Found problems: 6530
2022 Stanford Mathematics Tournament, 1
If $x$, $y$, and $z$ are real numbers such that $x^2+2y^2+3z^2=96$, what is the maximum possible value of $x+2y+3z$?
2002 AMC 12/AHSME, 18
If $a,b,c$ are real numbers such that $a^2+2b=7$, $b^2+4c=-7$, and $c^2+6a=-14$, find $a^2+b^2+c^2$.
$\textbf{(A) }14\qquad\textbf{(B) }21\qquad\textbf{(C) }28\qquad\textbf{(D) }35\qquad\textbf{(E) }49$
2011 Baltic Way, 4
Let $a,b,c,d$ be non-negative reals such that $a+b+c+d=4$. Prove the inequality
\[\frac{a}{a^3+8}+\frac{b}{b^3+8}+\frac{c}{c^3+8}+\frac{d}{d^3+8}\le\frac{4}{9}\]
2024 Regional Olympiad of Mexico West, 6
We say that a triangle of sides $a,b,c$ is [i] virtual[/i] if such measures satisfy
$$\begin{cases}
a^{2024}+b^{2024}> c^{2024},\\
b^{2024}+c^{2024}> a^{2024},\\
c^{2024}+a^{2024}> b^{2024}
\end{cases}$$
Find the number of ordered triples $(a,b,c)$ such that $a,b,c$ are integers between $1$ and $2024$ (inclusive) and $a,b,c$ are the sides of a [i]virtual [/i] triangle.
1993 Rioplatense Mathematical Olympiad, Level 3, 4
$x$ and $y$ are real numbers such that $6 -x$, $3 + y^2$, $11 + x$, $14 - y^2$ are greater than zero.
Find the maximum of the function $$f(x,y) = \sqrt{(6 -x)(3 + y^2)} + \sqrt{(11 + x)(14 - y^2)}.$$
2012 Gulf Math Olympiad, 2
Prove that if $a, b, c$ are positive real numbers, then the least possible value of \[6a^3 + 9b^3 + 32c^3 + \frac{1}{4abc}\]
is $6$. For which values of $a, b$ and $c$ is equality attained?
2017 China National Olympiad, 6
Given an integer $n \geq2$ and real numbers $a,b$ such that $0<a<b$. Let $x_1,x_2,\ldots, x_n\in [a,b]$ be real numbers. Find the maximum value of $$\frac{\frac{x^2_1}{x_2}+\frac{x^2_2}{x_3}+\cdots+\frac{x^2_{n-1}}{x_n}+\frac{x^2_n}{x_1}}{x_1+x_2+\cdots +x_{n-1}+x_n}.$$
2015 Middle European Mathematical Olympiad, 1
Prove that for all positive real numbers $a$, $b$, $c$ such that $abc=1$ the following inequality holds:
$$\frac{a}{2b+c^2}+\frac{b}{2c+a^2}+\frac{c}{2a+b^2}\le \frac{a^2+b^2+c^2}3.$$
2019 Canadian Mathematical Olympiad Qualification, 8
For $t \ge 2$, defi ne $S(t)$ as the number of times $t$ divides into $t!$. We say that a positive integer $t$ is a [i]peak[/i] if $S(t) > S(u)$ for all values of $u < t$.
Prove or disprove the following statement:
For every prime $p$, there is an integer $k$ for which $p$ divides $k$ and $k$ is a peak.
2011 Vietnam National Olympiad, 1
Prove that if $x>0$ and $n\in\mathbb N,$ then we have
\[\frac{x^n(x^{n+1}+1)}{x^n+1}\leq\left(\frac {x+1}{2}\right)^{2n+1}.\]
2012 India IMO Training Camp, 3
Let $\mathbb{R}^{+}$ denote the set of all positive real numbers. Find all functions $f:\mathbb{R}^{+}\longrightarrow \mathbb{R}$ satisfying
\[f(x)+f(y)\le \frac{f(x+y)}{2}, \frac{f(x)}{x}+\frac{f(y)}{y}\ge \frac{f(x+y)}{x+y},\]
for all $x, y\in \mathbb{R}^{+}$.
2003 National High School Mathematics League, 7
The solution set for inequality $|x|^3-2x^2-4|x|+3<0$ is________.
V Soros Olympiad 1998 - 99 (Russia), 11.3
For what a from the interval $[0,\pi]$ do there exist $a$ and $b$ that are not simultaneously equal to zero, for which the inequality
$$a \cos x + b \cos 2x \le 0$$ is satisfied for all $x$ belonging to the segment $[a, \pi]$?
