1,\dots,2023$.
a) Find the minimum possible value of $$a_0a_1+a_1a_2+\dots+a_{2023}a_{2024}$$
b) Does there exist a real number $C$ such that $$a_0a_1-a_1a_2+a_2a_3-a_3a_4+\dots+a_{2022}a_{2023}-a_{2023}a_{2024} \ge C$$ for all real numbers $a_0,a_1,\dots,a_2024$ such that $\left|a_{i+1}-a_i\right| \le 1$ for $i=0,1,\dots,2023$.
2012 Romania Team Selection Test, 3
Let $m$ and $n$ be two positive integers for which $m<n$. $n$ distinct points $X_1,\ldots , X_n$ are in the interior of the unit disc and at least one of them is on its border. Prove that we can find $m$ distinct points $X_{i_1},\ldots , X_{i_m}$ so that the distance between their center of gravity and the center of the circle is at least $\frac{1}{1+2m(1- 1/n)}$.
2022 Cyprus JBMO TST, 3
If $a,b,c$ are positive real numbers with $abc=1$, prove that
(a) \[2\left(\frac{ab}{a+b}+\frac{bc}{b+c}+\frac{ca}{c+a}\right) \geqslant \frac{9}{ab+bc+ca}\]
(b)\[2\left(\frac{ab}{a+b}+\frac{bc}{b+c}+\frac{ca}{c+a}\right) \geqslant \frac{9}{a^2 b+b^2 c+c^2 a}\]
2006 Pre-Preparation Course Examination, 1
Show that for a triangle we have \[ \max \{am_a,bm_b,cm_c\} \leq sR \] where $m_a$ denotes the length of median of side $BC$ and $s$ is half of the perimeter of the triangle.
2010 India National Olympiad, 3
Find all non-zero real numbers $ x, y, z$ which satisfy the system of equations:
\[ (x^2 \plus{} xy \plus{} y^2)(y^2 \plus{} yz \plus{} z^2)(z^2 \plus{} zx \plus{} x^2) \equal{} xyz\]
\[ (x^4 \plus{} x^2y^2 \plus{} y^4)(y^4 \plus{} y^2z^2 \plus{} z^4)(z^4 \plus{} z^2x^2 \plus{} x^4) \equal{} x^3y^3z^3\]
2009 Mathcenter Contest, 3
Let $x,y,z>0$ Prove that $$\frac{x^2+2}{\sqrt{z^2+xy}}+\dfrac{y^2+2}{\sqrt{x ^2+yz}}+\dfrac{z^2+2}{\sqrt{y^2+zx}}\geq 6$$.
[i](nooonuii)[/i]
2007 Hanoi Open Mathematics Competitions, 8
Let a; b; c be positive integers. Prove that
$$ \frac{(b+c-a)^2}{(b+c)^2+a^2} + \frac{(c+a-b)^2}{(c+a)^2+b^2} + \frac{(a+b-c)^2}{(a+b)^2+c^2} \geq \frac{3}{5}$$
1965 Bulgaria National Olympiad, Problem 2
Prove the inequality:
$$(1+\sin^2\alpha)^n+(1+\cos^2\alpha)^n\ge2\left(\frac32\right)^n$$is true for every natural number $n$. When does equality hold?
2020 Turkey EGMO TST, 6
$x,y,z$ are positive real numbers such that:
$$xyz+x+y+z=6$$
$$xyz+2xy+yz+zx+z=10$$
Find the maximum value of:
$$(xy+1)(yz+1)(zx+1)$$
Russian TST 2018, P1
Let $a,b,c{}$ be positive real numbers. Prove that \[108\cdot(ab+bc+ca)\leqslant(\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a})^4.\]
2023 Moldova Team Selection Test, 9
Let $ n $ $(n\geq2)$ be an integer. Find the greatest possible value of the expression $$E=\frac{a_1}{1+a_1^2}+\frac{a_2}{1+a_2^2}+\ldots+\frac{a_n}{1+a_n^2}$$ if the positive real numbers $a_1,a_2,\ldots,a_n$ satisfy $a_1+a_2+\ldots+a_n=\frac{n}{2}.$ What are the values of $a_1,a_2,\ldots,a_n$ when the greatest value is achieved?
2009 Romania National Olympiad, 3
Let be a natural number $ n, $ a permutation $ \sigma $ of order $ n, $ and $ n $ nonnegative real numbers $ a_1,a_2,\ldots , a_n. $ Prove the following inequality.
$$ \left( a_1^2+a_{\sigma (1)} \right)\left( a_2^2+a_{\sigma (2)} \right)\cdots \left( a_n^2+a_{\sigma (n)} \right)\ge \left( a_1^2+a_1 \right)\left( a_2^2+a_{2} \right)\cdots \left( a_n^2+a_n \right) $$
Kvant 2025, M2827
It is known about positive numbers $a, b, c$ that it is possible to form a triangle from segments of length $a^{2024}, b^{2024}, c^{2024}$. Prove that it is possible to reduce one of the numbers $a, b, c$ by $2024$ times and obtain the numbers $a', b', c'$ so that segments with lengths $a', b', c'$ can also be formed into a triangle.
[i]L. Shatunov[/i]
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 $X$.
2005 China Team Selection Test, 2
Let $n$ be a positive integer, and $x$ be a positive real number. Prove that $$\sum_{k=1}^{n} \left( x \left[\frac{k}{x}\right] - (x+1)\left[\frac{k}{x+1}\right]\right) \leq n,$$ where $[x]$ denotes the largest integer not exceeding $x$.
2003 China Team Selection Test, 3
Suppose $A\subset \{(a_1,a_2,\dots,a_n)\mid a_i\in \mathbb{R},i=1,2\dots,n\}$. For any $\alpha=(a_1,a_2,\dots,a_n)\in A$ and $\beta=(b_1,b_2,\dots,b_n)\in A$, we define
\[ \gamma(\alpha,\beta)=(|a_1-b_1|,|a_2-b_2|,\dots,|a_n-b_n|), \] \[ D(A)=\{\gamma(\alpha,\beta)\mid\alpha,\beta\in A\}. \] Please show that $|D(A)|\geq |A|$.
1990 China National Olympiad, 3
A function $f(x)$ defined for $x\ge 0$ satisfies the following conditions:
i. for $x,y\ge 0$, $f(x)f(y)\le x^2f(y/2)+y^2f(x/2)$;
ii. there exists a constant $M$($M>0$), such that $|f(x)|\le M$ when $0\le x\le 1$.
Prove that $f(x)\le x^2$.
2010 Math Prize For Girls Problems, 9
Lynnelle took 10 tests in her math class at Stanford. Her score on each test was an integer from 0 through 100. She noticed that, for every four consecutive tests, her average score on those four tests was at most 47.5. What is the largest possible average score she could have on all 10 tests?