Found problems: 6530
2013 Romania National Olympiad, 4
a)Prove that $\frac{1}{2}+\frac{1}{3}+...+\frac{1}{{{2}^{m}}}<m$, for any $m\in {{\mathbb{N}}^{*}}$.
b)Let ${{p}_{1}},{{p}_{2}},...,{{p}_{n}}$ be the prime numbers less than ${{2}^{100}}$. Prove that
$\frac{1}{{{p}_{1}}}+\frac{1}{{{p}_{2}}}+...+\frac{1}{{{p}_{n}}}<10$
2010 District Olympiad, 4
Let $ f: [0,1]\rightarrow \mathbb{R}$ a derivable function such that $ f(0)\equal{}f(1)$, $ \int_0^1f(x)dx\equal{}0$ and $ f^{\prime}(x) \neq 1\ ,\ (\forall)x\in [0,1]$.
i)Prove that the function $ g: [0,1]\rightarrow \mathbb{R}\ ,\ g(x)\equal{}f(x)\minus{}x$ is strictly decreasing.
ii)Prove that for each integer number $ n\ge 1$, we have:
$ \left|\sum_{k\equal{}0}^{n\minus{}1}f\left(\frac{k}{n}\right)\right|<\frac{1}{2}$
2022 Belarusian National Olympiad, 9.2
Prove the inequality
$$\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\ldots+\frac{1}{2022!}>\frac{1^2}{2!}+\frac{2^2}{3!}+\frac{3^2}{4!}+\ldots+\frac{2022^2}{2023!}$$.
2014 Mexico National Olympiad, 6
Let $d(n)$ be the number of positive divisors of a positive integer $n$ (including $1$ and $n$). Find all values of $n$ such that $n + d(n) = d(n)^2$.
2016 Danube Mathematical Olympiad, 3
3. Let n > 1 be an integer and $a_1, a_2, . . . , a_n$ be positive reals with sum 1.
a) Show that there exists a constant c ≥ 1/2 so that
$\sum \frac{a_k}{1+(a_0+a_1+...+a_{k-1})^2}\geq c$,
where $a_0 = 0$.
b) Show that ’the best’ value of c is at least $\frac{\pi}{4}$.
2005 Bosnia and Herzegovina Junior BMO TST, 1
Non-negative real numbers $x, y, z$ satisfy the following relations:
$3x + 5y + 7z = 10$ and $x + 2y + 5z = 6$.
Find the minimum and maximum of $w = 2x - 3y + 4z$.
2010 Kazakhstan National Olympiad, 4
For $x;y \geq 0$ prove the inequality:
$\sqrt{x^2-x+1} \sqrt{y^2-y+1}+ \sqrt{x^2+x+1} \sqrt{y^2+y+1} \geq 2(x+y)$
2004 France Team Selection Test, 3
Each point of the plane with two integer coordinates is the center of a disk with radius $ \frac {1} {1000}$.
Prove that there exists an equilateral triangle whose vertices belong to distinct disks.
Prove that such a triangle has side-length greater than 96.
2019 Bulgaria National Olympiad, 1
Let $f(x)=x^2+bx+1,$ where $b$ is a real number. Find the number of integer solutions to the inequality $f(f(x)+x)<0.$
2004 Romania National Olympiad, 2
The sidelengths of a triangle are $a,b,c$.
(a) Prove that there is a triangle which has the sidelengths $\sqrt a,\sqrt b,\sqrt c$.
(b) Prove that $\displaystyle \sqrt{ab}+\sqrt{bc}+\sqrt{ca} \leq a+b+c < 2 \sqrt{ab} + 2 \sqrt{bc} + 2 \sqrt{ca}$.
2004 Thailand Mathematical Olympiad, 18
Find positive reals $a, b, c$ which maximizes the value of $abc$ subject to the constraint that $b(a^2 + 2) + c(a + 2) = 12$.
1992 Baltic Way, 18
Show that in a non-obtuse triangle the perimeter of the triangle is always greater than two times the diameter of the circumcircle.
2012 Belarus Team Selection Test, 3
Given a polynomial $P(x)$ with positive real coefficients.
Prove that $P(1)P(xy) \ge P(x)P(y)$ for all $x\ge1, y \ge 1$.
