Found problems: 6530
2019 China Team Selection Test, 1
Given complex numbers $x,y,z$, with $|x|^2+|y|^2+|z|^2=1$. Prove that: $$|x^3+y^3+z^3-3xyz| \le 1$$
2009 Switzerland - Final Round, 3
Let $a, b, c, d$ be positive real numbers. Prove the following inequality and determine all cases in which the equality holds :
$$\frac{a - b}{b + c}+\frac{b - c}{c + d}+\frac{c - d}{d + a}+\frac{d - a}{a + b} \ge 0.$$
2009 Danube Mathematical Competition, 5
Let $\sigma, \tau$ be two permutations of the quantity $\{1, 2,. . . , n\}$.
Prove that there is a function $f: \{1, 2,. . . , n\} \to \{-1, 1\}$ such that for any $1 \le i \le j \le n$,
we have $\left|\sum_{k=i}^{j} f(\sigma (k)) \right| \le 2$ and $\left|\sum_{k=i}^{j} f(\tau (k))\right| \le 2$
2019 Moldova Team Selection Test, 6
Let $a,b,c \ge 0$ such that $a+b+c=1$ and $s \ge 5$.
Prove that $s(a^2+b^2+c^2) \le 3(s-3)(a^3+b^3+c^3)+1$
2023 China Western Mathematical Olympiad, 5
Let $a_1,a_2,\cdots,a_{100}\geq 0$ such that $\max\{a_{i-1}+a_i,a_i+a_{i+1}\}\geq i $ for any $2\leq i\leq 99.$ Find the minimum of $a_1+a_2+\cdots+a_{100}.$
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 $x.$
Oliforum Contest IV 2013, 5
Let $x,y,z$ be distinct positive integers such that $(y+z)(z+x)=(x+y)^2$ . Show that \[x^2+y^2>8(x+y)+2(xy+1).\] (Paolo Leonetti)
1976 Dutch Mathematical Olympiad, 5
$f(k) = k + \left[ \frac{n}{k}\right ] $,$k \in \{1,2,..., n\}$, $k_0 =\left[ \sqrt{n} \right] + 1$.
Prove that $f(k_0) < f(k)$ if $k \in \{1,2,..., n\}$
1982 Bulgaria National Olympiad, Problem 4
If $x_1,x_2,\ldots,x_n$ are arbitrary numbers from the interval $[0,2]$, prove that
$$\sum_{i=1}^n\sum_{j=1}^n|x_i-x_j|\le n^2$$When is the equality attained?
1998 National High School Mathematics League, 2
Let $a_1,a_2,\cdots,a_n,b_1,b_2,\cdots,b_n$ are real numbers in $[1,2]$. If $\sum_{i=1}^{n}a_i^2=\sum_{i=1}^{n}b_i^2$, prove that
$$\sum_{i=1}^{n}\frac{a_i^3}{b_i}\leq\frac{17}{10}\sum_{i=1}^{n}a_i^2.$$
Find when the equality holds.
2007 Iran Team Selection Test, 1
Find all polynomials of degree 3, such that for each $x,y\geq 0$: \[p(x+y)\geq p(x)+p(y)\]
2021 China Team Selection Test, 1
Given positive integers $m$ and $n$. Let $a_{i,j} ( 1 \le i \le m, 1 \le j \le n)$ be non-negative real numbers, such that
$$ a_{i,1} \ge a_{i,2} \ge \cdots \ge a_{i,n} \text{ and } a_{1,j} \ge a_{2,j} \ge \cdots \ge a_{m,j} $$
holds for all $1 \le i \le m$ and $1 \le j \le n$. Denote
$$ X_{i,j}=a_{1,j}+\cdots+a_{i-1,j}+a_{i,j}+a_{i,j-1}+\cdots+a_{i,1},$$
$$ Y_{i,j}=a_{m,j}+\cdots+a_{i+1,j}+a_{i,j}+a_{i,j+1}+\cdots+a_{i,n}.$$
Prove that
$$ \prod_{i=1}^{m} \prod_{j=1}^{n} X_{i,j} \ge \prod_{i=1}^{m} \prod_{j=1}^{n} Y_{i,j}.$$
2024 Vietnam Team Selection Test, 4
Let $\alpha \in (1, +\infty)$ be a real number, and let $P(x) \in \mathbb{R}[x]$ be a monic polynomial with degree $24$, such that
(i) $P(0) = 1$.
(ii) $P(x)$ has exactly $24$ positive real roots that are all less than or equal to $\alpha$.
Show that $|P(1)| \le \left( \frac{19}{5}\right)^5 (\alpha-1)^{24}$.
