Found problems: 2529
Let $ ABC$ be a triangle, and let $ P$, $ Q$, $ R$ be three points in the interiors of the sides $ BC$, $ CA$, $ AB$ of this triangle. Prove that the area of at least one of the three triangles $ AQR$, $ BRP$, $ CPQ$ is less than or equal to one quarter of the area of triangle $ ABC$.
[i]Alternative formulation:[/i] Let $ ABC$ be a triangle, and let $ P$, $ Q$, $ R$ be three points on the segments $ BC$, $ CA$, $ AB$, respectively. Prove that
$ \min\left\{\left|AQR\right|,\left|BRP\right|,\left|CPQ\right|\right\}\leq\frac14\cdot\left|ABC\right|$,
where the abbreviation $ \left|P_1P_2P_3\right|$ denotes the (non-directed) area of an arbitrary triangle $ P_1P_2P_3$.
Let $a$, $b$, $c$ be positive real numbers such that $\dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c} = a+b+c$. Prove that:
\[\frac{1}{(2a+b+c)^2}+\frac{1}{(a+2b+c)^2}+\frac{1}{(a+b+2c)^2}\leq \frac{3}{16}.\]
[i]Proposed by Juhan Aru, Estonia[/i]
$n$ is a natural number. for every positive real numbers $x_{1},x_{2},...,x_{n+1}$ such that $x_{1}x_{2}...x_{n+1}=1$ prove that:
$\sqrt[x_{1}]{n}+...+\sqrt[x_{n+1}]{n} \geq n^{\sqrt[n]{x_{1}}}+...+n^{\sqrt[n]{x_{n+1}}}$
Let $ n $ be an integer greater than or equal to $ 2 $. Prove that if the real numbers $ a_1 , a_2 , \cdots , a_n $ satisfy $ a_1 ^2 + a_2 ^2 + \cdots + a_n ^ 2 = n $, then
\[\sum_{1 \le i < j \le n} \frac{1}{n- a_i a_j} \le \frac{n}{2} \]
must hold.
Let $ABC$ be a triangle with circumcircle $\Omega$ and let $G$ be the centroid of triangle $ABC$. Extend $AG, BG$ and $CG$ to meet the circle $\Omega$ again in $A_1, B_1$ and $C_1$. Suppose $\angle BAC = \angle A_1 B_1 C_1, \angle ABC = \angle A_1 C_1 B_1$ and $ \angle ACB = B_1 A_1 C_1$. Prove that $ABC$ and $A_1 B_1 C_1$ are equilateral triangles.
Consider an acute-angled triangle $ABC$. Let $P$ be the foot of the altitude of triangle $ABC$ issuing from the vertex $A$, and let $O$ be the circumcenter of triangle $ABC$. Assume that $\angle C \geq \angle B+30^{\circ}$. Prove that $\angle A+\angle COP < 90^{\circ}$.
Let $x_1, x_2, \ldots, x_n$ be positive reals; for any positive integer $k$, let $S_k=x_1^k+x_2^k+\ldots+x_n^k$.
(a) Given that $S_1<S_2$, show that $S_1, S_2, S_3, \ldots$ is strictly increasing.
(b) Prove that there exists a positive integer $n$ and positive reals $x_1, x_2, \ldots, x_n$, such that $S_1>S_2$ and $S_1, S_2, S_3, \ldots$ is not strictly decreasing.
Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of
$$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$
[i]Israel[/i]
4. Let $a \geq b \geq c \geq d>0$. Show that
\[
\frac{b^3}{a} + \frac{c^3}{b} + \frac{d^3}{c} + \frac{a^3}{d} + 3 \left( ab+bc+cd+da \right) \geq
4 {\left( a^2 + b^2 + c^2 +d^2 \right)}.
\]
Other problems (in Korean) are also available at https://www.facebook.com/KoreanMathOlympiad
Let $a,b,c>0$ such that $a+b+c=3$. Prove that :$$\frac{ab}{ab+a+b}+\frac{bc}{bc+b+c}+\frac{ca}{ca+c+a}+\frac{1}{9}\left(\frac{(a-b)^2}{ab+a+b}+\frac{(b-c)^2}{bc+b+c}+\frac{(c-a)^2}{ca+c+a}\right)\leq1.$$
A polynomial $p(x)$ of degree $n\ge 2$ has exactly $n$ real roots, counted with multiplicity. We know that the coefficient of $x^n$ is $1$, all the roots are less than or equal to $1$, and $p(2)=3^n$. What values can $p(1)$ take?
