Found problems: 6530
1988 IMO Longlists, 55
Suppose $\alpha_i > 0, \beta_i > 0$ for $1 \leq i \leq n, n > 1$ and that \[ \sum^n_{i=1} \alpha_i = \sum^n_{i=1} \beta_i = \pi. \] Prove that \[ \sum^n_{i=1} \frac{\cos(\beta_i)}{\sin(\alpha_i)} \leq \sum^n_{i=1} \cot(\alpha_i). \]
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)
2019 Jozsef Wildt International Math Competition, W. 4
If $x, y, z, t > 1$ then: $$\left(\log _{zxt}x\right)^2+\left(\log _{xyt}y\right)^2+\left(\log _{xyz}z\right)^2+\left(\log _{yzt}t\right)^2>\frac{1}{4}$$
2010 Romania National Olympiad, 2
Consider $v,w$ two distinct non-zero complex numbers. Prove that
\[|zw+\bar{w}|\le |zv+\bar{v}|,\]
for any $z\in\mathbb{C},|z|=1$, if and only if there exists $k\in [-1,1]$ such that $w=kv$.
[i]Dan Marinescu[/i]
MIPT Undergraduate Contest 2019, 1.5 & 2.5
Prove the inequality
$$\sum _{k = 1} ^n (x_k - x_{k-1})^2 \geq 4 \sin ^2 \frac{\pi}{2n} \cdot \sum ^n _{k = 0} x_k ^2$$
for any sequence of real numbers $x_0, x_1, ..., x_n$ for which $x_0 = x_n = 0.$
2006 Taiwan National Olympiad, 1
Positive reals $a,b,c$ satisfy $abc=1$. Prove that
$\displaystyle 1+ \frac{3}{a+b+c} \ge \frac{6}{ab+bc+ca}$.
1997 Junior Balkan MO, 1
Show that given any 9 points inside a square of side 1 we can always find 3 which form a triangle with area less than $\frac 18$.
[i]Bulgaria[/i]
1977 AMC 12/AHSME, 5
The set of all points $P$ such that the sum of the (undirected) distances from $P$ to two fixed points $A$ and $B$ equals the distance between $A$ and $B$ is
$\textbf{(A) }\text{the line segment from }A\text{ to }B\qquad$
$\textbf{(B) }\text{the line passing through }A\text{ and }B\qquad$
$\textbf{(C) }\text{the perpendicular bisector of the line segment from }A\text{ to }B\qquad$
$\textbf{(D) }\text{an elllipse having positive area}\qquad$
$\textbf{(E) }\text{a parabola}$
2009 Germany Team Selection Test, 3
Let $ a$, $ b$, $ c$, $ d$ be positive real numbers such that $ abcd \equal{} 1$ and $ a \plus{} b \plus{} c \plus{} d > \dfrac{a}{b} \plus{} \dfrac{b}{c} \plus{} \dfrac{c}{d} \plus{} \dfrac{d}{a}$. Prove that
\[ a \plus{} b \plus{} c \plus{} d < \dfrac{b}{a} \plus{} \dfrac{c}{b} \plus{} \dfrac{d}{c} \plus{} \dfrac{a}{d}\]
[i]Proposed by Pavel Novotný, Slovakia[/i]
2004 Junior Balkan Team Selection Tests - Romania, 2
Let $ABC$ be a triangle with side lengths $a,b,c$, such that $a$ is the longest side. Prove that $\angle BAC = 90^\circ$ if and only if
\[ (\sqrt { a+b } + \sqrt { a-b} )(\sqrt {a+c } + \sqrt { a-c } ) = (a+b+c) \sqrt 2. \]
2008 All-Russian Olympiad, 5
The numbers from $ 51$ to $ 150$ are arranged in a $ 10\times 10$ array. Can this be done in such a way that, for any two horizontally or vertically adjacent numbers $ a$ and $ b$, at least one of the equations $ x^2 \minus{} ax \plus{} b \equal{} 0$ and $ x^2 \minus{} bx \plus{} a \equal{} 0$ has two integral roots?
2013 Hanoi Open Mathematics Competitions, 12
If $f(x) = ax^2 + bx + c$ satisfies the condition $|f(x)| < 1; \forall x \in [-1, 1]$,
prove that the equation $f(x) = 2x^2 - 1$ has two real roots.
2012 JBMO TST - Turkey, 1
Let $a, b, c$ be the side-lengths of a triangle, $r$ be the inradius and $r_a, r_b, r_c$ be the corresponding exradius. Show that
\[ \frac{a+b+c}{\sqrt{a^2+b^2+c^2}} \leq 2 \cdot \frac{\sqrt{{r_a}^2+{r_b}^2+{r_c}^2}}{r_a+r_b+r_c-3r} \]
2013 Greece Team Selection Test, 3
Find the largest possible value of $M$ for which $\frac{x}{1+\frac{yz}{x}}+\frac{y}{1+\frac{zx}{y}}+\frac{z}{1+\frac{xy}{z}}\geq M$ for all $x,y,z>0$ with $xy+yz+zx=1$
2006 AMC 12/AHSME, 10
In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is 15. What is the greatest possible perimeter of the triangle?
