The incircle of triangle $ABC$ touches $BC$ at $D$ and $AB$ at $F$, intersects the line $AD$ again at $H$ and the line $CF$ again at $K$. Prove that $\frac{FD\times HK}{FH\times DK}=3$

In a game, a participant chooses a nine-digit positive integer $\overline{ABCDEFGHI}$ with distinct non-zero digits. The score of the participant is $A^{B^{C^{D^{E^{F^{G^{H^{I}}}}}}}}$. Which nine-digit number should be chosen in order to maximise the score?

Let $t\ge\frac{1}{2}$ be a real number and $n$ a positive integer. Prove that \[t^{2n}\ge (t-1)^{2n}+(2t-1)^n\]

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a continuous function such that the limit $$g(x)=\lim_{h\rightarrow 0}{\frac{f(x+h)-2f(x)+f(x-h)}{h^2}}$$ exists for all real $x$. Prove that $g(x)$ is constant if and only if $f(x)$ is a polynomial function whose degree is at most $2$.

Given a positive integer $n (n>2004)$, we put 1, 2, 3, …,$n^2$ into squares of an $n\times n$ chessboard with one number in a square. A square is called a “good square” if the square satisfies following conditions: 1) There are at least 2004 squares that are in the same row with the square such that any number within these 2004 squares is less than the number within the square. 2) There are at least 2004 squares that are in the same column with the square such that any number within these 2004 squares is less than the number within the square. Find the maximum value of the number of the “good square”.

$f(x) = \frac{12x+9}{19x+86}, \,\, x \ne -\frac{86}{19}$ Prove that $\exists ! \,\,\, {x_o \in R} \,\,\, \forall h_1,h_2 \in R [f(x_0+h_1)f(x_0-h_1)=f(x_0+h_2)f(x_0-h_2)]$, and calculate $x_0$.

In acute angled $\triangle ABC$, $AB > AC$, points $E, F$ lie on $AC, AB$ respectively, satisfying $BF+CE = BC$. Let $I_B, I_C$ be the excenters of $\triangle ABC$ opposite $B, C$ respectively, $EI_C, FI_B$ intersect at $T$, and let $K$ be the midpoint of arc $BAC$. Let $KT$ intersect the circumcircle of $\triangle ABC$ at $K,P$. Show $T,F,P,E$ concyclic.

The entries in a $ 3\times3$ array include all the digits from 1 through 9, arranged so that the entries in every row and column are in increasing order. How many such arrays are there? $ \textbf{(A)}\ 18\qquad\textbf{(B)}\ 24\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 42\qquad\textbf{(E)}\ 60$

The sequence $(\chi_n) _{n=1}^{\infty}$ is defined as follows \begin{align*} \chi_{n+1}=\chi_n + \chi _{\lceil \frac{n}{2} \rceil} ~, \chi_1 =1 \end{align*} Prove that none of the terms of this sequence are divisible by $4$

Assume that a face of a convex polyhedron $ P$ has a common edge with every other face. Show that there exists a simple closed polygon that consists of edges of $ P$ and passes through all vertices. [i]L .Lovasz[/i]

(a) If $\displaystyle ABC$ is a triangle and $\displaystyle M$ is a point from its plane, then prove that \[ \displaystyle AM \sin A \leq BM \sin B + CM \sin C . \] (b) Let $\displaystyle A_1,B_1,C_1$ be points on the sides $\displaystyle (BC),(CA),(AB)$ of the triangle $\displaystyle ABC$, such that the angles of $\triangle A_1 B_1 C_1$ are $\widehat{A_1} = \alpha, \widehat{B_1} = \beta, \widehat{C_1} = \gamma$. Prove that \[ \displaystyle \sum A A_1 \sin \alpha \leq \sum BC \sin \alpha . \] [i]Dan Ştefan Marinescu, Viorel Cornea[/i]

Let $O$ be the circumcentre of the acute triangle $ABC$ and let lines $AO$ and $BC$ intersect at point $K$. On sides $AB$ and $AC$, points $L$ and $M$ are chosen such that $|KL|= |KB|$ and $|KM| = |KC|$. Prove that segments $LM$ and $BC$ are parallel.

A right circular cylinder is inscribed in a sphere of radius $\sqrt3$. What is the height of the cylinder when its volume is maximal?

$(SWE 3)$ Find the natural number $n$ with the following properties: $(1)$ Let $S = \{P_1, P_2, \cdots\}$ be an arbitrary finite set of points in the plane, and $r_j$ the distance from $P_j$ to the origin $O.$ We assign to each $P_j$ the closed disk $D_j$ with center $P_j$ and radius $r_j$. Then some $n$ of these disks contain all points of $S.$ $(2)$ $n$ is the smallest integer with the above property.

