Found problems: 6530
1961 Putnam, B1
Let $a_1 , a_2 , a_3 ,\ldots$ be a sequence of positive real numbers, define $s_n = \frac{a_1 +a_2 +\ldots+a_n }{n}$ and $r_n = \frac{a_{1}^{-1} +a_{2}^{-1} +\ldots+a_{n}^{-1} }{n}.$ Given that $\lim_{n\to \infty} s_n $ and $\lim_{n\to \infty} r_n $ exist, prove that the product of these limits is not less than $1.$
1999 Croatia National Olympiad, Problem 3
a,b,c are positive & abc=1, then show that
a^(b+c) .b^(c+a) .c^(a+b) ≤ 1
2016 Romania National Olympiad, 3
Let be a real number $ a, $ and a nondecreasing function $ f:\mathbb{R}\longrightarrow\mathbb{R} . $ Prove that $ f $ is continuous in $ a $ if and only if there exists a sequence $ \left( a_n \right)_{n\ge 1} $ of real positive numbers such that
$$ \int_a^{a+a_n} f(x)dx+\int_a^{a-a_n} f(x)dx\le\frac{a_n}{n} , $$
for all natural numbers $ n. $
[i]Dan Marinescu[/i]
Taiwan TST 2015 Round 1, 1
Let $a,b,c,d$ be any real numbers such that $a+b+c+d=0$, prove that
\[1296(a^7+b^7+c^7+d^7)^2\le637(a^2+b^2+c^2+d^2)^7\]
2025 India National Olympiad, P4
Let $n\ge 3$ be a positive integer. Find the largest real number $t_n$ as a function of $n$ such that the inequality
\[\max\left(|a_1+a_2|, |a_2+a_3|, \dots ,|a_{n-1}+a_{n}| , |a_n+a_1|\right) \ge t_n \cdot \max(|a_1|,|a_2|, \dots ,|a_n|)\]
holds for all real numbers $a_1, a_2, \dots , a_n$ .
[i]Proposed by Rohan Goyal and Rijul Saini[/i]
2000 Korea - Final Round, 3
The real numbers $a,b,c,x,y,$ and $z$ are such that $a>b>c>0$ and $x>y>z>0$. Prove that
\[\frac {a^2x^2}{(by+cz)(bz+cy)}+\frac{b^2y^2}{(cz+ax)(cx+az)}+\frac{c^2z^2}{(ax+by)(ay+bx)}\ge \frac{3}{4}\]
1994 Baltic Way, 3
Find the largest value of the expression
\[xy+x\sqrt{1-x^2}+y\sqrt{1-y^2}-\sqrt{(1-x^2)(1-y^2)}\]
2010 AMC 10, 23
Each of 2010 boxes in a line contains a single red marble, and for $ 1 \le k \le 2010$, the box in the $ kth$ position also contains $ k$ white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops when she first draws a red marble. Let $ P(n)$ be the probability that Isabella stops after drawing exactly $ n$ marbles. What is the smallest value of $ n$ for which $ P(n) < \frac {1}{2010}$?
$ \textbf{(A)}\ 45 \qquad
\textbf{(B)}\ 63 \qquad
\textbf{(C)}\ 64 \qquad
\textbf{(D)}\ 201 \qquad
\textbf{(E)}\ 1005$
2019 Irish Math Olympiad, 9
Suppose $x, y, z$ are real numbers such that $x^2 + y^2 + z^2 + 2xyz = 1$. Prove that $8xyz \le 1$, with equality if and only if $(x, y,z)$ is one of the following:
$$\left( \frac12, \frac12, \frac12 \right) , \left( -\frac12, -\frac12, \frac12 \right), \left(- \frac12, \frac12, -\frac12 \right), \left( \frac12,- \frac12, - \frac12 \right)$$
VMEO III 2006 Shortlist, N10
The notation $\phi (n)$ is the number of positive integers smaller than $n$ and coprime with $n$, $\pi (n)$ is the number of primes that do not exceed $n$. Prove that for any natural number $n > 1$, we have
$$\phi (n) \ge \frac{\pi (n)}{2}$$
2011 India IMO Training Camp, 2
Let the real numbers $a,b,c,d$ satisfy the relations $a+b+c+d=6$ and $a^2+b^2+c^2+d^2=12.$ Prove that
\[36 \leq 4 \left(a^3+b^3+c^3+d^3\right) - \left(a^4+b^4+c^4+d^4 \right) \leq 48.\]
[i]Proposed by Nazar Serdyuk, Ukraine[/i]
2018 Hanoi Open Mathematics Competitions, 1
Let $a, b$, and $c$ be distinct positive integers such that $a + 2b + 3c < 12$.
