Found problems: 6530
2008 Moldova Team Selection Test, 2
Let $ a_1,\ldots,a_n$ be positive reals so that $ a_1\plus{}a_2\plus{}\ldots\plus{}a_n\le\frac n2$. Find the minimal value of
$ \sqrt{a_1^2\plus{}\frac1{a_2^2}}\plus{}\sqrt{a_2^2\plus{}\frac1{a_3^2}}\plus{}\ldots\plus{}\sqrt{a_n^2\plus{}\frac1{a_1^2}}$.
2000 Tournament Of Towns, 3
Prove the inequality $$ 1^k+2^k+...+n^k \le \frac{n^{2k}-(n-1)^k}{n^k-(n-1)^k}$$
(L Emelianov)
1976 Czech and Slovak Olympiad III A, 5
Let $\mathbf{P}_1,\mathbf{P}_2$ be convex polygons with perimeters $o_1,o_2,$ respectively. Show that if $\mathbf P_1\subseteq\mathbf P_2,$ then $o_1\le o_2.$
1958 AMC 12/AHSME, 17
If $ x$ is positive and $ \log{x} \ge \log{2} \plus{} \frac{1}{2}\log{x}$, then:
$ \textbf{(A)}\ {x}\text{ has no minimum or maximum value}\qquad \\
\textbf{(B)}\ \text{the maximum value of }{x}\text{ is }{1}\qquad \\
\textbf{(C)}\ \text{the minimum value of }{x}\text{ is }{1}\qquad \\
\textbf{(D)}\ \text{the maximum value of }{x}\text{ is }{4}\qquad \\
\textbf{(E)}\ \text{the minimum value of }{x}\text{ is }{4}$
2009 District Olympiad, 1
Let $ f:[0,\infty )\longrightarrow [0,\infty ) $ a nonincreasing function that satisfies the inequality:
$$ \int_0^x f(t)dt <1,\quad\forall x\ge 0. $$ Prove the following affirmations:
[b]a)[/b] $ \exists \lim_{x\to\infty} \int_0^x f(t)dt \in\mathbb{R} . $
[b]b)[/b] $ \lim_{x\to\infty} xf(x) =0. $
2011 ISI B.Stat Entrance Exam, 1
Let $x_1, x_2, \cdots , x_n$ be positive reals with $x_1+x_2+\cdots+x_n=1$. Then show that
\[\sum_{i=1}^n \frac{x_i}{2-x_i} \ge \frac{n}{2n-1}\]
2014 Iran MO (2nd Round), 3
Let $ x,y,z $ be three non-negative real numbers such that \[x^2+y^2+z^2=2(xy+yz+zx). \] Prove that \[\dfrac{x+y+z}{3} \ge \sqrt[3]{2xyz}.\]
2016 IMO Shortlist, A8
Find the largest real constant $a$ such that for all $n \geq 1$ and for all real numbers $x_0, x_1, ... , x_n$ satisfying $0 = x_0 < x_1 < x_2 < \cdots < x_n$ we have
\[\frac{1}{x_1-x_0} + \frac{1}{x_2-x_1} + \dots + \frac{1}{x_n-x_{n-1}} \geq a \left( \frac{2}{x_1} + \frac{3}{x_2} + \dots + \frac{n+1}{x_n} \right)\]
1998 Romania Team Selection Test, 3
Find all positive integers $(x, n)$ such that $x^{n}+2^{n}+1$ divides $x^{n+1}+2^{n+1}+1$.
2010 Contests, 3
What is the biggest shadow that a cube of side length $1$ can have, with the sun at its peak?
Note: "The biggest shadow of a figure with the sun at its peak" is understood to be the biggest possible area of the orthogonal projection of the figure on a plane.
2009 VJIMC, Problem 4
Let $k,m,n$ be positive integers such that $1\le m\le n$ and denote $S=\{1,2,\ldots,n\}$. Suppose that $A_1,A_2,\ldots,A_k$ are $m$-element subsets of $S$ with the following property: for every $i=1,2,\ldots,k$ there exists a partition $S=S_{1,i}\cup S_{2,i}\cup\ldots\cup S_{m,i}$ (into pairwise disjoint subsets) such that
(i) $A_i$ has precisely one element in common with each member of the above partition.
(ii) Every $A_j,j\ne i$ is disjoint from at least one member of the above partition.
Show that $k\le\binom{n-1}{m-1}$.
1996 Romania Team Selection Test, 9
Let $ n\geq 3 $ be an integer and let $ x_1,x_2,\ldots,x_{n-1} $ be nonnegative integers such that
\begin{eqnarray*} \ x_1 + x_2 + \cdots + x_{n-1} &=& n \\ x_1 + 2x_2 + \cdots + (n-1)x_{n-1} &=& 2n-2. \end{eqnarray*}
Find the minimal value of $ F(x_1,x_2,\ldots,x_n) = \sum_{k=1}^{n-1} k(2n-k)x_k $.
