Found problems: 6530
2019 Moroccan TST, 2
Let $a>1$ be a real number. Prove that for all $n\in\mathbb{N}*$ that :
$\frac{a^n-1}{n}\ge \sqrt{a}^{n+1}-\sqrt{a}^{n-1}$
2002 IMC, 12
Let $f:\mathbb{R}^{n}\rightarrow \mathbb{R}$ be a convex function whose gradient $\nabla f$
exists at every point of $\mathbb{R}^{n}$ and satisfies the condition
$$\exists L>0\; \forall x_{1},x_{2}\in \mathbb{R}^{n}:\;\; ||\nabla f(x_{1})-\nabla f(x_{2})||\leq L||x_{1}-x_{2}||.$$
Prove that
$$ \forall x_{1},x_{2}\in \mathbb{R}^{n}:\;\; ||\nabla f(x_{1})-\nabla f(x_{2})||^{2}\leq L\langle\nabla f(x_{1})-\nabla f(x_{2}), x_{1}-x_{2}\rangle. $$
2015 European Mathematical Cup, 2
Let $a,b,c$ be positive real numbers such that $abc=1$. Prove that $$\frac{a+b+c+3}{4}\geqslant \frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}.$$
[i]Dimitar Trenevski[/i]
2016 China Girls Math Olympiad, 4
Let $n$ is a positive integers ,$a_1,a_2,\cdots,a_n\in\{0,1,\cdots,n\}$ . For the integer $j$ $(1\le j\le n)$ ,define $b_j$ is the number of elements in the set $\{i|i\in\{1,\cdots,n\},a_i\ge j\}$ .For example ：When $n=3$ ,if $a_1=1,a_2=2,a_3=1$ ,then $b_1=3,b_2=1,b_3=0$ .
$(1)$ Prove that $$\sum_{i=1}^{n}(i+a_i)^2\ge \sum_{i=1}^{n}(i+b_i)^2.$$
$(2)$ Prove that $$\sum_{i=1}^{n}(i+a_i)^k\ge \sum_{i=1}^{n}(i+b_i)^k,$$
for the integer $k\ge 3.$
1996 APMO, 5
Let $a$, $b$, $c$ be the lengths of the sides of a triangle. Prove that
\[ \sqrt{a+b-c} + \sqrt{b+c-a} + \sqrt{c+a-b} \leq \sqrt{a} + \sqrt{b} + \sqrt{c} \]
and determine when equality occurs.
2008 Hong Kong TST, 1
Let $ \alpha_1$, $ \alpha_2$, $ \ldots$, $ \alpha_{2008}$ be real numbers. Find the maximum value of \[ \sin\alpha_1\cos\alpha_2 \plus{} \sin\alpha_2\cos\alpha_3 \plus{} \cdots \plus{} \sin\alpha_{2007}\cos\alpha_{2008} \plus{} \sin\alpha_{2008}\cos\alpha_1\]
2010 Rioplatense Mathematical Olympiad, Level 3, 2
Find the minimum and maximum values of $ S=\frac{a}{b}+\frac{c}{d} $ where $a$, $b$, $c$, $d$ are positive integers satisfying $a + c = 20202$ and $b + d = 20200$.
2011 Romania Team Selection Test, 4
Let $ABCDEF$ be a convex hexagon of area $1$, whose opposite sides are parallel. The lines $AB$, $CD$ and $EF$ meet in pairs to determine the vertices of a triangle. Similarly, the lines $BC$, $DE$ and $FA$ meet in pairs to determine the vertices of another triangle. Show that the area of at least one of these two triangles is at least $3/2$.
2019 Jozsef Wildt International Math Competition, W. 34
Let $a$, $b$, $c$ be positive real numbers and let $m$, $n$ $(m \geq n)$ be positive integers. Prove that$$\frac{a^{n-1}b^{n-1}c^{m-n-1}}{a^{m+n}+b^{m+n}+a^nb^nc^{m-n}}+\frac{b^{n-1}bc^{n-1}a^{m-n-1}}{b^{m+n}+c^{m+n}+b^nc^na^{m-n}}+\frac{c^{n-1}a^{n-1}b^{m-n-1}}{c^{m+n}+a^{m+n}+c^na^nb^{m-n}}\leq \frac{1}{abc}$$
2016 All-Russian Olympiad, 7
All russian olympiad 2016,Day 2 ,grade 9,P8 :
Let $a, b, c, d$ be are positive numbers such that $a+b+c+d=3$ .Prove that$$\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{1}{d^2}\le\frac{1}{a^2b^2c^2d^2}$$
All russian olympiad 2016,Day 2,grade 11,P7 :
Let $a, b, c, d$ be are positive numbers such that $a+b+c+d=3$ .Prove that
$$\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}+\frac{1}{d^3}\le\frac{1}{a^3b^3c^3d^3}$$
Russia national 2016
2001 Korea - Final Round, 3
Let $x_1,x_2, \cdots,x_n$ and $y_1,y_2, \cdots ,y_n$ be arbitrary real numbers satisfying $x_1^2+x_2^2+\cdots+x_n^2=y_1^2+y_2^2+\cdots+y_n^2=1$. Prove that
\[(x_1y_2-x_2y_1)^2 \le 2\left|1-\sum_{k=1}^n x_ky_k\right|\]
and find all cases of equality.