(K. Gorodnin)
1999 Tuymaada Olympiad, 4
Prove the inequality
\[
{x\over y^2-z}+{y\over z^2-x}+{z\over x^2-y} > 1,
\]
where $2 < x, y, z < 4.$
[i]Proposed by A. Golovanov[/i]
2013 Kosovo National Mathematical Olympiad, 5
Let $P$ be a point inside or outside (but not on) of a triangle $ABC$. Prove that $PA +PB +PC$ is greater than half of the perimeter of the triangle
1979 Czech And Slovak Olympiad IIIA, 4
Let $n$ be any natural number. Find all $n$-tuples of real numbers $x_1\le x_2\le ... \le x_n$, for which holds
$$\left(\sum_{i=1}^n x_i\right)^2 \le n \sum_{i=1}^n x_i x_{n-i+1}.$$
2024 USAMO, 6
Let $n > 2$ be an integer and let $\ell \in \{1, 2,\dots, n\}$. A collection $A_1,\dots,A_k$ of (not necessarily distinct) subsets of $\{1, 2,\dots, n\}$ is called $\ell$-large if $|A_i| \ge \ell$ for all $1 \le i \le k$. Find, in terms of $n$ and $\ell$, the largest real number $c$ such that the inequality
\[ \sum_{i=1}^k\sum_{j=1}^k x_ix_j\frac{|A_i\cap A_j|^2}{|A_i|\cdot|A_j|}\ge c\left(\sum_{i=1}^k x_i\right)^2 \]
holds for all positive integer $k$, all nonnegative real numbers $x_1,x_2,\dots,x_k$, and all $\ell$-large collections $A_1,A_2,\dots,A_k$ of subsets of $\{1,2,\dots,n\}$.
[i]Proposed by Titu Andreescu and Gabriel Dospinescu[/i]
2009 Kosovo National Mathematical Olympiad, 3
Let $a,b$ and $c$ be the sides of a triangle, prove that
$\frac {a}{b+c}+\frac {b}{c+a}+\frac {c}{a+b}<2$.
1987 IMO Longlists, 27
Find, with proof, the smallest real number $C$ with the following property:
For every infinite sequence $\{x_i\}$ of positive real numbers such that $x_1 + x_2 +\cdots + x_n \leq x_{n+1}$ for $n = 1, 2, 3, \cdots$, we have
\[\sqrt{x_1}+\sqrt{x_2}+\cdots+\sqrt{x_n} \leq C \sqrt{x_1+x_2+\cdots+x_n} \qquad \forall n \in \mathbb N.\]
2010 Malaysia National Olympiad, 8
Show that \[\log_{a}bc+\log_bca+\log_cab \ge 4(\log_{ab}c+\log_{bc}a+\log_{ca}b)\] for all $a,b,c$ greater than 1.
2015 Azerbaijan National Olympiad, 2
Let $a,b$ and $c$ be the length of sides of a triangle.Then prove that $S\le\frac{a^2+b^2+c^2}{6}$ where $S$ is the area of triangle.
2005 Italy TST, 2
$(a)$ Prove that in a triangle the sum of the distances from the centroid to the sides is not less than three times the inradius, and find the cases of equality.
$(b)$ Determine the points in a triangle that minimize the sum of the distances to the sides.
2003 Iran MO (3rd Round), 15
Assume $m\times n$ matrix which is filled with just 0, 1 and any two row differ in at least $n/2$ members, show that $m \leq 2n$.
( for example the diffrence of this two row is only in one index
110
100)
[i]Edited by Myth[/i]
2001 Moldova National Olympiad, Problem 6
Set $a_n=\frac{2n}{n^4+3n^2+4},n\in\mathbb N$. Prove that $\frac14\le a_1+a_2+\ldots+a_n\le\frac12$ for all $n$.
2017 Kazakhstan National Olympiad, 2
For positive reals $x,y,z\ge \frac{1}{2}$ with $x^2+y^2+z^2=1$, prove this inequality holds
$$(\frac{1}{x}+\frac{1}{y}-\frac{1}{z})(\frac{1}{x}-\frac{1}{y}+\frac{1}{z})\ge 2$$