Find all possible digits $x, y, z$ such that the number $\overline{13xy45z}$ is divisible by $792.$

Click for solution since $792=8\cdot 9\cdot 11$, we find that $1+3+x+y+4+5+z\equiv 0\pmod 9\iff x+y+z\equiv 5\pmod 9$ $(1)$ $1-3+x-y+4-5+z\equiv 0\pmod{11}\iff x-y+z\equiv 3\pmod{11}$ $(2)$ $400+50+z\equiv 0\pmod 8\iff z\equiv 6\pmod 8$ thus $z=6$ so we get from $(2)$, $x-y\equiv-3\pmod{11}$ case 1: if $x-y=-3$ from $(1)$ we get $2y-3\equiv-1\pmod 9\iff y\equiv 1\pmod 9\iff y=1$ so $x=-2$ which is impossible. case 2: $x-y=8\Longrightarrow 2y+8\equiv 8\pmod 9\iff y\equiv 0\pmod 9\Longrightarrow y=0$ so $x=8$ and we easily see that $792|1380456$

The lines joining the incenter of a triangle to the vertices divide the triangle into three triangles. If one of these triangles is similar to the initial one,determine the angles of the triangle.

If $k, l, m$ are positive integers with $\frac{1}{k}+\frac{1}{l}+\frac{1}{m}<1$, find the maximum possible value of $\frac{1}{k}+\frac{1}{l}+\frac{1}{m}$.

The circumradius $R$ of a triangle with side lengths $a, b, c$ satisfies $R =\frac{a\sqrt{bc}}{b+c}$. Find the angles of the triangle.

Let $a \not = 0, b, c$ be real numbers. If $x_{1}$ is a root of the equation $ax^{2}+bx+c = 0$ and $x_{2}$ a root of $-ax^{2}+bx+c = 0$, show that there is a root $x_{3}$ of $\frac{a}{2}\cdot x^{2}+bx+c = 0$ between $x_{1}$ and $x_{2}$.

Let $U$ be the incenter of a triangle $ABC$ and $O_{1}, O_{2}, O_{3}$ be the circumcenters of the triangles $BCU, CAU, ABU$ , respectively. Prove that the circumcircles of the triangles $ABC$ and $O_{1}O_{2}O_{3}$ have the same center.

If $a, b, c$ are real numbers greater than $1$, prove that for any real number $r$ \[(\log_{a}bc)^{r}+(\log_{b}ca)^{r}+(\log_{c}ab)^{r}\geq 3 \cdot 2^{r}. \]

Click for solution Let $ ln(a) = x$ ,$ ln(b) = y$ and $ ln(c) = z$ ,$ x,y,z > 0$ so the inequality to prove is \[ \sum \left(\frac {x + y}{z}\right)^{r}\geq 3\cdot 2^{r}\] Since $ r>0$ By AM-GM we have \[ \frac {1}{3}\sum \left(\frac {x + y}{z}\right)^{r}\geq \sqrt [3]{\left (\frac {(x + y)(y + z)(x + z)}{xyz}\right )^{r}}\geq 2^{r}\] and because $ r > 0$ the last one is equivalent to \[ (x + y)(y + z)(x + z)\geq 8xyz\]

Show that in any set of eleven integers there are six whose sum is divisible by $6$.

Find all positive integer solutions of the equation $k!l! = k!+l!+m!.$

The incircle of a triangle $ABC$ touches $AC, BC$ , and $AB$ at $M , N$, and $R$, respectively. Let $S$ be a point on the smaller arc $MN$ and $t$ be the tangent to this arc at $S$ . The line $t$ meets $NC$ at $P$ and $MC$ at $Q$. Prove that the lines $AP, BQ, SR, MN$ have a common point.

Find the locus of points inside a trihedral angle such that the sum of their distances from the faces of the trihedral angle has a fixed positive value $a$.

The vertices of a regular $2005$-gon are colored red, white and blue. Whenever two vertices of different colors stand next to each other, we are allowed to recolor them into the third color. (a) Prove that there exists a finite sequence of allowed recolorings after which all the vertices are of the same color. (b) Is that color uniquely determined by the initial coloring?

A sequence $(a_{n})$ is defined by $a_{1}= 1$ and $a_{n}= a_{1}a_{2}...a_{n-1}+1$ for $n \geq 2.$ Find the smallest real number $M$ such that $\sum_{n=1}^{m}\frac{1}{a_{n}}<M\; \forall m\in\mathbb{N}$.

Let $P(x)$ be a monic polynomial of degree $n$ with nonnegative coefficients and the free term equal to $1$. Prove that if all the roots of $P(x)$ are real, then $P(x) \geq (x+1)^{n}$ holds for every $x \geq 0$.

Show that there is a unique positive integer which consists of the digits $2$ and $5$, having $2005$ digits and divisible by $2^{2005}$.

Let $P$ and $Q$ be points on the sides $BC$ and $CD$ of a convex quadrilateral $ABCD$, respectively, such that $\angle{BAP}=\angle{ DAQ}$. Prove that the triangles $ABP$ and $ADQ$ have equal area if and only if the line joining their orthocenters is perpendicular to $AC.$