Found problems: 6530
2003 USAMO, 5
Let $ a$, $ b$, $ c$ be positive real numbers. Prove that
\[ \dfrac{(2a \plus{} b \plus{} c)^2}{2a^2 \plus{} (b \plus{} c)^2} \plus{} \dfrac{(2b \plus{} c \plus{} a)^2}{2b^2 \plus{} (c \plus{} a)^2} \plus{} \dfrac{(2c \plus{} a \plus{} b)^2}{2c^2 \plus{} (a \plus{} b)^2} \le 8.
\]
2020 Thailand TSTST, 2
Let $x, y, z$ be positive real numbers such that $x^2+y^2+z^2=3$. Prove that $$\frac{x+1}{z+x+1}+\frac{y+1}{x+y+1}+\frac{z+1}{y+z+1}\geq\frac{(xy+yz+zx+\sqrt{xyz})^2}{(x+y)(y+z)(z+x)}.$$
2021 OMpD, 3
Let $a$ and $b$ be positive real numbers, with $a < b$ and let $n$ be a positive integer. Prove that for all real numbers $x_1, x_2, \ldots , x_n \in [a, b]$:
$$ |x_1 - x_2| + |x_2 - x_3| + \cdots + |x_{n-1} - x_n| + |x_n - x_1| \leq \frac{2(b - a)}{b + a}(x_1 + x_2 + \cdots + x_n)$$
And determine for what values of $n$ and $x_1, x_2, \ldots , x_n$ the equality holds.
2014 Math Prize For Girls Problems, 14
A triangle has area 114 and sides of integer length. What is the perimeter of the triangle?
2004 Croatia National Olympiad, Problem 2
If $a,b,c$ are positive numbers, prove the inequality
$$\frac{a^2}{(a+b)(a+c)}+\frac{b^2}{(b+c)(b+a)}+\frac{c^2}{(c+a)(c+b)}\ge\frac34.$$
1978 Bundeswettbewerb Mathematik, 1
Let $a, b, c$ be sides of a triangle. Prove that
$$\frac{1}{3} \leq \frac{a^2 +b^2 +c^2 }{(a+b+c)^2 } < \frac{1}{2}$$
and show that $\frac{1}{2}$ cannot be replaced with a smaller number.
2008 Pan African, 1
Let $x$ and $y$ be two positive reals. Prove that $xy\le\frac{x^{n+2}+y^{n+2}}{x^n+y^n}$ for all non-negative integers $n$.
the 14th XMO, P1
Nonnegative reals $x_1$, $x_2$, $\dots$, $x_n$ satisfies $x_1+x_2+\dots+x_n=n$. Let $||x||$ be the distance from $x$ to the nearest integer of $x$ (e.g. $||3.8||=0.2$, $||4.3||=0.3$). Let $y_i = x_i ||x_i||$. Find the maximum value of $\sum_{i=1}^n y_i^2$.
2016 China National Olympiad, 1
Let $a_1,a_2,\cdots, a_{31} ;b_1,b_2, \cdots, b_{31}$ be positive integers such that
$a_1< a_2<\cdots< a_{31}\leq2015$ , $ b_1< b_2<\cdots<b_{31}\leq2015$ and $a_1+a_2+\cdots+a_{31}=b_1+b_2+\cdots+b_{31}.$
Find the maximum value of $S=|a_1-b_1|+|a_2-b_2|+\cdots+|a_{31}-b_{31}|.$
1993 USAMO, 5
Let $ \, a_{0}, a_{1}, a_{2},\ldots\,$ be a sequence of positive real numbers satisfying $ \, a_{i\minus{}1}a_{i\plus{}1}\leq a_{i}^{2}\,$ for $ i \equal{} 1,2,3,\ldots\; .$ (Such a sequence is said to be [i]log concave[/i].) Show that for each $ \, n > 1,$
\[ \frac{a_{0}\plus{}\cdots\plus{}a_{n}}{n\plus{}1}\cdot\frac{a_{1}\plus{}\cdots\plus{}a_{n\minus{}1}}{n\minus{}1}\geq\frac{a_{0}\plus{}\cdots\plus{}a_{n\minus{}1}}{n}\cdot\frac{a_{1}\plus{}\cdots\plus{}a_{n}}{n}.\]
1990 ITAMO, 4
Let $a,b,c$ be side lengths of a triangle with $a+b+c = 1$. Prove that $a^2 +b^2 +c^2 +4abc \le \frac12$ .
