Found problems: 6530
2018 Mexico National Olympiad, 1
Let $A$ and $B$ be two points on a line $\ell$, $M$ the midpoint of $AB$, and $X$ a point on segment $AB$ other than $M$. Let $\Omega$ be a semicircle with diameter $AB$. Consider a point $P$ on $\Omega$ and let $\Gamma$ be the circle through $P$ and $X$ that is tangent to $AB$. Let $Q$ be the second intersection point of $\Omega$ and $\Gamma$. The internal angle bisector of $\angle PXQ$ intersects $\Gamma$ at a point $R$. Let $Y$ be a point on $\ell$ such that $RY$ is perpendicular to $\ell$. Show that $MX > XY$
2013 National Olympiad First Round, 27
For how many pairs $(a,b)$ from $(1,2)$, $(3,5)$, $(5,7)$, $(7,11)$, the polynomial $P(x)=x^5+ax^4+bx^3+bx^2+ax+1$ has exactly one real root?
$
\textbf{(A)}\ 4
\qquad\textbf{(B)}\ 3
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 1
\qquad\textbf{(E)}\ 0
$
2003 China Western Mathematical Olympiad, 3
The non-negative numbers $ x_1, x_2, \ldots, x_5$ satisfy $ \sum_{i \equal{} 1}^5 \frac {1}{1 \plus{} x_i} \equal{} 1$. Prove that $ \sum_{i \equal{} 1}^5 \frac {x_i}{4 \plus{} x_i^2} \leq 1$.
1975 USAMO, 2
Let $ A,B,C,$ and $ D$ denote four points in space and $ AB$ the distance between $ A$ and $ B$, and so on. Show that \[ AC^2\plus{}BD^2\plus{}AD^2\plus{}BC^2 \ge AB^2\plus{}CD^2.\]
2010 Danube Mathematical Olympiad, 5
Let $n\ge3$ be a positive integer. Find the real numbers $x_1\ge0,\ldots,x_n\ge 0$, with $x_1+x_2+\ldots +x_n=n$, for which the expression \[(n-1)(x_1^2+x_2^2+\ldots+x_n^2)+nx_1x_2\ldots x_n\] takes a minimal value.
2023 OlimphÃada, 1
Let $n \geq 2023$ be an integer. For each real $x$, we say that $\lfloor x \rceil$ is the closest integer to $x$, and if there are two closest integers then it is the greater of the two. Suppose there is a positive real $a$ such that $$\lfloor an \rceil = n + \bigg\lfloor\frac{n}{a} \bigg\rceil.$$
Show that $|a^2 - a - 1| < \frac{n\varphi+1}{n^2}$.
1991 Vietnam Team Selection Test, 2
For a positive integer $ n>2$, let $ \left(a_{1}, a_{2}, \ldots, a_{n}\right)$ be a sequence of $ n$ positive reals which is either non-decreasing (this means, we have $ a_{1}\leq a_{2}\leq \ldots \leq a_{n}$) or non-increasing (this means, we have $ a_{1}\geq a_{2}\geq \ldots \geq a_{n}$), and which satisfies $ a_{1}\neq a_{n}$. Let $ x$ and $ y$ be positive reals satisfying $ \frac{x}{y}\geq \frac{a_{1}-a_{2}}{a_{1}-a_{n}}$. Show that:
\[ \frac{a_{1}}{a_{2}\cdot x+a_{3}\cdot y}+\frac{a_{2}}{a_{3}\cdot x+a_{4}\cdot y}+\ldots+\frac{a_{n-1}}{a_{n}\cdot x+a_{1}\cdot y}+\frac{a_{n}}{a_{1}\cdot x+a_{2}\cdot y}\geq \frac{n}{x+y}. \]
2022 Turkey MO (2nd round), 3
Let $a_1, a_2, \cdots, a_{2022}$ be nonnegative real numbers such that $a_1+a_2+\cdots +a_{2022}=1$. Find the maximum number of ordered pairs $(i, j)$, $1\leq i,j\leq 2022$, satisfying $$a_i^2+a_j\ge \frac 1{2021}.$$
2013 Irish Math Olympiad, 10
Let $a,b,c $ be real numbers and let $x=a+b+c,y=a^2+b^2+c^2,z=a^3+b^3+c^3$
and $S=2x^3-9xy+9z .$
$(a)$ Prove that $S$ is unchanged when $a,b,c$ are replaced by $a+t,b+t,c+t $ , respectively , for any real number $t$.
