Found problems: 6530
2014 Contests, 1
Prove that for $\forall$ $a,b,c\in [\frac{1}{3},3]$ the following inequality is true:
$\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}\geq \frac{7}{5}$.
2009 China National Olympiad, 1
Given an integer $ n > 3.$ Let $ a_{1},a_{2},\cdots,a_{n}$ be real numbers satisfying $ min |a_{i} \minus{} a_{j}| \equal{} 1, 1\le i\le j\le n.$ Find the minimum value of $ \sum_{k \equal{} 1}^n|a_{k}|^3.$
2011 AIME Problems, 13
Point $P$ lies on the diagonal $AC$ of square $ABCD$ with $AP>CP$. Let $O_1$ and $O_2$ be the circumcenters of triangles $ABP$ and $CDP$ respectively. Given that $AB=12$ and $\angle O_1 P O_2 = 120^\circ$, then $AP=\sqrt{a}+\sqrt{b}$ where $a$ and $b$ are positive integers. Find $a+b$.
2014 Contests, 3
Let $a$, $b$, $c$ and $d$ be real numbers such that no two of them are equal,
\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}=4\] and $ac=bd$. Find the maximum possible value of
\[\frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}.\]
2018 Moldova EGMO TST, 5
Let $a$ and $b$ be real numbers such that $a + b = 1$. Prove the inequality
$$\sqrt{1+5a^2} + 5\sqrt{2+b^2} \geq 9.$$
[i]Proposed by Baasanjav Battsengel[/i]
2018 Vietnam National Olympiad, 3
An investor has two rectangular pieces of land of size $120\times 100$.
a. On the first land, she want to build a house with a rectangular base of size $25\times 35$ and nines circular flower pots with diameter $5$ outside the house. Prove that even the flower pots positions are chosen arbitrary on the land, the remaining land is still sufficient to build the desired house.
b. On the second land, she want to construct a polygonal fish pond such that the distance from an arbitrary point on the land, outside the pond, to the nearest pond edge is not over $5$. Prove that the perimeter of the pond is not smaller than $440-20\sqrt{2}$.
2001 China Team Selection Test, 2
Let ${a_n}$ be a non-increasing sequence of positive numbers. Prove that if for $n \ge 2001$, $na_{n} \le 1$, then for any positive integer $m \ge 2001$ and $x \in \mathbb{R}$, the following inequality holds:
$\left | \sum_{k=2001}^{m} a_{k} \sin kx \right | \le 1 + \pi$
1995 China National Olympiad, 1
Let $a_1,a_2,\cdots ,a_n; b_1,b_2,\cdots ,b_n (n\ge 3)$ be real numbers satisfying the following conditions:
(1) $a_1+a_2+\cdots +a_n= b_1+b_2+\cdots +b_n $;
(2) $0<a_1=a_2, a_i+a_{i+1}=a_{i+2}$ ($i=1,2,\cdots ,n-2$);
(3) $0<b_1\le b_2, b_i+b_{i+1}\le b_{i+2}$ ($i=1,2,\cdots ,n-2$).
Prove that $a_{n-1}+a_n\le b_{n-1}+b_n$.
1976 AMC 12/AHSME, 23
For integers $k$ and $n$ such that $1\le k<n$, let $C^n_k=\frac{n!}{k!(n-k)!}$. Then $\left(\frac{n-2k-1}{k+1}\right)C^n_k$ is an integer
$\textbf{(A) }\text{for all }k\text{ and }n\qquad$
$\textbf{(B) }\text{for all even values of }k\text{ and }n,\text{ but not for all }k\text{ and }n\qquad$
$\textbf{(C) }\text{for all odd values of }k\text{ and }n,\text{ but not for all }k\text{ and }n\qquad$
$\textbf{(D) }\text{if }k=1\text{ or }n-1,\text{ but not for all odd values }k\text{ and }n\qquad$
$\textbf{(E) }\text{if }n\text{ is divisible by }k,\text{ but not for all even values }k\text{ and }n$
2006 ISI B.Stat Entrance Exam, 7
for any positive integer $n$ greater than $1$, show that
\[2^n<\binom{2n}{n}<\frac{2^n}{\prod\limits_{i=0}^{n-1} \left(1-\frac{i}{n}\right)}\]
2020 USA IMO Team Selection Test, 5
Find all integers $n \ge 2$ for which there exists an integer $m$ and a polynomial $P(x)$ with integer coefficients satisfying the following three conditions: [list] [*]$m > 1$ and $\gcd(m,n) = 1$; [*]the numbers $P(0)$, $P^2(0)$, $\ldots$, $P^{m-1}(0)$ are not divisible by $n$; and [*]$P^m(0)$ is divisible by $n$. [/list] Here $P^k$ means $P$ applied $k$ times, so $P^1(0) = P(0)$, $P^2(0) = P(P(0))$, etc.
