Found problems: 6530
2019 Latvia Baltic Way TST, 4
Let $P(x)$ be a polynomial with degree $n$ and real coefficients. For all $0 \le y \le 1$ holds $\mid p(y) \mid \le 1$. Prove that $p(-\frac{1}{n}) \le 2^{n+1} -1$
2017 China Team Selection Test, 1
Let $n \geq 4$ be a natural and let $x_1,\ldots,x_n$ be non-negative reals such that $x_1 + \cdots + x_n = 1$. Determine the maximum value of $x_1x_2x_3 + x_2x_3x_4 + \cdots + x_nx_1x_2$.
2010 Albania Team Selection Test, 4
With $\sigma (n)$ we denote the sum of natural divisors of the natural number $n$. Prove that, if $n$ is the product of different prime numbers of the form $2^k-1$ for $k \in \mathbb{N}$($Mersenne's$ prime numbers) , than $\sigma (n)=2^m$, for some $m \in \mathbb{N}$. Is the inverse statement true?
MathLinks Contest 5th, 6.3
Let $x, y, z$ be three positive numbers such that $(x + y-z) \left( \frac{1}{x}+ \frac{1}{y}- \frac{1}{z} \right)=4$.
Find the minimal value of the expression $$E(x, y, z) = (x^4 + y^4 + z^4) \left( \frac{1}{x^4}+ \frac{1}{y^4}+ \frac{1}{z^4} \right) .$$
2022 Harvard-MIT Mathematics Tournament, 4
Suppose $n \ge 3$ is a positive integer. Let $a_1 < a_2 < ... < a_n$ be an increasing sequence of positive real numbers, and let $a_{n+1} = a_1$. Prove that $$\sum_{k=1}^{n}\frac{a_k}{a_{k+1}}>\sum_{k=1}^{n}\frac{a_{k+1}}{a_k}$$
2003 Irish Math Olympiad, 1
If $a,b,c$ are the sides of a triangle whose perimeter is equal to 2 then prove that:
a) $abc+\frac{28}{27}\geq ab+bc+ac$;
b) $abc+1<ab+bc+ac$
See also [url]http://www.mathlinks.ro/Forum/viewtopic.php?t=47939&view=next[/url] (problem 1)
:)
2009 Middle European Mathematical Olympiad, 5
Let $ x$, $ y$, $ z$ be real numbers satisfying $ x^2\plus{}y^2\plus{}z^2\plus{}9\equal{}4(x\plus{}y\plus{}z)$. Prove that
\[ x^4\plus{}y^4\plus{}z^4\plus{}16(x^2\plus{}y^2\plus{}z^2) \ge 8(x^3\plus{}y^3\plus{}z^3)\plus{}27\]
and determine when equality holds.
2018 Caucasus Mathematical Olympiad, 6
Given a convex quadrilateral $ABCD$ with $\angle BCD=90^\circ$. Let $E$ be the midpoint of $AB$. Prove that $2EC \leqslant AD+BD$.
1954 Moscow Mathematical Olympiad, 269
a) Given $100$ numbers $a_1, ..., a_{100}$ such that $\begin{cases}
a_1 - 3a_2 + 2a_3 \ge 0, \\
a_2 - 3a_3 + 2a_4 \ge 0, \\
a_3 - 3a_4 + 2a_5 \ge 0, \\
... \\
a_{99} - 3a_{100} + 2a_1 \ge 0, \\
a_{100} - 3a_1 + 2a_2 \ge 0 \end{cases}$
prove that the numbers are equal.
b) Given numbers $a_1=1, ..., a_{100}$ such that $\begin{cases}
a_1 - 4a_2 + 3a_3 \ge 0, \\
a_2 - 4a_3 + 3a_4 \ge 0, \\
a_3 - 4a_4 + 3a_5 \ge 0, \\
... \\
a_{99} - 4a_{100} + 3a_1 \ge 0, \\
a_{100} - 4a_1 + 3a_2 \ge 0 \end{cases}$
Find $a_2, a_3, ... , a_{100}.$
2007 Finnish National High School Mathematics Competition, 1
Show: when a prime number is divided by $30,$ the remainder is either $1$ or a prime number. Is a similar claim true, when the divisor is $60$ or $90$?
2007 All-Russian Olympiad Regional Round, 8.4
On the chessboard, $ 32$ black pawns and $ 32$ white pawns are arranged. In every move, a pawn can capture another pawn of the opposite color, moving diagonally to an adjacent square where the captured one stands. White pawns move only in upper-left or upper-right directions, while black ones can move in down-left or in down-right directions only; the captured pawn is removed from the board. A pawn cannot move without capturing an opposite pawn. Find the least possible number of pawns which can stay on the chessboard.
