Found problems: 6530
2004 Unirea, 3
[b]a)[/b] Prove that for any natural numbers $ n, $ the inequality
$$ e^{2-1/n} >\prod_{k=1}^n (1+1/k^2) $$
holds.
[b]b)[/b] Prove that the sequence $ \left( a_n \right)_{n\ge 1} $ with $ a_1=1 $ and defined by the recursive relation $ a_{n+1}=\frac{2}{n^2}\sum_{k=1}^n ka_k $ is nondecreasing. Is it convergent?
2005 Germany Team Selection Test, 3
Let $ABC$ be a triangle with area $S$, and let $P$ be a point in the plane. Prove that $AP+BP+CP\geq 2\sqrt[4]{3}\sqrt{S}$.
1975 Chisinau City MO, 97
Find the smallest value of the expression $(x-1) (x -2) (x -3) (x - 4) + 10$.
2002 Korea - Final Round, 3
Let $p_n$ be the $n^{\mbox{th}}$ prime counting from the smallest prime $2$ in increasing order. For example, $p_1=2, p_2=3, p_3 =5, \cdots$
(a) For a given $n \ge 10$, let $r$ be the smallest integer satisfying
\[2\le r \le n-2, \quad n-r+1 < p_r\]
and define $N_s=(sp_1p_2\cdots p_{r-1})-1$ for $s=1,2,\ldots, p_r$. Prove that there exists $j, 1\le j \le p_r$, such that none of $p_1,p_2,\cdots, p_n$ divides $N_j$.
(b) Using the result of (a), find all positive integers $m$ for which
\[p_{m+1}^2 < p_1p_2\cdots p_m\]
2005 Junior Balkan Team Selection Tests - Romania, 4
Let $a,b,c$ be positive numbers such that $a+b+c \geq \dfrac 1a + \dfrac 1b + \dfrac 1c$. Prove that
\[ a+b+c \geq \frac 3{abc}. \]
2010 AIME Problems, 6
Let $ P(x)$ be a quadratic polynomial with real coefficients satisfying \[x^2 \minus{} 2x \plus{} 2 \le P(x) \le 2x^2 \minus{} 4x \plus{} 3\] for all real numbers $ x$, and suppose $ P(11) \equal{} 181$. Find $ P(16)$.
2014 Saint Petersburg Mathematical Olympiad, 3
$N$ in natural. There are natural numbers from $N^3$ to $N^3+N$ on the board. $a$ numbers was colored in red, $b$ numbers was colored in blue. Sum of red numbers in divisible by sum of blue numbers. Prove, that $b|a$
2014 Contests, 1
Prove that for positive reals $a$,$b$,$c$ so that $a+b+c+abc=4$, \[\left (1+\dfrac{a}{b}+ca \right )\left (1+\dfrac{b}{c}+ab \right)\left (1+\dfrac{c}{a}+bc \right) \ge 27\] holds.
2010 USAMO, 3
The 2010 positive numbers $a_1, a_2, \ldots , a_{2010}$ satisfy the inequality $a_ia_j \le i+j$ for all distinct indices $i, j$. Determine, with proof, the largest possible value of the product $a_1a_2\ldots a_{2010}$.
1996 Iran MO (3rd Round), 1
Let $a,b,c,d$ be positive real numbers. Prove that
\[\frac{a}{b+2c+3d}+\frac{b}{c+2d+3a}+\frac{c}{d+2a+3b}+\frac{d}{a+2b+3c} \geq \frac{2}{3}.\]
2015 Caucasus Mathematical Olympiad, 2
The equation $(x+a) (x+b) = 9$ has a root $a+b$. Prove that $ab\le 1$.
2022-23 IOQM India, 15
Let $x,y$ be real numbers such that $xy=1$. Let $T$ and $t$ be the largest and smallest values of the expression \\
$\hspace{2cm} \frac{(x+y)^2-(x-y)-2}{(x+y)^2+(x-y)-2}$\\.
\\
If $T+t$ can be expressed in the form $\frac{m}{n}$ where $m,n$ are nonzero integers with $GCD(m,n)=1$, find the value of $m+n$.