$(b)$ Prove that $ (3y-x^2)^3\ge S^2 .$
2006 Turkey Team Selection Test, 3
If $x,y,z$ are positive real numbers and $xy+yz+zx=1$ prove that
\[ \frac{27}{4} (x+y)(y+z)(z+x) \geq ( \sqrt{x+y} +\sqrt{ y+z} + \sqrt{z+x} )^2 \geq 6 \sqrt 3. \]
2008 China Team Selection Test, 6
Find the maximal constant $ M$, such that for arbitrary integer $ n\geq 3,$ there exist two sequences of positive real number $ a_{1},a_{2},\cdots,a_{n},$ and $ b_{1},b_{2},\cdots,b_{n},$ satisfying
(1):$ \sum_{k \equal{} 1}^{n}b_{k} \equal{} 1,2b_{k}\geq b_{k \minus{} 1} \plus{} b_{k \plus{} 1},k \equal{} 2,3,\cdots,n \minus{} 1;$
(2):$ a_{k}^2\leq 1 \plus{} \sum_{i \equal{} 1}^{k}a_{i}b_{i},k \equal{} 1,2,3,\cdots,n, a_{n}\equiv M$.
2016 Federal Competition For Advanced Students, P2, 4
Let $a,b,c\ge-1$ be real numbers with $a^3+b^3+c^3=1$.
Prove that $a+b+c+a^2+b^2+c^2\le4$, and determine the cases of equality.
(Proposed by Karl Czakler)
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 \[ 1989p \leq j < k < 1989 \plus{} 1989p.\]
2004 Thailand Mathematical Olympiad, 6
Let $a, b, c > 0$ satisfy $a + b + c \ge \frac{1}{a} +\frac{1}{b} +\frac{1}{c}$. Prove that $a^3 + b^3 + c^3 \ge a + b + c$
2003 National High School Mathematics League, 13
Prove that $2\sqrt{1+x}+\sqrt{2x-3}+\sqrt{15-3x}<2\sqrt{19}$, where $\frac{3}{2}\leq x\leq5$.
2016 Dutch IMO TST, 1
Prove that for all positive reals $a, b,c$ we have: $a +\sqrt{ab}+ \sqrt[3]{abc}\le \frac43 (a + b + c)$
2016 Czech And Slovak Olympiad III A, 4
For positive numbers $a, b, c$ holds $(a + c) (b^2 + a c) = 4a$.
Determine the maximum value of $b + c$ and find all triplets of numbers $(a, b, c)$ for which expression takes this value
2014 China Team Selection Test, 4
For any real numbers sequence $\{x_n\}$ ,suppose that $\{y_n\}$ is a sequence such that:
$y_1=x_1, y_{n+1}=x_{n+1}-(\sum\limits_{i = 1}^{n} {x^2_i})^{ \frac{1}{2}}$ ${(n \ge 1})$ .
Find the smallest positive number $\lambda$ such that for any real numbers sequence $\{x_n\}$ and all positive integers $m$ , have $\frac{1}{m}\sum\limits_{i = 1}^{m} {x^2_i}\le\sum\limits_{i = 1}^{m} {\lambda^{m-i}y^2_i} .$
(High School Affiliated to Nanjing Normal University )
2013 Ukraine Team Selection Test, 5
For positive $x, y$, and $z$ that satisfy the condition $xyz = 1$, prove the inequality
$$\sqrt[3]{\frac{x+y}{2z}}+\sqrt[3]{\frac{y+z}{2x}}+\sqrt[3]{\frac{z+x}{2y}}\le \frac{5(x+y+z)+9}{8}$$
2021 Kosovo National Mathematical Olympiad, 4
Let $M$ be the midpoint of segment $BC$ of $\triangle ABC$. Let $D$ be a point such that $AD=AB$, $AD\perp AB$ and points $C$ and $D$ are on different sides of $AB$. Prove that: $$\sqrt{AB\cdot AC+BC\cdot AM}\geq\frac{\sqrt{2}}{2}CD.$$
2004 China Team Selection Test, 2
Find the largest positive real $ k$, such that for any positive reals $ a,b,c,d$, there is always:
\[ (a\plus{}b\plus{}c) \left[ 3^4(a\plus{}b\plus{}c\plus{}d)^5 \plus{} 2^4(a\plus{}b\plus{}c\plus{}2d)^5 \right] \geq kabcd^3\]
2018 Pan-African Shortlist, A4
Let $a$, $b$, $c$ and $d$ be non-zero pairwise different real numbers such that
$$
\frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} = 4 \text{ and } ac = bd.
$$
Show that
$$
\frac{a}{c} + \frac{b}{d} + \frac{c}{a} + \frac{d}{b} \leq -12
$$
and that $-12$ is the maximum.
2018 IMO Shortlist, A7
Find the maximal value of
\[S = \sqrt[3]{\frac{a}{b+7}} + \sqrt[3]{\frac{b}{c+7}} + \sqrt[3]{\frac{c}{d+7}} + \sqrt[3]{\frac{d}{a+7}},\]
where $a$, $b$, $c$, $d$ are nonnegative real numbers which satisfy $a+b+c+d = 100$.
[i]Proposed by Evan Chen, Taiwan[/i]
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]
IV Soros Olympiad 1997 - 98 (Russia), 11.2
Find all values of the parameter $a$ for which there are exactly $1998$ integers $x$ satisfying the inequality $$x^2 -\pi x +a < 0.$$