1994 USAMO, 4
Let $\, a_1, a_2, a_3, \ldots \,$ be a sequence of positive real numbers satisfying $\, \sum_{j=1}^n a_j \geq \sqrt{n} \,$ for all $\, n \geq 1$. Prove that, for all $\, n \geq 1, \,$ \[ \sum_{j=1}^n a_j^2 > \frac{1}{4} \left( 1 + \frac{1}{2} + \cdots + \frac{1}{n} \right). \]
2025 Thailand Mathematical Olympiad, 3
Let $a,b,c,x,y,z$ be positive real numbers such that $ay+bz+cx \le az+bx+cy$. Prove that $$ \frac{xy}{ax+bx+cy}+\frac{yz}{by+cy+az}+\frac{zx}{cz+az+bx} \le \frac{x+y+z}{a+b+c}$$
2023 All-Russian Olympiad Regional Round, 9.9
Find the largest real $m$, such that for all positive real $a, b, c$ with sum $1$, the inequality $\sqrt{\frac{ab} {ab+c}}+\sqrt{\frac{bc} {bc+a}}+\sqrt{\frac{ca} {ca+b}} \geq m$ is satisfied.
2004 India IMO Training Camp, 1
Prove that in any triangle $ABC$,
\[ 0 < \cot { \left( \frac{A}{4} \right)} - \tan{ \left( \frac{B}{4} \right) } - \tan{ \left( \frac{C}{4} \right) } - 1 < 2 \cot { \left( \frac{A}{2} \right) }. \]
1976 Yugoslav Team Selection Test, Problem 3
Find the minimum and maximum values of the function
$$f(x,y,z,t)=\frac{ax^2+by^2}{ax+by}+\frac{az^2+bt^2}{az+bt},~(a>0,b>0),$$given that $x+z=y+t=1$, and $x,y,z,t\ge0$.
1985 Iran MO (2nd round), 7
Let $a,b$ and $c$ be real numbers with $b,c >0.$ Prove that if $ a<b \ ( a>b),$ then
\[\frac{a+c}{b+c} > \frac ab \qquad ( \frac{a+c}{b+c} < \frac ab) \]
And then prove that $\frac{a+c}{b+c}$ is between $1$ and $\frac ab.$
2002 AMC 12/AHSME, 6
For how many positive integers $ m$ does there exist at least one positive integer $ n$ such that $ m\cdot n \le m \plus{} n$?
$ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}$ infinitely many
2002 China Team Selection Test, 1
Let $P_n(x)=a_0 + a_1x + \cdots + a_nx^n$, with $n \geq 2$, be a real-coefficient polynomial. Prove that if there exists $a > 0$ such that
\begin{align*}
P_n(x) = (x + a)^2 \left( \sum_{i=0}^{n-2} b_i x^i \right),
\end{align*}
where $b_i$ are positive real numbers, then there exists some $i$, with $1 \leq i \leq n-1$, such that \[a_i^2 - 4a_{i-1}a_{i+1} \leq 0.\]
2017 Romania National Olympiad, 3
Let $n \in N, n\ge 2$, and $a_1, a_2, ..., a_n, b_1, b_2, ..., b_n$ be real positive numbers such that
$$\frac{a_1}{b_1} \le \frac{a_2}{b_2} \le ... \le\frac{a_n}{b_n}.$$
Find the largest real $c$ so that $$(a_1-b_1c)x_1+(a_2-b_2c)x_2+...+(a_n-b_nc)x_n \ge 0,$$
for every $x_1, x_2,..., x_n > 0$, with $x_1\le x_2\le ...\le x_n$.
2005 India IMO Training Camp, 3
For real numbers $a,b,c,d$ not all equal to $0$ , define a real function $f(x) = a +b\cos{2x} + c\sin{5x} +d \cos{8x}$. Suppose $f(t) = 4a$ for some real $t$. prove that there exist a real number $s$ s.t. $f(s)<0$
2006 ITAMO, 5
Consider the inequality
\[(a_1+a_2+\dots+a_n)^2\ge 4(a_1a_2+a_2a_3+\cdots+a_na_1).\]
a) Find all $n\ge 3$ such that the inequality is true for positive reals.
b) Find all $n\ge 3$ such that the inequality is true for reals.
2014 JBMO Shortlist, 9
Let $n$ a positive integer and let $x_1, \ldots, x_n, y_1, \ldots, y_n$ real positive numbers such that $x_1+\ldots+x_n=y_1+\ldots+y_n=1$. Prove that:
$$|x_1-y_1|+\ldots+|x_n-y_n|\leq 2-\underset{1\leq i\leq n}{min} \;\dfrac{x_i}{y_i}-\underset{1\leq i\leq n}{min} \;\dfrac{y_i}{x_i}$$
1996 Romania Team Selection Test, 14
Let $ x,y,z $ be real numbers. Prove that the following conditions are equivalent:
(i) $ x,y,z $ are positive numbers and $ \dfrac 1x + \dfrac 1y + \dfrac 1z \leq 1 $;
(ii) $ a^2x+b^2y+c^2z>d^2 $ holds for every quadrilateral with sides $ a,b,c,d $.
2015 Stars Of Mathematics, 4
Let $S$ be a finite set of points in the plane,situated in general position(any three points in $S$ are not collinear),and let $$D(S,r)=\{\{x,y\}:x,y\in S,\text{dist}(x,y)=r\},$$ where $R$ is a positive real number,and $\text{dist}(x,y)$ is the euclidean distance between points $x$ and $y$.Prove that $$\sum_{r>0}|D(S,r)|^2\le\frac{3|S|^2(|S|-1)}{4}.$$