[i]Carl Schildkraut[/i]
2014 South East Mathematical Olympiad, 5
Let $\triangle ABC $ and $\triangle A'B'C'$ are acute triangles.Prove that\[Max\{cotA'(cotB+cotC),cotB'(cotC+cotA),cotC'(cotA+cotB)\}\ge \frac{2}{3}.\]
2023 Taiwan TST Round 3, 4
Find all positive integers $a$, $b$ and $c$ such that $ab$ is a square, and
\[a+b+c-3\sqrt[3]{abc}=1.\]
[i]Proposed by usjl[/i]
2021 Switzerland - Final Round, 4
Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of
$$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$
[i]Israel[/i]
2005 Italy TST, 2
$(a)$ Prove that in a triangle the sum of the distances from the centroid to the sides is not less than three times the inradius, and find the cases of equality.
$(b)$ Determine the points in a triangle that minimize the sum of the distances to the sides.
2013 Tuymaada Olympiad, 3
For every positive real numbers $a$ and $b$ prove the inequality
\[\displaystyle \sqrt{ab} \leq \dfrac{1}{3} \sqrt{\dfrac{a^2+b^2}{2}}+\dfrac{2}{3} \dfrac{2}{\dfrac{1}{a}+\dfrac{1}{b}}.\]
[i]A. Khabrov[/i]
2018 Serbia National Math Olympiad, 3
Let $n$ be a positive integer. There are given $n$ lines such that no two are parallel and no three meet at a single point.
a) Prove that there exists a line such that the number of intersection points of these $n$ lines on both of its sides is at least
$$\left \lfloor \frac{(n-1)(n-2)}{10} \right \rfloor.$$
Notice that the points on the line are not counted.
b) Find all $n$ for which there exists a configurations where the equality is achieved.
2016 India National Olympiad, P2
For positive real numbers $a,b,c$ which of the following statements necessarily implies $a=b=c$: (I) $a(b^3+c^3)=b(c^3+a^3)=c(a^3+b^3)$, (II) $a(a^3+b^3)=b(b^3+c^3)=c(c^3+a^3)$ ? Justify your answer.
2022 Greece National Olympiad, 3
The positive real numbers $a,b,c,d$ satisfy the equality
$$a+bc+cd+db+\frac{1}{ab^2c^2d^2}=18.$$
Find the maximum possible value of $a$.
2007 Germany Team Selection Test, 1
Prove the inequality:
\[\sum_{i < j}{\frac {a_{i}a_{j}}{a_{i} \plus{} a_{j}}}\leq \frac {n}{2(a_{1} \plus{} a_{2} \plus{}\cdots \plus{} a_{n})}\cdot \sum_{i < j}{a_{i}a_{j}}\]
for positive reals $ a_{1},a_{2},\ldots,a_{n}$.
[i]Proposed by Dusan Dukic, Serbia[/i]
2014 Peru IMO TST, 8
Let $x, y, z$ be real numbers such that $$\displaystyle{\begin{cases} x^2+y^2+z^2+(x+y+z)^2=9 \\ xyz \leq \frac{15}{32} \end{cases}} $$
Find the maximum possible value of $x.$
1988 All Soviet Union Mathematical Olympiad, 480
Find the minimum value of $\frac{xy}{z} + \frac{yz}{x} +\frac{ zx}{y}$ for positive reals $x, y, z$ with $x^2 + y^2 + z^2 = 1$.
2010 China Western Mathematical Olympiad, 5
Let $k$ be an integer and $k > 1$. Define a sequence $\{a_n\}$ as follows:
$a_0 = 0$,
$a_1 = 1$, and
$a_{n+1} = ka_n + a_{n-1}$ for $n = 1,2,...$.
Determine, with proof, all possible $k$ for which there exist non-negative integers $l,m (l \not= m)$ and positive integers $p,q$ such that $a_l + ka_p = a_m + ka_q$.
2010 Czech And Slovak Olympiad III A, 6
Find the minimum of the expression $\frac{a + b + c}{2} -\frac{[a, b] + [b, c] + [c, a]}{a + b + c}$ where the variables $a, b, c$ are any integers greater than $1$ and $[x, y]$ denotes the least common multiple of numbers $x, y$.
2011 Serbia National Math Olympiad, 1
Let $n \ge 2$ be integer. Let $a_0$, $a_1$, ... $a_n$ be sequence of positive reals such that:
$(a_{k-1}+a_k)(a_k+a_{k+1})=a_{k-1}-a_{k+1}$, for $k=1, 2, ..., n-1$.
Prove $a_n< \frac{1}{n-1}$.