Found problems: 6530
2001 Moldova National Olympiad, Problem 1
Real numbers $b>a>0$ are given. Find the number $r$ in $[a,b]$ which minimizes the value of $\max\left\{\left|\frac{r-x}x\right||a\le x\le b\right\}$.
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 $a \ge b \ge c \Rightarrow A \ge B \ge C$.
2021 Romania National Olympiad, 2
Prove that for all positive real numbers $a,b,c$ the following inequality holds:
\[(a+b+c)\left(\frac1a+\frac1b+\frac1c\right)\ge\frac{2(a^2+b^2+c^2)}{ab+bc+ca}+7\]
and determine all cases of equality.
[i]Lucian Petrescu[/i]
2009 Today's Calculation Of Integral, 503
Prove the following inequality.
\[ \frac{2}{2\plus{}e^{\frac 12}}<\int_0^1 \frac{dx}{1\plus{}xe^{x}}<\frac{2\plus{}e}{2(1\plus{}e)}\]
2014 Saudi Arabia GMO TST, 4
Let $a_1 \ge a_2 \ge ... \ge a_n > 0$ be real numbers. Prove that
$$a_1a_2(a_1 - a_2) + a_2a_3(a_2 - a_3) +...+ a_{n-1}a_n(a_{n-1} - a_n) \ge a_1a_n(a_1 - a_n)$$
2004 Baltic Way, 17
Consider a rectangle with sidelengths 3 and 4, pick an arbitrary inner point on each side of this rectangle. Let $x, y, z$ and $u$ denote the side lengths of the quadrilateral spanned by these four points. Prove that $25 \leq x^2+y^2+z^2+u^2 \leq 50$.
1963 IMO, 3
In an $n$-gon $A_{1}A_{2}\ldots A_{n}$, all of whose interior angles are equal, the lengths of consecutive sides satisfy the relation
\[a_{1}\geq a_{2}\geq \dots \geq a_{n}. \]
Prove that $a_{1}=a_{2}= \ldots= a_{n}$.
2018 China Girls Math Olympiad, 1
Let $a\le 1$ be a real number. Sequence $\{x_n\}$ satisfies $x_0=0, x_{n+1}= 1-a\cdot e^{x_n}$, for all $n\ge 1$, where $e$ is the natural logarithm. Prove that for any natural $n$, $x_n\ge 0$.
2025 China National Olympiad, 6
Let $a_1, a_2, \ldots, a_n$ be real numbers such that $\sum_{i=1}^n a_i = n$, $\sum_{i = 1}^n a_i^2 = 2n$, $\sum_{i=1}^n a_i^3 = 3n$.
(i) Find the largest constant $C$, such that for all $n \geqslant 4$, \[ \max \left\{ a_1, a_2, \ldots, a_n \right\} - \min \left\{ a_1, a_2, \ldots, a_n \right\} \geqslant C. \]
(ii) Prove that there exists a positive constant $C_2$, such that \[ \max \left\{ a_1, a_2, \ldots, a_n \right\} - \min \left\{ a_1, a_2, \ldots, a_n \right\} \geqslant C + C_2 n^{-\frac 32}, \]where $C$ is the constant determined in (i).
2025 AIME, 12
The set of points in $3$-dimensional coordinate space that lie in the plane $x+y+z=75$ whose coordinates satisfy the inequalities $$x-yz<y-zx<z-xy$$forms three disjoint convex regions. Exactly one of those regions has finite area. The area of this finite region can be expressed in the form $a\sqrt{b},$ where $a$ and $b$ are positive integers and $b$ is not divisible by the square of any prime. Find $a+b.$
2018 Iran Team Selection Test, 2
Determine the least real number $k$ such that the inequality
$$\left(\frac{2a}{a-b}\right)^2+\left(\frac{2b}{b-c}\right)^2+\left(\frac{2c}{c-a}\right)^2+k \geq 4\left(\frac{2a}{a-b}+\frac{2b}{b-c}+\frac{2c}{c-a}\right)$$
holds for all real numbers $a,b,c$.
[i]Proposed by Mohammad Jafari[/i]
1957 AMC 12/AHSME, 45
If two real numbers $ x$ and $ y$ satisfy the equation $ \frac{x}{y} \equal{} x \minus{} y$, then:
$ \textbf{(A)}\ {x \ge 4}\text{ and }{x \le 0}\qquad \\
\textbf{(B)}\ {y}\text{ can equal }{1}\qquad \\
\textbf{(C)}\ \text{both }{x}\text{ and }{y}\text{ must be irrational}\qquad \\
\textbf{(D)}\ {x}\text{ and }{y}\text{ cannot both be integers}\qquad \\
\textbf{(E)}\ \text{both }{x}\text{ and }{y}\text{ must be rational}$
2016 Korea Winter Program Practice Test, 4
Let $x,y,z \ge 0$ be real numbers such that $(x+y-1)^2+(y+z-1)^2+(z+x-1)^2=27$.