2014 ELMO Shortlist, 6
Let $a,b,c$ be positive reals such that $a+b+c=ab+bc+ca$. Prove that \[ (a+b)^{ab-bc}(b+c)^{bc-ca}(c+a)^{ca-ab} \ge a^{ca}b^{ab}c^{bc}. \][i]Proposed by Sammy Luo[/i]
2011 Stars Of Mathematics, 3
For a given integer $n\geq 3$, determine the range of values for the expression
\[ E_n(x_1,x_2,\ldots,x_n) := \dfrac {x_1} {x_2} + \dfrac {x_2} {x_3} + \cdots + \dfrac {x_{n-1}} {x_n} + \dfrac {x_n} {x_1}\]
over real numbers $x_1,x_2,\ldots,x_n \geq 1$ satisfying $|x_k - x_{k+1}| \leq 1$ for all $1\leq k \leq n-1$.
Do also determine when the extremal values are achieved.
(Dan Schwarz)
2010 Victor Vâlcovici, 3
$ A',B',C' $ are the feet of the heights of an acute-angled triangle $ ABC. $ Calculate
$$ \frac{\text{area} (ABC)}{\text{area}\left( A'B'C'\right)} , $$
knowing that $ ABC $ and $ A'B'C' $ have the same center of mass.
[i]Carmen[/i] and [i]Viorel Botea[/i]
2001 Miklós Schweitzer, 6
Let $I\subset \mathbb R$ be a non-empty open interval, $\varepsilon\geq 0$ and $f\colon I\rightarrow\mathbb R$ a function satisfying the
$$f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)+\varepsilon t(1-t)|x-y|$$
inequality for all $x,y\in I$ and $t\in [0,1]$. Prove that there exists a convex $g\colon I\rightarrow\mathbb R$ function, such that the function $l :=f-g$ has the $\varepsilon$-Lipschitz property, that is
$$|l(x)-l(y)|\leq \varepsilon|x-y|\text{ for all }x,y\in I$$
Ukrainian TYM Qualifying - geometry, X.12
Inside the convex polygon $A_1A_2...A_n$ , there is a point $M$ such that $\sum_{k=1}^n \overrightarrow {A_kM} = \overrightarrow{0}$. Prove that $nP\ge 4d$, where $P$ is the perimeter of the polygon, and $d=\sum_{k=1}^n |\overrightarrow {A_kM}|$ . Investigate the question of the achievement of equality in this inequality.
2025 International Zhautykov Olympiad, 1
Let $a, b$ be positive reals such that $a^3 + b^3 = ab + 1$. Prove that \[(a-b)^2 + a + b \geq 2\]
PEN H Problems, 85
Find all integer solutions to $2(x^5 +y^5 +1)=5xy(x^2 +y^2 +1)$.
1989 IMO Longlists, 61
Prove for $ 0 < k \leq 1$ and $ a_i \in \mathbb{R}^\plus{},$ $ i \equal{} 1,2 \ldots, n$ the following inequality holds:
\[ \left( \frac{a_1}{a_2 \plus{} \ldots \plus{} a_n} \right)^k \plus{} \ldots \plus{} \left( \frac{a_n}{a_1 \plus{} \ldots \plus{} a_{n\minus{}1}} \right)^k \geq \frac{n}{(n\minus{}1)^k}.\]
2008 Junior Balkan Team Selection Tests - Moldova, 9
Find all triplets $ (x,y,z)$, that satisfy:
$ \{\begin{array}{c}\ \ x^2 - 2x - 4z = 3\
y^2 - 2y - 2x = - 14 \
z^2 - 4y - 4z = - 18 \end{array}$
2011 Turkey MO (2nd round), 3
$x,y,z$ positive real numbers such that $xyz=1$
Prove that:
$\frac{1}{x+y^{20}+z^{11}}+\frac{1}{y+z^{20}+x^{11}}+\frac{1}{z+x^{20}+y^{11}}\leq1$
2019 Mediterranean Mathematics Olympiad, 4
Let $P$ be a point in the interior of an equilateral triangle with height $1$, and let $x,y,z$ denote the distances from $P$ to the three sides of the triangle. Prove that
\[ x^2+y^2+z^2 ~\ge~ x^3+y^3+z^3 +6xyz \]
PEN J Problems, 11
Prove that ${d((n^2 +1)}^2)$ does not become monotonic from any given point onwards.
2014 Contests, 3
Find all polynomials $P(x)$ with real coefficients that satisfy \[P(x\sqrt{2})=P(x+\sqrt{1-x^2})\]for all real $x$ with $|x|\le 1$.