Found problems: 6530
2009 Indonesia TST, 3
Let $ n \ge 2009$ be an integer and define the set:
\[ S \equal{} \{2^x|7 \le x \le n, x \in \mathbb{N}\}.
\]
Let $ A$ be a subset of $ S$ and the sum of last three digits of each element of $ A$ is $ 8$. Let $ n(X)$ be the number of elements of $ X$. Prove that
\[ \frac {28}{2009} < \frac {n(A)}{n(S)} < \frac {82}{2009}.
\]
2004 Federal Competition For Advanced Students, P2, 1
Prove without using advanced (differential) calculus:
(a) For any real numbers a,b,c,d it holds that $a^6+b^6+c^6+d^6-6abcd \ge -2$.
When does equality hold?
(b) For which natural numbers $k$ does some inequality of the form $a^k +b^k +c^k +d^k -kabcd \ge M_k$ hold for all real $a,b,c,d$? For each such $k$,
2009 Indonesia TST, 3
Let $ x,y,z$ be real numbers. Find the minimum value of $ x^2\plus{}y^2\plus{}z^2$ if $ x^3\plus{}y^3\plus{}z^3\minus{}3xyz\equal{}1$.
2008 Iran MO (3rd Round), 2
Find the smallest real $ K$ such that for each $ x,y,z\in\mathbb R^{ \plus{} }$:
\[ x\sqrt y \plus{} y\sqrt z \plus{} z\sqrt x\leq K\sqrt {(x \plus{} y)(y \plus{} z)(z \plus{} x)}
\]
2012 Turkey Junior National Olympiad, 3
Let $a, b, c$ be positive real numbers satisfying $a^3+b^3+c^3=a^4+b^4+c^4$. Show that
\[ \frac{a}{a^2+b^3+c^3}+\frac{b}{a^3+b^2+c^3}+\frac{c}{a^3+b^3+c^2} \geq 1 \]
2011 Kyrgyzstan National Olympiad, 4
Given equation ${a^5} - {a^3} + a = 2$, with real $a$ . Prove that $3 < {a^6} < 4$.
2006 China Girls Math Olympiad, 7
Given that $x_{i}>0$, $i = 1, 2, \cdots, n$, $k \geq 1$. Show that: \[\sum_{i=1}^{n}\frac{1}{1+x_{i}}\cdot \sum_{i=1}^{n}x_{i}\leq \sum_{i=1}^{n}\frac{x_{i}^{k+1}}{1+x_{i}}\cdot \sum_{i=1}^{n}\frac{1}{x_{i}^{k}}\]
2002 China Team Selection Test, 3
Seventeen football fans were planning to go to Korea to watch the World Cup football match. They selected 17 matches. The conditions of the admission tickets they booked were such that
- One person should book at most one admission ticket for one match;
- At most one match was same in the tickets booked by every two persons;
- There was one person who booked six tickets.
How many tickets did those football fans book at most?
2016 India Regional Mathematical Olympiad, 2
Let \(a,b,c\) be positive real numbers such that \(\dfrac{ab}{1+bc}+\dfrac{bc}{1+ca}+\dfrac{ca}{1+ab}=1\). Prove that $$\dfrac{1}{a^3}+\dfrac{1}{b^3}+\dfrac{1}{c^3} \ge 6\sqrt{2}$$
2008 JBMO Shortlist, 2
Find all real numbers $ a,b,c,d$ such that \[ \left\{\begin{array}{cc}a \plus{} b \plus{} c \plus{} d \equal{} 20, \\
ab \plus{} ac \plus{} ad \plus{} bc \plus{} bd \plus{} cd \equal{} 150. \end{array} \right.\]
2008 Putnam, A6
Prove that there exists a constant $ c>0$ such that in every nontrivial finite group $ G$ there exists a sequence of length at most $ c\ln |G|$ with the property that each element of $ G$ equals the product of some subsequence. (The elements of $ G$ in the sequence are not required to be distinct. A [i]subsequence[/i] of a sequence is obtained by selecting some of the terms, not necessarily consecutive, without reordering them; for example, $ 4,4,2$ is a subesequence of $ 2,4,6,4,2,$ but $ 2,2,4$ is not.)