Find the maximum and minimum of $x^4+y^4+z^4$
1988 National High School Mathematics League, 12
$a,b$ are real numbers, satisfying that $\frac{1}{a}+\frac{1}{b}=1$. Prove that for any $n\in\mathbb{Z}_+$, $(a+b)^{2n}-a^n-b^n\geq2^{2n}-2^{n+1}$.
2013 Estonia Team Selection Test, 3
Let $x_1,..., x_n$ be non-negative real numbers, not all of which are zeros.
(i) Prove that
$$1 \le \frac{\left(x_1+\frac{x_2}{2}+\frac{x_3}{3}+...+\frac{x_n}{n}\right)(x_1+2x_2+3x_3+...+nx_n)}{(x_1+x_2+x_3+...+x_n)^2} \le \frac{(n+1)^2}{4n}$$
(ii) Show that, for each $n > 1$, both inequalities can hold as equalities.
2007 Cuba MO, 9
Let $O$ be the circumcircle of $\triangle ABC$, with $AC=BC$ end let $D=AO\cap BC$. If $BD$ and $CD$ are integer numbers and $AO-CD$ is prime, determine such three numbers.
1971 Bulgaria National Olympiad, Problem 6
In a triangular pyramid $SABC$ one of the plane angles with vertex $S$ is a right angle and the orthogonal projection of $S$ on the base plane $ABC$ coincides with the orthocenter of the triangle $ABC$. Let $SA=m$, $SB=n$, $SC=p$, $r$ is the inradius of $ABC$. $H$ is the height of the pyramid and $r_1,r_2,r_3$ are radii of the incircles of the intersections of the pyramid with the plane passing through $SA,SB,SC$ and the height of the pyramid. Prove that
(a) $m^2+n^2+p^2\ge18r^2$;
(b) $\frac{r_1}H,\frac{r_2}H,\frac{r_3}H$ are in the range $(0.4,0.5)$.
2017 Kazakhstan NMO, Problem 2
For positive reals $x,y,z\ge \frac{1}{2}$ with $x^2+y^2+z^2=1$, prove this inequality holds
$$(\frac{1}{x}+\frac{1}{y}-\frac{1}{z})(\frac{1}{x}-\frac{1}{y}+\frac{1}{z})\ge 2$$
2024 Indonesia TST, A
Given real numbers $x,y,z$ which satisfies
$$|x+y+z|+|xy+yz+zx|+|xyz| \le 1$$
Show that $max\{ |x|,|y|,|z|\} \le 1$.
1975 Putnam, B3
Let $n$ be a positive integer. Let $S=\{a_1,\ldots, a_{k}\}$ be a finite collection of at least $n$ not necessarily distinct positive real numbers. Let
$$f(S)=\left(\sum_{i=1}^{k} a_{i}\right)^{n}$$ and
$$g(S)=\sum_{1\leq i_{1}<\ldots<i_{n} \leq k} a_{i_{1}}\cdot\ldots\cdot a_{i_{n}}.$$ Determine $\sup_{S} \frac{g(S)}{f(S)}$.
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.
2011 IMAC Arhimede, 6
Let $a$, $b$ and $c$ be positive real numbers such that $a+b+c=1$. Prove that
$\frac{a}{a^3+b^2c+c^2b} + \frac{b}{b^3+c^2a+a^2c} + \frac{c}{c^3+a^2b+b^2a} \le 1+\frac{8}{27abc}$
2006 Balkan MO, 1
Let $ a$, $ b$, $ c$ be positive real numbers. Prove the inequality
\[ \frac{1}{a\left(b+1\right)}+\frac{1}{b\left(c+1\right)}+\frac{1}{c\left(a+1\right)}\geq \frac{3}{1+abc}. \]
1976 IMO Longlists, 24
Let $0 \le x_1 \le x_2\le\cdots\le x_n \le 1$. Prove that for all $A \ge 1$, there exists an interval $I$ of length $2\sqrt[n]{A}$ such that for all $x \in I$,
\[|(x - x_1)(x - x_2) \cdots (x -x_n)| \le A.\]
2001 Austrian-Polish Competition, 3
Let $a,b,c$ be sides of a triangle. Prove that
\[ 2 < \frac{a+b}{c} + \frac{b+c}{a} + \frac{c+a}{b} - \frac{a^3+b^3+c^3}{abc}\leq 3 \]