Which of the following inequalities must be true?
A. $a + b + c < 7$
B. $a- b + c < 4$
C. $b + c- a < 3$
D. $a + b- c <5 $
E. $5a + 3b + c < 27$
2002 VJIMC, Problem 4
The numbers $1,2,\ldots,n$ are assigned to the vertices of a regular $n$-gon in an arbitrary order. For each edge, compute the product of the two numbers at the endpoints and sum up these products. What is the smallest possible value of this sum?
2016 Greece JBMO TST, 1
a) Prove that, for any real $x>0$, it is true that $x^3-3x\ge -2$ .
b) Prove that, for any real $x,y,z>0$, it is true that
$$\frac{x^2y}{z}+\frac{y^2z}{x}+\frac{z^2x}{y}+2\left(\frac{y}{xz}+\frac{z}{xy}+\frac{x}{yz} \right)\ge 9$$ . When we have equality ?
2013 Chile National Olympiad, 3
Given a finite sequence of real numbers $a_1,a_2,...,a_n$ such that $$a_1 + a_2 + ... + a_n > 0.$$
Prove that there is at least one index $ i$ such that $$a_i > 0, a_i + a_{i+1} > 0, ..., a_i + a_{i+1} + ...+ a_n > 0.$$
2021 Kyiv City MO Round 1, 10.4
Positive real numbers $a, b, c$ satisfy $a^2 + b^2 + c^2 + a + b + c = 6$. Prove the following inequality:
$$2(\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}) \geq 3 + \frac{1}{a} + \frac{1}{b} + \frac{1}{c}$$
[i]Proposed by Oleksii Masalitin[/i]
1997 Putnam, 4
Let $a_{m,n}$ denote the coefficient of $x^n$ in the expansion $(1+x+x^2)^n$. Prove the inequality for all integers $k\ge 0$ :
\[ 0\le \sum_{\ell=0}^{\left\lfloor{\frac{2k}{3}}\right\rfloor} (-1)^{\ell} a_{k-\ell,\ell}\le 1 \]
2008 Moldova National Olympiad, 12.1
Consider the equation $ x^4 \minus{} 4x^3 \plus{} 4x^2 \plus{} ax \plus{} b \equal{} 0$, where $ a,b\in\mathbb{R}$. Determine the largest value $ a \plus{} b$ can take, so that the given equation has two distinct positive roots $ x_1,x_2$ so that $ x_1 \plus{} x_2 \equal{} 2x_1x_2$.
2016 Romania National Olympiad, 4
Determine all functions $f: \mathbb R \to \mathbb R$ which satisfy the inequality
$$f(a^2) - f(b^2) \leq \left( f(a) + b\right)\left( a - f(b)\right),$$
for all $a,b \in \mathbb R$.
2004 AIME Problems, 12
Let $S$ be the set of ordered pairs $(x, y)$ such that $0<x\le 1$, $0<y\le 1$, and $\left[\log_2{\left(\frac 1x\right)}\right]$ and $\left[\log_5{\left(\frac 1y\right)}\right]$ are both even. Given that the area of the graph of $S$ is $m/n$, where $m$ and $n$ are relatively prime positive integers, find $m+n$. The notation $[z]$ denotes the greatest integer that is less than or equal to $z$.
2014 Contests, 1a
Assume that $x, y \ge 0$. Show that $x^2 + y^2 + 1 \le \sqrt{(x^3 + y + 1)(y^3 + x + 1)}$.
1957 Kurschak Competition, 3
What is the largest possible value of $|a_1 - 1| + |a_2-2|+...+ |a_n- n|$ where $a_1, a_2,..., a_n$ is a permutation of $1,2,..., n$?
1998 Swedish Mathematical Competition, 2
$ABC$ is a triangle. Show that $c \ge (a+b) \sin \frac{C}{2}$
1992 AIME Problems, 10
Consider the region $A$ in the complex plane that consists of all points $z$ such that both $\frac{z}{40}$ and $\frac{40}{\overline{z}}$ have real and imaginary parts between $0$ and $1$, inclusive. What is the integer that is nearest the area of $A$?
1985 Polish MO Finals, 4
$P$ is a point inside the triangle $ABC$ is a triangle. The distance of $P$ from the lines $BC, CA, AB$ is $d_a, d_b, d_c$ respectively. If $r$ is the inradius, show that $$\frac{2}{ \frac{1}{d_a} + \frac{1}{d_b} + \frac{1}{d_c}} < r < \frac{d_a + d_b + d_c}{2}$$