2011 Junior Balkan Team Selection Tests - Romania, 1
Determine
a) the smallest number
b) the biggest number
$n \ge 3$ of non-negative integers $x_1, x_2, ... , x_n$, having the sum $2011$ and satisfying:
$x_1 \le | x_2 - x_3 | , x_2 \le | x_3 - x_4 | , ... , x_{n-2} \le | x_{n-1} -x_n | , x_{n-1} \le | x_n - x_1 |$ and $x_n \le | x_1 - x_2 | $.
1984 Spain Mathematical Olympiad, 3
If $p$ and $q$ are positive numbers with $p+q = 1$,
knowing that any real numbers $x,y$ satisfy $(x-y)^2 \ge 0$, show that
$\frac{x+y}{2} \ge \sqrt{xy}$,
$\frac{x^2+y^2}{2} \ge \big(\frac{x+y}{2}\big)^2$,
$\big(p+\frac{1}{p}\big)^2+\big(q+\frac{1}{q}\big)^2 \ge \frac{25}{2}$
VI Soros Olympiad 1999 - 2000 (Russia), 10.8
There are $100$ positive numbers $a_1$, $a_2$, $...$, $a_{100}$ such that $$\frac{1}{a_1+1}+\frac{1}{a_2+1}+...+\frac{1}{a_{100}+1} \le 1.$$ Prove that $$a_1 \cdot a_2\cdot ... \cdot a_{100} \ge 99^{100}.$$
2009 Serbia National Math Olympiad, 5
Let $x$, $y$, $z$ be arbitrary positive numbers such that $xy+yz+zx=x+y+z$.
Prove that $$\frac{1}{x^2+y+1} + \frac{1}{y^2+z+1} + \frac{1}{z^2+x+1} \leq 1$$.
When does equality occur?
[i]Proposed by Marko Radovanovic[/i]
1992 Taiwan National Olympiad, 6
Find the greatest positive integer $A$ with the following property: For every permutation of $\{1001,1002,...,2000\}$ , the sum of some ten consecutive terms is great than or equal to $A$.
2007 Singapore Team Selection Test, 2
Prove the inequality
\[\sum_{i<j} \frac{a_ia_j}{a_i \plus{} a_j} \le \frac{n}{2(a_1 \plus{} a_2 \plus{}\cdots \plus{} a_n)}\sum_{i<j} a_ia_j\]
for all positive real numbers $ a_1, a_2,\ldots , a_n$.
Oliforum Contest II 2009, 2
Define $ \phi$ the positive real root of $ x^2 \minus{} x \minus{} 1$ and let $ a,b,c,d$ be positive real numbers such that $ (a \plus{} 2b)^2 \equal{} 4c^2 \plus{} 1$.
Show that $ \displaystyle 2d^2 \plus{} a^2\left(\phi \minus{} \frac {1}{2}\right) \plus{} b^2\left(\frac {1}{\phi \minus{} 1} \plus{} 2\right) \plus{} 2 \ge 4(c \minus{} d) \plus{} 2\sqrt {d^2 \plus{} 2d}$ and find all cases of equality.
[i](A.Naskov)[/i]
2001 Estonia National Olympiad, 4
If $x$ and $y$ are nonnegative real numbers with $x+y= 2$, show that $x^2y^2(x^2+y^2)\le 2$.
1989 IMO Longlists, 13
Let $ n \leq 44, n \in \mathbb{N}.$ Prove that for any function $ f$ defined over $ \mathbb{N}^2$ whose images are in the set $ \{1, 2, \ldots , n\},$ there are four ordered pairs $ (i, j), (i, k), (l, j),$ and $ (l, k)$ such that \[ f(i, j) \equal{} f(i, k) \equal{} f(l, j) \equal{} f(l, k),\] in which $ i, j, k, l$ are chosen in such a way that there are natural numbers $ m, p$ that satisfy \[ 1989m \leq i < l < 1989 \plus{} 1989m\] and \[ 1989p \leq j < k < 1989 \plus{} 1989p.\]
1971 Spain Mathematical Olympiad, 4
Prove that in every triangle with sides $a, b, c$ and opposite angles $A, B, C$, is fulfilled (measuring the angles in radians) $$\frac{a A+bB+cC}{a+b+c} \ge \frac{\pi}{3}$$
Hint: Use $a \ge b \ge c \Rightarrow A \ge B \ge C$.
1999 Czech and Slovak Match, 6
Prove that for any integer $n \ge 3$, the least common multiple of the numbers $1,2, ... ,n$ is greater than $2^{n-1}$.
2011 China Team Selection Test, 2
Let $S$ be a set of $n$ points in the plane such that no four points are collinear. Let $\{d_1,d_2,\cdots ,d_k\}$ be the set of distances between pairs of distinct points in $S$, and let $m_i$ be the multiplicity of $d_i$, i.e. the number of unordered pairs $\{P,Q\}\subseteq S$ with $|PQ|=d_i$. Prove that $\sum_{i=1}^k m_i^2\leq n^3-n^2$.