Found problems: 6530
1996 Tournament Of Towns, (519) 2
(a) Prove that
$$3-\frac{2}{(n-1)!} < \frac{2^2-2}{2!}+\frac{2^2-2}{3!}+...+\frac{n^2-2}{n!}<3$$
(b) Find some positive integers $a$, $b$ and $c$ such that for any $n > 2$,
$$b-\frac{c}{(n-2)!} < \frac{2^3-a}{2!}+\frac{3^3-a}{3!}+...+\frac{n^3-a}{n!}<b$$
(V Senderov, NB Vassiliev)
2021 Moldova EGMO TST, 1
Postive real numbers $a, b, c$ satisfy $abc=1$. Show that $$\frac{a^3+a^2}{1+bc}+\frac{b^3+b^2}{1+ca}+\frac{c^3+c^2}{1+ab}\geq3.$$
2011 Canadian Students Math Olympiad, 3
Find the largest $C \in \mathbb{R}$ such that
\[\frac{x+z}{(x-z)^2} +\frac{x+w}{(x-w)^2} +\frac{y+z}{(y-z)^2}+\frac{y+w}{(y-w)^2} + \sum_{cyc} \frac{1}{x} \ge \frac{C}{x+y+z+w}\]
where $x,y,z,w \in \mathbb{R^+}$.
[i]Author: Hunter Spink[/i]
1996 Putnam, 2
Prove the inequality for all positive integer $n$ :
\[ \left(\frac{2n-1}{e}\right)^{\frac{2n-1}{2}}<1\cdot 3\cdot 5\cdots (2n-1)<\left(\frac{2n+1}{e}\right)^{\frac{2n+1}{2}} \]
2015 Peru IMO TST, 14
Let $ n$ be a positive integer and let $ a_1,a_2,\ldots,a_n$ be positive real numbers such that:
\[ \sum^n_{i \equal{} 1} a_i \equal{} \sum^n_{i \equal{} 1} \frac {1}{a_i^2}.
\]
Prove that for every $ i \equal{} 1,2,\ldots,n$ we can find $ i$ numbers with sum at least $ i$.
1987 IMO Longlists, 26
Prove that if $x, y, z$ are real numbers such that $x^2+y^2+z^2 = 2$, then
\[x + y + z \leq xyz + 2.\]
1989 Federal Competition For Advanced Students, 3
Let $ a$ be a real number. Prove that if the equation $ x^2\minus{}ax\plus{}a\equal{}0$ has two real roots $ x_1$ and $ x_2$, then: $ x_1^2\plus{}x_2^2 \ge 2(x_1\plus{}x_2).$
2008 International Zhautykov Olympiad, 3
Let $ a, b, c$ be positive integers for which $ abc \equal{} 1$. Prove that
$ \sum \frac{1}{b(a\plus{}b)} \ge \frac{3}{2}$.
2019 VJIMC, 4
Determine the largest constant $K\geq 0$ such that $$\frac{a^a(b^2+c^2)}{(a^a-1)^2}+\frac{b^b(c^2+a^2)}{(b^b-1)^2}+\frac{c^c(a^2+b^2)}{(c^c-1)^2}\geq K\left (\frac{a+b+c}{abc-1}\right)^2$$ holds for all positive real numbers $a,b,c$ such that $ab+bc+ca=abc$.
[i]Proposed by Orif Ibrogimov (Czech Technical University of Prague).[/i]
1953 Putnam, A3
$a, b, c$ are real, and the sum of any two is greater than the third.
Show that $\frac{2(a + b + c)(a^2 + b^2 + c^2)}{3} > a^3 + b^3 + c^3 + abc$.
2004 239 Open Mathematical Olympiad, 4
Let the sum of positive reals $a,b,c$ be equal to 1. Prove an inequality \[
\sqrt{{ab}\over {c+ab}}+\sqrt{{bc}\over {a+bc}}+\sqrt{{ac}\over {b+ac}}\le 3/2
\].
[b]proposed by Fedor Petrov[/b]
2014 India IMO Training Camp, 2
Let $a,b$ be positive real numbers.Prove that $(1+a)^{8}+(1+b)^{8}\geq 128ab(a+b)^{2}$.
2014 Math Prize For Girls Problems, 14
A triangle has area 114 and sides of integer length. What is the perimeter of the triangle?
2023 JBMO Shortlist, A7
Let $a_1,a_2,a_3,\ldots,a_{250}$ be real numbers such that $a_1=2$ and
$$a_{n+1}=a_n+\frac{1}{a_n^2}$$
for every $n=1,2, \ldots, 249$. Let $x$ be the greatest integer which is less than
$$\frac{1}{a_1}+\frac{1}{a_2}+\ldots+\frac{1}{a_{250}}$$
How many digits does $x$ have?