2017 Korea USCM, 7
Prove the following inequality holds if $\{a_n\}$ is a deceasing sequence of positive reals, and $0<\theta<\frac{\pi}{2}$.
$$\left|\sum_{n=1}^{2017} a_n \cos n\theta \right| \leq \frac{\pi a_1}{\theta}$$
VI Soros Olympiad 1999 - 2000 (Russia), 10.9
Prove that for any $\lambda > 3$ there is a number $x$ for which $$\sin x + \sin (\lambda x) \ge 1.8.$$
2010 BAMO, 5
Let $a$, $b$, $c$, $d$ be positive real numbers such that $abcd=1$. Prove that
$1/[(1/2 +a+ab+abc)^{1/2}]+ 1/[(1/2+b+bc+bcd)^{1/2}] + 1/[(1/2+c+cd+cda)^{1/2}] + 1/[1(1/2+d+da+dab)^{1/2}]$ is greater than or equal to $2^{1/2}$.
2006 AIME Problems, 15
Given that a sequence satisfies $x_0=0$ and $|x_k|=|x_{k-1}+3|$ for all integers $k\ge 1,$ find the minimum possible value of $|x_1+x_2+\cdots+x_{2006}|$.
2005 Poland - Second Round, 3
In space are given $n\ge 2$ points, no four of which are coplanar. Some of these points are connected by segments. Let $K$ be the number of segments $(K>1)$ and $T$ be the number of formed triangles. Prove that $9T^2<2K^3$.
2019 LIMIT Category B, Problem 11
Let $S=\{1,2,\ldots,10\}$. Three numbers are chosen with replacement from $S$. If the chosen numbers denote the lengths of sides of a triangle, then the probability that they will form a triangle is:
$\textbf{(A)}~\frac{101}{200}$
$\textbf{(B)}~\frac{99}{200}$
$\textbf{(C)}~\frac12$
$\textbf{(D)}~\frac{110}{200}$
2020 Jozsef Wildt International Math Competition, W12
If $m,n,p,q\in\mathbb N,m,n,p,q\ge4$ then prove that:
$$4^n(4^n+1)+4^m(4^m+1)+4^p(4^p+1)+4^q(4^q+1)\ge4mnpq(mnpq+1)$$
[i]Proposed by Daniel Sitaru[/i]
2009 Iran MO (2nd Round), 2
Let $ a_1<a_2<\cdots<a_n $ be positive integers such that for every distinct $1\leq{i,j}\leq{n}$ we have $ a_j-a_i $ divides $ a_i $.
Prove that
\[ ia_j\leq{ja_i} \qquad \text{ for } 1\leq{i}<j\leq{n} \]
1985 Poland - Second Round, 1
Inside the triangle $ABC$, the point $P$ is chosen. Let $ a, b, c $ be the lengths of the sides $ BC $, $ CA $, $ AB $, respectively, and $ x, y, z $ the distances of the point $ P $ from the vertices $ B, C, A $. Prove that if
$$ x^2 + xy + y^2 = a^2 $$
$$y^2 + yz + z^2 = b^2 $$
$$z^2 + zx + x^2 = c^2$$
this
$$ a^2 + ab + b^2 > c^2.$$
2011 IMO Shortlist, 7
On a square table of $2011$ by $2011$ cells we place a finite number of napkins that each cover a square of $52$ by $52$ cells. In each cell we write the number of napkins covering it, and we record the maximal number $k$ of cells that all contain the same nonzero number. Considering all possible napkin configurations, what is the largest value of $k$?
[i]Proposed by Ilya Bogdanov and Rustem Zhenodarov, Russia[/i]
2007 Kyiv Mathematical Festival, 5
Let $a,b,c>0$ and $abc\ge1.$ Prove that
a) $\left(a+\frac{1}{a+1}\right)\left(b+\frac{1}{b+1}\right) \left(c+\frac{1}{c+1}\right)\ge\frac{27}{8}.$
b)$27(a^{3}+a^{2}+a+1)(b^{3}+b^{2}+b+1)(c^{3}+c^{2}+c+1)\ge$
$\ge 64(a^{2}+a+1)(b^{2}+b+1)(c^{2}+c+1).$
[hide="Generalization"]$n^{3}(a^{n}+\ldots+a+1)(b^{n}+\ldots+b+1)(c^{n}+\ldots+c+1)\ge$
$\ge (n+1)^{3}(a^{n-1}+\ldots+a+1)(b^{n-1}+\ldots+b+1)(c^{n-1}+\ldots+c+1),\ n\ge1.$ [/hide]
1985 IMO Longlists, 37
Prove that a triangle with angles $\alpha, \beta, \gamma$, circumradius $R$, and area $A$ satisfies
\[\tan \frac{ \alpha}{2}+\tan \frac{ \beta}{2}+\tan \frac{ \gamma}{2} \leq \frac{9R^2}{4A}.\]
[hide="Remark."]Remark. Can we determine [i]all[/i] of equality cases ?[/hide]