If $x_{1}, x_{2}, . . . , x_{n}$ are positive numbers, prove the inequality $\frac{x_{1}^{3}}{x_{1}^{2}+x_{1}x_{2}+x_{2}^{2}}+\frac{x_{2}^{3}}{x_{2}^{2}+x_{2}x_{3}+x_{3}^{2}}+...+\frac{x_{n}^{3}}{x_{n}^{2}+x_{n}x_{1}+x_{1}^{2}}\geq\frac{x_{1}+x_{2}+...+x_{n}}{3}$.
For $a,b,c$ positive reals find the minimum value of \[ \frac{a^2+b^2}{c^2+ab}+\frac{b^2+c^2}{a^2+bc}+\frac{c^2+a^2}{b^2+ca}. \]
Let $ n$ be a positive integer such that $ n\geq 3$. Let $ a_1$, $ a_2$, ..., $ a_n$ and $ b_1$, $ b_2$, ..., $ b_n$ be $ 2n$ positive real numbers satisfying the equations
\[ a_1 \plus{} a_2 \plus{} ... \plus{} a_n \equal{} 1, \quad \text{and} \quad b_1^2 \plus{} b_2^2 \plus{} ... \plus{} b_n^2 \equal{} 1.\]
Prove the inequality
\[a_1\left(b_1 \plus{} a_2\right) \plus{} a_2\left(b_2 \plus{} a_3\right) \plus{} ... \plus{} a_{n \minus{} 1}\left(b_{n \minus{} 1} \plus{} a_n\right) \plus{} a_n\left(b_n \plus{} a_1\right) < 1.\]
Let $\alpha , \beta ,\gamma$ be the angles of a triangle $ABC$ of perimeter $ 2p $ and $R$ is the radius of its circumscribed circle.
$(a)$ Prove that
\[\cot^{2}\alpha +\cot^{2}\beta+\cot^{2}\gamma\geq 3\left(9\cdot \frac{R^{2}}{p^{2}} - 1\right).\]
$(b)$ When do we have equality?
prove that for any sequence of nonnegative numbers $(a_n)$, $$\liminf_{n\to\infty} (n^2(4a_n(1-a_{n-1})-1))\leq\frac{1}{4}$$
Let $a$ and $b$ be real numbers greater than 1 such that $ab=100$. The maximum possible value of $a^{(\log_{10}b)^2}$ can be written in the form $10^x$ for some real number $x$. Find $x$.
Across all $x \in \mathbb{R}$, find the maximum value of the expression $$\sin x + \sin 3x + \sin 5x.$$
If $x$, $y$ are positive reals such that $x + y = 2$ show that $x^3y^3(x^3+ y^3) \leq 2$.
Let $a,b,c,d >0$ and $abcd=1$. Prove that:
\[ \frac{1}{(1+a)^2}+\frac{1}{(1+b)^2}+\frac{1}{(1+c)^2}+\frac{1}{(1+d)^2} \geq 1 \]
$x_{1}, x_{2}, \cdots, x_{n}$ are positive numbers such that $\sum_{i=1}^{n}x_{i}= 1$. Prove that \[\left( \sum_{i=1}^{n}\sqrt{x_{i}}\right) \left( \sum_{i=1}^{n}\frac{1}{\sqrt{1+x_{i}}}\right) \leq \frac{n^{2}}{\sqrt{n+1}}\]
Let $a, b>0$. Prove that:$$ (a^3+b^3+a^3b^3)(\frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{a^3b^3} ) +27 \ge 6(a+b+\frac{1}{a} +\frac{1}{b} +\frac{a}{b} +\frac{b}{a}) $$
Let $a,b,c$ be real numbers with $0 < a< 1$, $0 < b < 1$, $0 < c < 1$, and $a+b + c = 2$.
Prove that $\dfrac{a}{1-a} \cdot \dfrac{b}{1-b} \cdot \dfrac{c}{1-c} \geq 8$.
Let $a_1$, $a_2$, $\cdots$, $a_n$, $b_1$, $b_2$, $\cdots$, $b_n$ be positive real numbers such that $a_1 + a_2 + \cdots + a_n = b_1 + b_2 + \cdots + b_n$. Show that
\[ \frac{a_1^2}{a_1 + b_1} + \frac{a_2^2}{a_2 + b_2} + \cdots + \frac{a_n^2}{a_n + b_n} \geq \frac{a_1 + a_2 + \cdots + a_n}{2} \]
$x_{1}, x_{2}, \cdots, x_{n}$ are positive numbers such that $\sum_{i=1}^{n}x_{i}= 1$. Prove that \[\left( \sum_{i=1}^{n}\sqrt{x_{i}}\right) \left( \sum_{i=1}^{n}\frac{1}{\sqrt{1+x_{i}}}\right) \leq \frac{n^{2}}{\sqrt{n+1}}\]