Found problems: 85335
The circle $\Omega$ with center $I_A$, is the $A$-excircle of triangle $ABC$. Which is tangent to $AB,AC$ at $F,E$ respectivly. Point $D$ is the reflection of $A$ through $I_AB$. Lines $DI_A$ and $EF$ meet at $K$. Prove that ,circumcenter of $DKE$ , midpoint of $BC$ and $I_A$ are collinear.
The sum of two natural numbers is $17,402.$ One of the two numbers is divisible by $10.$ If the units digit of that number is erased, the other number is obtained. What is the difference of these two numbers?
$\textbf{(A) }10,272 \qquad \textbf{(B) }11,700 \qquad \textbf{(C) }13,362 \qquad \textbf{(D) }14,238 \qquad \textbf{(E) }15,426$
Let $f:S\to\mathbb{R}$ be the function from the set of all right triangles into the set of real numbers, defined by $f(\Delta ABC)=\frac{h}{r}$, where $h$ is the height with respect to the hypotenuse and $r$ is the inscribed circle's radius. Find the image, $Im(f)$, of the function.
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 \[ \sum^n_{i=1} \frac{\cos(\beta_i)}{\sin(\alpha_i)} \leq \sum^n_{i=1} \cot(\alpha_i). \]
Let $x_1, x_2, \dots, x_n$ be different real numbers. Prove that
\[\sum_{1 \leqslant i \leqslant n} \prod_{j \neq i} \frac{1-x_{i} x_{j}}{x_{i}-x_{j}}=\left\{\begin{array}{ll}
0, & \text { if } n \text { is even; } \\
1, & \text { if } n \text { is odd. }
\end{array}\right.\]
Evaluate $ \displaystyle I \equal{} \int_0^{\frac\pi4}\left(\sin^62x \plus{} \cos^62x\right)\cdot \ln(1 \plus{} \tan x)\text{d}x$.
Colour a $20000\times 20000$ square grid using 2000 different colours with 1 colour in each square. Two squares are neighbours if they share a vertex. A path is a sequence of squares so that 2 successive squares are neighbours. Mark $k$ of the squares. For each unmarked square $x$, there is exactly 1 marked square $y$ of the same colour so that $x$ and $y$ are connected by a path of squares of the same colour. For any 2 marked squares of the same colour, any path connecting them must pass through squares of all the colours. Find the maximum value of $k$.
Suppose that $ a_1$, $ a_2$, ..., $ a_n$ are positive real numbers such that $ a_1 \leq a_2 \leq \dots \leq a_n$. Let
\[ {{a_1 \plus{} a_2 \plus{} \dots \plus{} a_n} \over n} \equal{} m; \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {{a_1^2 \plus{} a_2^2 \plus{} \dots \plus{} a_n^2} \over n} \equal{} 1.
\]
Suppose that, for some $ i$, we know $ a_i \leq m$. Prove that:
\[ n \minus{} i \geq n \left(m \minus{} a_i\right)^2
\]
A regular hexagon is inscribed in a circle. The ratio of the length of a side of the hexagon to the length of the shorter of the arcs intercepted by the side, is:
$ \textbf{(A)}\ 1: 1 \qquad \textbf{(B)}\ 1: 6 \qquad \textbf{(C)}\ 1: \pi \qquad \textbf{(D)}\ 3: \pi \qquad \textbf{(E)}\ 6: \pi$
Let $v_1,v_2,\ldots,v_n$ vectors in $\mathbb{R}^2$ such that $|v_i|\leq 1$ for $1 \leq i \leq n$ and $\sum_{i=1}^n v_i=0$. Prove that there exists a permutation $\sigma$ of $(1,2,\ldots,n)$ such that $\left|\sum_{j=1}^k v_{\sigma(j)}\right| \leq\sqrt 5$ for every $k$, $1\leq k \leq n$.
[i]Remark[/i]: If $v = (x,y)\in \mathbb{R}^2$, $|v| = \sqrt{x^2 + y^2}$.
$a,b,m,k\in \mathbb{Z}$, $a,b,m>2,k>1$, for which $k^n a+b$ is an $m$-th power of a natural number for $\forall n\in \mathbb{N}$. Prove that $b$ is an $m$-th power of a non-negative integer.
Let $A_1A_2 . . . A_{12}$ be a regular dodecagon. Equilateral triangles $\vartriangle A_1A_2B_1$, $\vartriangle A_2A_3B_2$, $. . . $, and $\vartriangle A_{12}A_1B_{12}$ are drawn such that points $B_1$, $B_2$,$ . . . $, and B_{12} lie outside dodecagon $A_1A_2 . . . A_{12}$. Then, equilateral triangles $\vartriangle A_1A_2C_1$, $\vartriangle A_2A_3C_2$, $. . .$ , and $\vartriangle A_{12}A_1C_{12}$ are drawn such that points $C_1$, $C_2$, $. . .$ , and $C_{12}$ lie inside dodecagon $A_1A_2 . . . A_{12}$. Compute the ratio of the area of dodecagon $B_1B_2 . . . B_{12}$ to the area of dodecagon $C_1C_2 . . . C_{12}$.