1999 German National Olympiad, 4
A convex polygon $P$ is placed inside a unit square $Q$. Prove that the perimeter of $P$ does not exceed $4$.
2010 Today's Calculation Of Integral, 525
Let $ a,\ b$ be real numbers satisfying $ \int_0^1 (ax\plus{}b)^2dx\equal{}1$.
Determine the values of $ a,\ b$ for which $ \int_0^1 3x(ax\plus{}b)\ dx$ is maximized.
2002 Regional Competition For Advanced Students, 4
Let $a_0, a_1, ..., a_{2002}$ be real numbers.
a) Show that the smallest of the values $a_k (1-a_{2002-k})$ ($0 \le k \le 2002$) the following applies:
it is smaller or equal to $1/4$.
b) Does this statement always apply to the smallest of the values $a_k (1-a_{2003-k})$ ($1 \le k \le 2002$) ?
c) Show for positive real numbers $a_0, a_1, ..., a_{2002}$ :
the smallest of the values $a_k (1-a_{2003-k})$ ($1 \le k \le 2002$) is less than or equal to $1/4$.
2018 Latvia Baltic Way TST, P4
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function that satisfies
$$\sqrt{2f(x)}-\sqrt{2f(x)-f(2x)}\ge 2$$
for all real $x$.
Prove for all real $x$:
[i](a)[/i] $f(x)\ge 4$;
[i](b)[/i] $f(x)\ge 7.$
1990 IMO Longlists, 59
Given eight real numbers $a_1 \leq a_2 \leq \cdots \leq a_7 \leq a_8$. Let $x = \frac{ a_1 + a_2 + \cdots + a_7 + a_8}{8}$, $y = \frac{ a_1^2 + a_2^2 + \cdots + a_7^2 + a_8^2}{8}$. Prove that
\[2 \sqrt{y-x^2} \leq a_8 - a_1 \leq 4 \sqrt{y-x^2}.\]
2012 Dutch IMO TST, 2
Let $a, b, c$ and $d$ be positive real numbers. Prove that
$$\frac{a - b}{b + c}+\frac{b - c}{c + d}+\frac{c - d}{d + a} +\frac{d - a}{a + b } \ge 0 $$
2005 Indonesia MO, 1
Let $ n$ be a positive integer. Determine the number of triangles (non congruent) with integral side lengths and the longest side length is $ n$.
2022 Austrian Junior Regional Competition, 1
Show that for all real numbers $x$ and $y$ with $x > -1$ and $y > -1$ and $x + y = 1$ the inequality
$$\frac{x}{y + 1} +\frac{y}{x + 1} \ge \frac23$$
holds. When does equality apply?
[i](Walther Janous)[/i]
2023 Thailand Mathematical Olympiad, 6
Let $a,b,c,x,y$ be positive real numbers such that $abc=1$. Prove that
$$\frac{a^5}{xc+yb}+\frac{b^5}{xa+yc}+\frac{c^5}{xb+ya}\geq \frac{9}{(x+y)(a^2+b^2+c^2)}.$$
2005 Poland - Second Round, 3
In space are given $n\ge 2$ points, no four of which are coplanar. Some of these points are connected by segments. Let $K$ be the number of segments $(K>1)$ and $T$ be the number of formed triangles. Prove that $9T^2<2K^3$.
2004 South East Mathematical Olympiad, 8
Determine the number of ordered quadruples $(x, y, z, u)$ of integers, such that
\[\dfrac{x-y}{x+y}+\dfrac{y-z}{y+z}+\dfrac{z-u}{z+u}>0 \textrm{ and } 1\le x,y,z,u\le 10.\]
2020 German National Olympiad, 3
Show that the equation
\[x(x+1)(x+2)\dots (x+2020)-1=0\]
has exactly one positive solution $x_0$, and prove that this solution $x_0$ satisfies
\[\frac{1}{2020!+10}<x_0<\frac{1}{2020!+6}.\]
2014 IFYM, Sozopol, 8
Prove that, if $a,b,c$ are sides of a triangle, then we have the following inequality:
$3(a^3 b+b^3 c+c^3 a)+2(ab^3+bc^3+ca^3 )\geq 5(a^2 b^2+a^2 c^2+b^2 c^2 )$.
2003 APMO, 4
Let $a,b,c$ be the sides of a triangle, with $a+b+c=1$, and let $n\ge 2$ be an integer. Show that
\[ \sqrt[n]{a^n+b^n}+\sqrt[n]{b^n+c^n}+\sqrt[n]{c^n+a^n}<1+\frac{\sqrt[n]{2}}{2}. \]