$ \textbf{(A) } 43 \qquad \textbf{(B) } 44 \qquad \textbf{(C) } 45 \qquad \textbf{(D) } 46 \qquad \textbf{(E) } 47$
2000 Moldova National Olympiad, Problem 2
Show that if real numbers $x<1<y$ satisfy the inequality
$$2\log x+\log(1-x)\ge3\log y+\log(y-1),$$then $x^3+y^3<2$.
1983 IMO Shortlist, 6
Suppose that ${x_1, x_2, \dots , x_n}$ are positive integers for which $x_1 + x_2 + \cdots+ x_n = 2(n + 1)$. Show that there exists an integer $r$ with $0 \leq r \leq n - 1$ for which the following $n - 1$ inequalities hold:
\[x_{r+1} + \cdots + x_{r+i} \leq 2i+ 1, \qquad \qquad \forall i, 1 \leq i \leq n - r; \]
\[x_{r+1} + \cdots + x_n + x_1 + \cdots+ x_i \leq 2(n - r + i) + 1, \qquad \qquad \forall i, 1 \leq i \leq r - 1.\]
Prove that if all the inequalities are strict, then $r$ is unique and that otherwise there are exactly two such $r.$
2008 India Regional Mathematical Olympiad, 6
Find the number of all integer-sided [i]isosceles obtuse-angled[/i] triangles with perimeter $ 2008$.
[16 points out of 100 for the 6 problems]
2003 China Team Selection Test, 1
$x$, $y$ and $z$ are positive reals such that $x+y+z=xyz$. Find the minimum value of:
\[ x^7(yz-1)+y^7(zx-1)+z^7(xy-1) \]
2014 Benelux, 4
Let $ABCD$ be a square. Consider a variable point $P$ inside the square for which $\angle BAP \ge 60^\circ.$ Let $Q$ be the intersection of the line $AD$ and the perpendicular to $BP$ in $P$. Let $R$ be the intersection of the line $BQ$ and the perpendicular to $BP$ from $C$.
[list]
[*] [b](a)[/b] Prove that $|BP|\ge |BR|$
[*] [b](b)[/b] For which point(s) $P$ does the inequality in [b](a)[/b] become an equality?[/list]
VI Soros Olympiad 1999 - 2000 (Russia), 9.10
Let $x, y, z$ be real numbers from interval $(0, 1)$. Prove that
$$\frac{1}{x(1-y)}+\frac{1}{y(1-x)}+\frac{1}{z(1-x)}\ge \frac{3}{xyz+(1-x)(1-y)(1-z)}$$
2009 Kyiv Mathematical Festival, 6
Let $\{a_1,...,a_n\}\subset \{-1,1\}$ and $a>0$ . Denote by $X$ and $Y$ the number of collections $\{\varepsilon_1,...,\varepsilon_n\}\subset \{-1,1\}$, such that $$max_{1\le k\le n}(\varepsilon_1a_1+...+\varepsilon_ka_k) >\alpha$$ and $$\varepsilon_1a_1+...+\varepsilon_na_n>a$$ respectively. Prove that $X\le 2Y$.
2006 Germany Team Selection Test, 2
In an acute triangle $ABC$, let $D$, $E$, $F$ be the feet of the perpendiculars from the points $A$, $B$, $C$ to the lines $BC$, $CA$, $AB$, respectively, and let $P$, $Q$, $R$ be the feet of the perpendiculars from the points $A$, $B$, $C$ to the lines $EF$, $FD$, $DE$, respectively.
Prove that $p\left(ABC\right)p\left(PQR\right) \ge \left(p\left(DEF\right)\right)^{2}$, where $p\left(T\right)$ denotes the perimeter of triangle $T$ .
[i]Proposed by Hojoo Lee, Korea[/i]
2008 Polish MO Finals, 1
In each cell of a matrix $ n\times n$ a number from a set $ \{1,2,\ldots,n^2\}$ is written --- in the first row numbers $ 1,2,\ldots,n$, in the second $ n\plus{}1,n\plus{}2,\ldots,2n$ and so on. Exactly $ n$ of them have been chosen, no two from the same row or the same column. Let us denote by $ a_i$ a number chosen from row number $ i$. Show that:
\[ \frac{1^2}{a_1}\plus{}\frac{2^2}{a_2}\plus{}\ldots \plus{}\frac{n^2}{a_n}\geq \frac{n\plus{}2}{2}\minus{}\frac{1}{n^2\plus{}1}\]
1992 China Team Selection Test, 3
For any $n,T \geq 2, n, T \in \mathbb{N}$, find all $a \in \mathbb{N}$ such that $\forall a_i > 0, i = 1, 2, \ldots, n$, we have
\[\sum^n_{k=1} \frac{a \cdot k + \frac{a^2}{4}}{S_k} < T^2 \cdot \sum^n_{k=1} \frac{1}{a_k},\] where $S_k = \sum^k_{i=1} a_i.$