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NEWS | ИНФОРМАЦИИ
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ARTICLES | СТАТИИ
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USEFUL INFORMATION | КОРИСНИ ИНФОРМАЦИИ
SELECTED PROBLEMS | ОДБРАНИ ЗАДАЧИ
- Find all triples of numbers $a, b, c$ satisfying $$\left(a^{2}+1\right)\left(b^{2}+2\right)\left(c^{2}+8\right)=32 a b c.$$
- Find the least value of the expression $$\frac{a^{8}}{\left(a^{2}+b^{2}\right)^{2}}+\frac{b^{8}}{\left(b^{2}+c^{2}\right)^{2}}+\frac{c^{8}}{\left(c^{2}+a^{2}\right)^{2}}$$ where $a, b, c$ are positive numbers satisfying the condition $a b+b c+c a=1$.
- Let $A B C$ be a triangle, right at $A$. Construct a square $E F G D$ so that $E$, $F$ lie on the side $B C$ and $G$, $D$ lie respectively on the sides $A C$, $A B$. Let $R_{1}$, $R_{2}$, $R_{3}$ be respectively the inradii of the triangles $B D E$, $C G F$, $A D G$. Prove that the area of $E F G D$ assumes its greatest value when and only when $$R_{1}^{2}+R_{2}^{2}=R_{3}^{2}.$$
- A chord $D E$ of the circumcircle of triangle $A B C$ cuts the incircle of $A B C$ at $M$ and $N$. Prove that $D E \geq 2 M N$.
- Each domino consists of two squares having a common side, on its first and second squares are marked $x$ and respectively $y$ dots. Thid momino is denoted by $(x, y)$ or $(y, x) .$ For every non ordered pair of number $x, y$, there's only one domino $(x, y)$. Divide the set of dominoes with $1 \leq x \leq 5,1 \leq y \leq 5$ into three groups, each group is presented as a closed circuit where the numbers $a, b, c, d$, $a \mid b] b \mid c$ are not necessarily distinct. $a|e| e \mid d\rfloor d$ How many such dividings are there?
- Prove that for every $x \in\left[0 ; \frac{\pi}{2}\right]$, holds $$\sin x \leq \frac{4}{\pi} x-\frac{4}{\pi^{2}} x^{2}.$$
- Prove that for every triangle $A B C$, hold $$\sqrt{3} \leq \frac{\cos (A / 2)}{1+\sin (A / 2)}+\frac{\cos (B / 2)}{1+\sin (B / 2)}+\frac{\cos (C / 2)}{1+\sin (C / 2)}<2$$
- Le $A B C D$ be a convex a quadrilateral, the diagonals $A C$ and $B D$ of which are orthogonal. The lines $B C$ and $A D$ intersect at $I$, the lines $A B$ and $C D$ intersect at $J$. Prove that the quadrilateral $B D I J$ is inscribable when and only when $A B \cdot C D=A D \cdot B C$.
- The tetrahedron $A B C D$ has its four altitudes concurrent at a point $H$ inside the tetrahedron. Let $M$, $N$, $P$ be respectively the midpoints of $B C$, $C D$, $D B$. Let $R$ and $R_{1}$ be respectively the circumradii of the tetrahedra $A B C D$ and $H M N P$. Prove that $R=2 R_{1}$.
- Find the the first three digits from the left side of the number $$1^{1}+2^{2}+3^{3}+\ldots+999^{999}+1000^{1000} .$$
- Prove that if the numbers $a, b, c, x, y, z$ satisfy the condition $$\frac{b z+c y}{x(-a x+b y+c z)}=\frac{c x+a z}{y(a x-b y+c z)}=\frac{a y+b x}{z(a x+b y-c z)}$$ then $$\frac{x}{a\left(b^{2}+c^{2}-a^{2}\right)}=\frac{y}{b\left(a^{2}+c^{2}-b^{2}\right)}=\frac{z}{c\left(a^{2}+b^{2}-c^{2}\right)}$$
- Given an integer $n>1$ such that $3^{n}-1$ is divisible by $n$, show that $n$ is even.
PROBLEM OF THE WEEK | ЗАДАЧА НА НЕДЕЛАТА
- PW 208. Let I be the incenter of triangle ABC and let $m_a, m_b, m_c$ be the measures of the medians of ABC issued respectively from A, B, C. Prove that
$\frac{\overline{IA}^2}{m_a^2}+\frac{\overline{IB}^2}{m_b^2}+\frac{\overline{IC}^2}{m_c^2} ≤ 43$.
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