2022 Turkey EGMO TST, 6
Let $x,y,z$ be positive real numbers satisfying the equations
$$xyz=1\text{ and }\frac yz(y-x^2)+\frac zx(z-y^2)+\frac xy(x-z^2)=0$$
What is the minimum value of the ratio of the sum of the largest and smallest numbers among $x,y,z$ to the median of them.
2013 India IMO Training Camp, 1
Let $a, b, c$ be positive real numbers such that $a + b + c = 1$. If $n$ is a positive integer then prove that
\[ \frac{(3a)^n}{(b + 1)(c + 1)} + \frac{(3b)^n}{(c + 1)(a + 1)} + \frac{(3c)^n}{(a + 1)(b + 1)} \ge \frac{27}{16} \,. \]
2012 Korea National Olympiad, 4
$a,b,c$ are positive numbers such that $ a^2 + b^2 + c^2 = 2abc + 1 $. Find the maximum value of
\[ (a-2bc)(b-2ca)(c-2ab) \]
2014 Saudi Arabia BMO TST, 5
Find all positive integers $n$ such that \[3^n+4^n+\cdots+(n+2)^n=(n+3)^n.\]
2008 China Team Selection Test, 6
Find the maximal constant $ M$, such that for arbitrary integer $ n\geq 3,$ there exist two sequences of positive real number $ a_{1},a_{2},\cdots,a_{n},$ and $ b_{1},b_{2},\cdots,b_{n},$ satisfying
(1):$ \sum_{k \equal{} 1}^{n}b_{k} \equal{} 1,2b_{k}\geq b_{k \minus{} 1} \plus{} b_{k \plus{} 1},k \equal{} 2,3,\cdots,n \minus{} 1;$
(2):$ a_{k}^2\leq 1 \plus{} \sum_{i \equal{} 1}^{k}a_{i}b_{i},k \equal{} 1,2,3,\cdots,n, a_{n}\equiv M$.
1998 Iran MO (2nd round), 1
If $a_1<a_2<\cdots<a_n$ be real numbers, prove that:
\[ a_1a_2^4+a_2a_3^4+\cdots+a_{n-1}a_n^4+a_na_1^4\geq a_2a_1^4+a_3a_2^4+\cdots+a_na_{n-1}^4+a_1a_n^4. \]
1990 APMO, 2
Let $a_1$, $a_2$, $\cdots$, $a_n$ be positive real numbers, and let $S_k$ be the sum of the products of $a_1$, $a_2$, $\cdots$, $a_n$ taken $k$ at a time. Show that
\[ S_k S_{n-k} \geq {n \choose k}^2 a_1 a_2 \cdots a_n \]
for $k = 1$, $2$, $\cdots$, $n - 1$.
2023 Turkey Team Selection Test, 6
Let $a,b,c,d$ be positive real numbers. What is the minimum value of $$ \frac{(a^2+b^2+2c^2+3d^2)(2a^2+3b^2+6c^2+6d^2)}{(a+b)^2(c+d)^2}$$
2006 MOP Homework, 3
Prove that the following inequality holds with the exception of finitely many positive integers $n$:
$\sum^{n}_{i=1}\sum^{n}_{j=1}gcd(i,j)>4n^2$.
2018 Taiwan TST Round 2, 5
An integer $n \geq 3$ is given. We call an $n$-tuple of real numbers $(x_1, x_2, \dots, x_n)$ [i]Shiny[/i] if for each permutation $y_1, y_2, \dots, y_n$ of these numbers, we have
$$\sum \limits_{i=1}^{n-1} y_i y_{i+1} = y_1y_2 + y_2y_3 + y_3y_4 + \cdots + y_{n-1}y_n \geq -1.$$
Find the largest constant $K = K(n)$ such that
$$\sum \limits_{1 \leq i < j \leq n} x_i x_j \geq K$$
holds for every Shiny $n$-tuple $(x_1, x_2, \dots, x_n)$.
1998 Bosnia and Herzegovina Team Selection Test, 2
For positive real numbers $x$, $y$ and $z$ holds $x^2+y^2+z^2=1$. Prove that $$\frac{x}{1+x^2}+\frac{y}{1+y^2}+\frac{z}{1+z^2} \leq \frac{3\sqrt{3}}{4}$$
2013 Sharygin Geometry Olympiad, 2
Let $ABCD$ is a tangential quadrilateral such that $AB=CD>BC$. $AC$ meets $BD$ at $L$. Prove that $\widehat{ALB}$ is acute.
[hide]According to the jury, they want to propose a more generalized problem is to prove $(AB-CD)^2 < (AD-BC)^2$, but this problem has appeared some time ago[/hide]
2025 Azerbaijan Junior NMO, 5
For positive real numbers $x;y;z$ satisfying $0<x,y,z<2$, find the biggest value the following equation could acquire:
$$(2x-yz)(2y-zx)(2z-xy)$$