[i]Proposed by Miroslav Marinov, Bulgaria[/i]
2013 India National Olympiad, 3
Let $a,b,c,d \in \mathbb{N}$ such that $a \ge b \ge c \ge d $. Show that the equation $x^4 - ax^3 - bx^2 - cx -d = 0$ has no integer solution.
2002 Greece Junior Math Olympiad, 4
Prove that $1\cdot2\cdot3\cdots 2002<\left(\frac{2003}{2}\right)^{2002}.$
2006 Junior Balkan Team Selection Tests - Romania, 1
Prove that $\frac{a^3}{bc} + \frac{b^3}{ca} + \frac{c^3}{ba} \ge a + b + c$, for all positive real numbers $a, b$, and $c$.
2001 239 Open Mathematical Olympiad, 2
For any positive numbers $ a_1 , a_2 , \dots, a_n $ prove the inequality $$\!
\left(\!1\!+\!\frac{1}{a_1(1+a_1)} \!\right)\!
\left(\!1\!+\!\frac{1}{a_2(1+a_2)} \! \right) \! \dots \!
\left(\!1\!+\!\frac{1}{a_n(1+a_n)} \! \right) \geq
\left(\!1\!+\!\frac{1}{p(1+p)} \! \right)^{\! n} \! ,$$
where $p=\sqrt[n]{a_1 a_2 \dots a_n}$.
2018 Junior Balkan Team Selection Tests - Romania, 2
Let $k > 2$ be a real number.
a) Prove that for all positive real numbers $x,y$ and $z$ the following inequality holds:
$$\sqrt{x + y }+\sqrt{y + z }+\sqrt{z + x} > 2\sqrt{\frac{(x + y)(y + z)(z + x)}{xy + yz + zx}}$$
b) Prove that there exist positive real numbers $x, y$ and $z$ such that
$$\sqrt{x + y }+\sqrt{y + z}+\sqrt{z + x} <k\sqrt{\frac{(x + y)(y + z)(z + x)}{xy + yz + zx}}$$
Leonard Giugiuc
2003 Turkey MO (2nd round), 2
Let $ABCD$ be a convex quadrilateral and $K,L,M,N$ be points on $[AB],[BC],[CD],[DA]$, respectively. Show that,
\[
\sqrt[3]{s_{1}}+\sqrt[3]{s_{2}}+\sqrt[3]{s_{3}}+\sqrt[3]{s_{4}}\leq 2\sqrt[3]{s}
\]
where $s_1=\text{Area}(AKN)$, $s_2=\text{Area}(BKL)$, $s_3=\text{Area}(CLM)$, $s_4=\text{Area}(DMN)$ and $s=\text{Area}(ABCD)$.
BIMO 2022, 1
Given an acute triangle $ABC$, mark $3$ points $X, Y, Z$ in the interior of the triangle. Let $X_1, X_2, X_3$ be the projections of $X$ to $BC, CA, AB$ respectively, and define the points $Y_i, Z_i$ similarly for $i=1, 2, 3$.
a) Suppose that $X_iY_i<X_iZ_i$ for all $i=1,2,3$, prove that $XY<XZ$.
b) Prove that this is not neccesarily true, if triangle $ABC$ is allowed to be obtuse.
[i]Proposed by Ivan Chan Kai Chin[/i]
III Soros Olympiad 1996 - 97 (Russia), 11.3
Prove that the equation x^3- x- 3 = 0 has a unique root. Which is greater, the root of this equation or $\sqrt[5]{13}$? (Use of a calculator is prohibited.)
2014 USAJMO, 4
Let $b\geq 2$ be an integer, and let $s_b(n)$ denote the sum of the digits of $n$ when it is written in base $b$. Show that there are infinitely many positive integers that cannot be represented in the form $n+s_b(n)$, where $n$ is a positive integer.
2014 District Olympiad, 1
Prove that:
[list=a][*]$\displaystyle\left( \frac{1}{2}\right) ^{3}+\left( \frac{2}{3}\right)^{3}+\left( \frac{5}{6}\right) ^{3}=1$
[*]$3^{33}+4^{33}+5^{33}<6^{33}$[/list]
2014 Middle European Mathematical Olympiad, 1
Determine the lowest possible value of the expression
\[ \frac{1}{a+x} + \frac{1}{a+y} + \frac{1}{b+x} + \frac{1}{b+y} \]
where $a,b,x,$ and $y$ are positive real numbers satisfying the inequalities
\[ \frac{1}{a+x} \ge \frac{1}{2} \] \[\frac{1}{a+y} \ge \frac{1}{2} \] \[ \frac{1}{b+x} \ge \frac{1}{2} \] \[ \frac{1}{b+y} \ge 1. \]