2021 Estonia Team Selection Test, 3
In the plane, there are $n \geqslant 6$ pairwise disjoint disks $D_{1}, D_{2}, \ldots, D_{n}$ with radii $R_{1} \geqslant R_{2} \geqslant \ldots \geqslant R_{n}$. For every $i=1,2, \ldots, n$, a point $P_{i}$ is chosen in disk $D_{i}$. Let $O$ be an arbitrary point in the plane. Prove that \[O P_{1}+O P_{2}+\ldots+O P_{n} \geqslant R_{6}+R_{7}+\ldots+R_{n}.\]
(A disk is assumed to contain its boundary.)
2004 China National Olympiad, 2
For a given positive integer $n\ge 2$, suppose positive integers $a_i$ where $1\le i\le n$ satisfy $a_1<a_2<\ldots <a_n$ and $\sum_{i=1}^n \frac{1}{a_i}\le 1$. Prove that, for any real number $x$, the following inequality holds
\[\left(\sum_{i=1}^n\frac{1}{a_i^2+x^2}\right)^2\le\frac{1}{2}\cdot\frac{1}{a_1(a_1-1)+x^2} \]
[i]Li Shenghong[/i]
2012 Germany Team Selection Test, 3
Let $a,b,c$ be positive real numbers with $a^2+b^2+c^2 \geq 3$. Prove that:
$$\frac{(a+1)(b+2)}{(b+1)(b+5)}+\frac{(b+1)(c+2)}{(c+1)(c+5)}+\frac{(c+1)(a+2)}{(a+1)(a+5)} \geq \frac{3}{2}.$$
2005 Romania Team Selection Test, 3
Let $P$ be a polygon (not necessarily convex) with $n$ vertices, such that all its sides and diagonals are less or equal with 1 in length. Prove that the area of the polygon is less than $\dfrac {\sqrt 3} 2$.
2013 Today's Calculation Of Integral, 863
For $0<t\leq 1$, let $F(t)=\frac{1}{t}\int_0^{\frac{\pi}{2}t} |\cos 2x|\ dx.$
(1) Find $\lim_{t\rightarrow 0} F(t).$
(2) Find the range of $t$ such that $F(t)\geq 1.$
2004 Harvard-MIT Mathematics Tournament, 1
How many ordered pairs of integers $(a,b)$ satisfy all of the following inequalities?
\begin{eqnarray*} a^2 + b^2 &<& 16 \\ a^2 + b^2 &<& 8a \\ a^2 + b^2 &<& 8b \end{eqnarray*}
1993 Tournament Of Towns, (398) 6
If it is known that the equation
$$x^4+ax^3+2x^2+bx+1=0$$
has a (real) root, prove the inequality
$$a^2+b^2 \ge 8.$$
(A Egorov)
2005 China Team Selection Test, 1
Let $a_{1}$, $a_{2}$, …, $a_{6}$; $b_{1}$, $b_{2}$, …, $b_{6}$ and $c_{1}$, $c_{2}$, …, $c_{6}$ are all permutations of $1$, $2$, …, $6$, respectively. Find the minimum value of $\sum_{i=1}^{6}a_{i}b_{i}c_{i}$.
2019 Nigerian Senior MO Round 4, 1
Let $f: N \to N$ be a function satisfying
(a) $1\le f(x)-x \le 2019$ $\forall x \in N$
(b) $f(f(x))\equiv x$ (mod $2019$) $\forall x \in N$
Show that $\exists x \in N$ such that $f^k(x)=x+2019 k, \forall k \in N$
2005 Taiwan TST Round 1, 1
Let $f(x)=Ax^2+Bx+C$, $g(x)=ax^2+bx+c$ be two quadratic polynomial functions with real coefficients that satisfy the relation \[|f(x)| \ge |g(x)|\] for all real $x$. Prove that $|b^2-4ac| \le |B^2-4AC|.$
My solution was nearly complete...
2001 IberoAmerican, 3
Let $S$ be a set of $n$ elements and $S_1,\ S_2,\dots,\ S_k$ are subsets of $S$ ($k\geq2$), such that every one of them has at least $r$ elements.
Show that there exists $i$ and $j$, with $1\leq{i}<j\leq{k}$, such that the number of common elements of $S_i$ and $S_j$ is greater or equal to: $r-\frac{nk}{4(k-1)}$
2001 Grosman Memorial Mathematical Olympiad, 2
If $x_1,x_2,...,x_{2001}$ are real numbers with $0 \le x_n \le 1$ for $n = 1,2,...,2001$, find the maximum value of
$$\left(\frac{1}{2001}\sum_{n=1}^{2001}x_n^2\right)-\left(\frac{1}{2001}\sum_{n=1}^{2001}x_n\right)^2$$
Where is this maximum attained?
2002 Moldova National Olympiad, 3
Let $ a,b> 0$ such that $ a\ne b$. Prove that:
$ \sqrt {ab} < \dfrac{a \minus{} b}{\ln a \minus{} \ln b} < \dfrac{a \plus{} b}{2}$
2011 Postal Coaching, 2
Let $\tau(n)$ be the number of positive divisors of a natural number $n$, and $\sigma(n)$ be their sum. Find the largest real number $\alpha$ such that
\[\frac{\sigma(n)}{\tau(n)}\ge\alpha \sqrt{n}\]
for all $n \ge 1$.