Two strictly ascending sequences of positive numbers are given. In each sequence, each number starting from the third one is the sum of two preceding ones. It is known that each of the sequences contains at least one number not present in the other sequence. What is the maximum quantity of numbers common for these two sequences?
Boris Frenkin
Let $f : \mathbb{Z} \to \mathbb{R}$ be a function satisfying $f(x) + f(y) + f(z) \ge 0$ for all integers $x, y, z$ with $x + y + z = 0$. Prove that
\[ f(-2017) + f(-2016) + \cdots + f(2016) + f(2017) \ge 0. \]
Consider the following table where initially all squares contain zeros:
$ \begin{tabular}{ | l | c | r| }
\hline
0 & 0 & 0 \\ \hline
0 & 0 & 0 \\ \hline
0 & 0 & 0 \\ \hline
\end{tabular}
$
To change the table, the following operation is allowed: a $2 \times 2$ square formed by adjacent squares is chosen, and a unit is added to all its numbers. Complete the following table, knowing that it was obtained by a sequence of permitted operations
$ \begin{tabular}{ | l | c | r| }
\hline
14 & & \\ \hline
19 & 36 & \\ \hline
& 16 & \\ \hline
\end{tabular}
$
Let $ABC$ be a triangle such that $|AB|=7$, $|BC|=8$, $|AC|=6$. Let $D$ be the midpoint of side $[BC]$. If the circle through $A$, $B$ and $D$ cuts $AC$ at $A$ and $E$, what is $|AE|$?
$
\textbf{(A)}\ \dfrac 23
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ \dfrac 32
\qquad\textbf{(D)}\ 2
\qquad\textbf{(E)}\ 3
$
Let $X_1, X_2, \ldots, X_{100}$ be a sequence of mutually distinct nonempty subsets of a set $S$. Any two sets $X_i$ and $X_{i+1}$ are disjoint and their union is not the whole set $S$, that is, $X_i\cap X_{i+1}=\emptyset$ and $X_i\cup X_{i+1}\neq S$, for all $i\in\{1, \ldots, 99\}$. Find the smallest possible number of elements in $S$.
Let $a$, $b$, $c$ be the distinct roots of the polynomial $P(x) = x^3 - 10x^2 + x - 2015$.
The cubic polynomial $Q(x)$ is monic and has distinct roots $bc-a^2$, $ca-b^2$, $ab-c^2$.
What is the sum of the coefficients of $Q$?
[i]Proposed by Evan Chen[/i]
A point $D$ is placed on the side $ BC$ of the triangle $ABC$. Circles are inscribed in the triangles $ABD$ and $ACD$, their common exterior tangent line (other than $BC$) intersects $AD$ at the point $K$. Prove that the length of $AK$ does not depend on the position of $D$. (An exterior tangent of two circles is one which is tangent to both circles but does not pass between them.)
(I Sharygin)
The polynomials $P$, $Q$, $R$ with real coefficients, one of which is degree $2$ and two of degree $3$, satisfy the equality $P^2+Q^2=R^2$. Prove that one of the polynomials of degree $3$ has three real roots.
A $9\times9$ chess board has two squares from opposite corners and its central square removed. Is it possible to cover the remaining squares using dominoes, where each domino covers two adjacent squares? Justify your answer.
What is the remainder of the number $1 \binom{2008}{0 }+2\binom{2008}{1}+ ...+2009\binom{2008}{2008}$ when divided by $2008$?
Find all pairs of positive integers $(p; q) $such that both the equations $x^2- px + q = 0 $ and $ x^2 -qx + p = 0 $ have integral solutions.
Let $BE$ and $CF$ be the altitudes of acute triangle $ABC$. Let $H$ be the orthocenter of $ABC$ and $M$ be the midpoint of side $BC$. The points of intersection of the midperpendicular line to $BC$ with segments $BE$ and $CF$ are denoted by $K$ and $L$ respectively. The point $Q$ is the orthocenter of triangle $KLH$. Prove that $Q$ belongs to the median $AM$.
(Bohdan Zheliabovskyi)
If $ \theta$ is an acute angle and $ \sin \frac12 \theta \equal{} \sqrt{\frac{x\minus{}1}{2x}}$, then $ \tan \theta$ equals
$ \textbf{(A)}\ x \qquad
\textbf{(B)}\ \frac1{x} \qquad
\textbf{(C)}\ \frac{\sqrt{x\minus{}1}}{x\plus{}1} \qquad
\textbf{(D)}\ \frac{\sqrt{x^2\minus{}1}}{x} \qquad
\textbf{(E)}\ \sqrt{x^2\minus{}1}$