2008 Romania National Olympiad, 3
Let $ a,b \in [0,1]$. Prove that \[ \frac 1{1\plus{}a\plus{}b} \leq 1 \minus{} \frac {a\plus{}b}2 \plus{} \frac {ab}3.\]
2018 India IMO Training Camp, 2
For an integer $n\ge 2$ find all $a_1,a_2,\cdots ,a_n, b_1,b_2,\cdots , b_n$ so that
(a) $0\le a_1\le a_2\le \cdots \le a_n\le 1\le b_1\le b_2\le \cdots \le b_n;$
(b) $\sum_{k=1}^n (a_k+b_k)=2n;$
(c) $\sum_{k=1}^n (a_k^2+b_k^2)=n^2+3n.$
2019 IMO Shortlist, A2
Let $u_1, u_2, \dots, u_{2019}$ be real numbers satisfying \[u_{1}+u_{2}+\cdots+u_{2019}=0 \quad \text { and } \quad u_{1}^{2}+u_{2}^{2}+\cdots+u_{2019}^{2}=1.\] Let $a=\min \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$ and $b=\max \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$. Prove that
\[
a b \leqslant-\frac{1}{2019}.
\]
PEN H Problems, 66
Let $b$ be a positive integer. Determine all $2002$-tuples of non-negative integers $(a_{1}, a_{2}, \cdots, a_{2002})$ satisfying \[\sum^{2002}_{j=1}{a_{j}}^{a_{j}}=2002{b}^{b}.\]
2015 Serbia National Math Olympiad, 5
Let $x,y,z$ be nonnegative positive integers.
Prove $\frac{x-y}{xy+2y+1}+\frac{y-z}{zy+2z+1}+\frac{z-x}{xz+2x+1}\ge 0$
2012 Austria Beginners' Competition, 3
Let $a$ and $b$ be two positive real numbers with $a \le 2b \le 4a$.
Prove that $4ab \le2 (a^2+ b^2) \le 5 ab$.
1989 All Soviet Union Mathematical Olympiad, 507
Find the least possible value of $(x + y)(y + z)$ for positive reals satisfying $(x + y + z) xyz = 1$.
1999 APMO, 4
Determine all pairs $(a,b)$ of integers with the property that the numbers $a^2+4b$ and $b^2+4a$ are both perfect squares.
2010 Contests, 2
Let the real numbers $a,b,c,d$ satisfy the relations $a+b+c+d=6$ and $a^2+b^2+c^2+d^2=12.$ Prove that
\[36 \leq 4 \left(a^3+b^3+c^3+d^3\right) - \left(a^4+b^4+c^4+d^4 \right) \leq 48.\]
[i]Proposed by Nazar Serdyuk, Ukraine[/i]
1990 IMO Longlists, 46
For each $P$ inside the triangle $ABC$, let $A(P), B(P)$, and $C(P)$ be the points of intersection of the lines $AP, BP$, and $CP$ with the sides opposite to $A, B$, and $C$, respectively. Determine $P$ in such a way that the area of the triangle $A(P)B(P)C(P)$ is as large as possible.
2012 CHKMO, 3
For any positive integer $n$ and real numbers $a_i>0$ ($i=1,2,...,n$), prove that
\[\sum_{k=1}^n \frac{k}{a_1^{-1}+a_2^{-1}+...+a_k^{-1}}\leq 2\sum_{k=1}^n a_k.\]
Discuss if the "$2$" at the right hand side of the inequality can or cannot be replaced by a smaller real number.
1986 IMO Longlists, 66
One hundred red points and one hundred blue points are chosen in the plane, no three of them lying on a line. Show that these points can be connected pairwise, red ones with blue ones, by disjoint line segments.
2006 Taiwan National Olympiad, 3
$a_1, a_2, ..., a_{95}$ are positive reals. Show that
$\displaystyle \sum_{k=1}^{95}{a_k} \le 94+ \prod_{k=1}^{95}{\max{\{1,a_k\}}}$
1961 IMO Shortlist, 2
Let $ a$, $ b$, $ c$ be the sides of a triangle, and $ S$ its area. Prove:
\[ a^{2} \plus{} b^{2} \plus{} c^{2}\geq 4S \sqrt {3}
\]
In what case does equality hold?