The tangent at $C$ to $\Omega$, the circumcircle of scalene triangle $ABC$ intersects $AB$ at $D$. Through point $D$, a line is drawn that intersects segments $AC$ and $BC$ at $K$ and $L$ respectively. On the segment $AB$ points $M$ and $N$ are marked such that $AC \parallel NL$ and $BC \parallel KM$. Lines $NL$ and $KM$ intersect at point $P$ lying inside the triangle $ABC$. Let $\omega$ be the circumcircle of $MNP$. Suppose $CP$ intersects $\omega$ again at $Q$. Show that $DQ$ is tangent to $\omega$.

Given positive integers $n,k$, $n \ge 2$. Find the minimum constant $c$ satisfies the following assertion: For any positive integer $m$ and a $kn$-regular graph $G$ with $m$ vertices, one could color the vertices of $G$ with $n$ different colors, such that the number of monochrome edges is at most $cm$.

Let $m$, $n$, and $x$ be positive integers. Prove that \[ \sum_{i = 1}^n \min\left(\left\lfloor \frac{x}{i} \right\rfloor, m \right) = \sum_{i = 1}^m \min\left(\left\lfloor \frac{x}{i} \right\rfloor, n \right). \] [i]Proposed by Yang Liu[/i]

The vertex $B$ of the triangle $ABC$ lies in the plane $P$. The plane of the triangle meets the plane in a line $L$. The angle between $L$ and $AB$ is a, and the angle between $L$ and $BC$ is $b$. The angle between the two planes is $c$. Angle $ABC$ is $90^o$. Show that $\sin^2c = \sin^2a + \sin^2b$. [img]https://cdn.artofproblemsolving.com/attachments/9/e/c0608e5408fd27a5f907a3488cce7dc2af6953.png[/img]

On each side of a parallelogram an arbitrary point is chosen. Each pair of chosen points on neighbouring sides (i.e. sides with a common vertex) are connected by a line segment. Prove that the centres of the circumscribed circles of the four triangles so created are themselves vertices of a parallelogram. (ED Kulanin)

Let $\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f:\mathbb{R}^+ \to \mathbb{R}^+$ such that \[x+f(yf(x)+1)=xf(x+y)+yf(yf(x))\] for all $x,y>0.$

A rectangle is partitioned into 5 regions as shown. Each region is to be painted a solid color - red, orange, yellow, blue, or green - so that regions that touch are painted different colors, and colors can be used more than once. How many different colorings are possible? [asy] size(5.5cm); draw((0,0)--(0,2)--(2,2)--(2,0)--cycle); draw((2,0)--(8,0)--(8,2)--(2,2)--cycle); draw((8,0)--(12,0)--(12,2)--(8,2)--cycle); draw((0,2)--(6,2)--(6,4)--(0,4)--cycle); draw((6,2)--(12,2)--(12,4)--(6,4)--cycle); [/asy] $\textbf{(A) }120\qquad\textbf{(B) }270\qquad\textbf{(C) }360\qquad\textbf{(D) }540\qquad\textbf{(E) }720$

Consider $64$ distinct natural numbers, at most equal to $2012$. Show that it is possible to choose four of them, denoted as $a,b,c,d$ such that $ a+b-c-d$ to be a multiple of $2013$

[b]p1.[/b] Find the smallest positive multiple of $9$ whose digits are all even. [b]p2.[/b] Anika writes down a positive real number $x$ in decimal form. When Nat erases everything to the left of the decimal point, the remaining value is one-fifth of x. Find the sum of all possible values of $x$. [b]p3.[/b] A star-like shape is formed by joining up the midpoints and vertices of a unit square, as shown in the diagram below. Compute the area of this shape. [img]https://cdn.artofproblemsolving.com/attachments/4/8/923b1bf26f6e9872b596e8c81ad1872137f362.png[/img] [b]p4.[/b] Benny and Daria are running a $200$ meter foot race, each at a different constant speed. When Daria finishes the race, she is $14$ meters ahead of Benny. The next time they race, Daria starts 14 meters behind Benny, who starts at the starting line. Both runners run at the same constant speed as in the first race. When Daria reaches the finish line, compute, in centimeters, how far she is ahead of Benny. [b]p5.[/b] In one semester, Ronald takes ten biology quizzes, earning a distinct integer score from $91$ to $100$ on each quiz. He notices that after the first three quizzes, the average of his three most recent scores always increased. Compute the number of ways Ronald could have earned the ten quiz scores. [b]p6.[/b] Ant and Ben are playing a game with stones. They begin with $Z$ stones on the ground. Ant and Ben take turns removing a prime number of stones. Ant moves first. The player who is first unable to make a valid move loses. Find the sum of all positive integers $Z \le 30$ such that Ben can guarantee a win with perfect play. [b]p7.[/b] Let $P$ be a point in a regular octagon as shown in the diagram below. The areas of three triangles are shown. Find the area of the octagon. [img]https://cdn.artofproblemsolving.com/attachments/0/9/6fde77eeafd04614046292175e4b1411158e85.png[/img] [b]p8.[/b] Find the number of ordered triples $(a, b, c)$ of nonnegative integers with $a \le b \le c$ for which $5a + 4b + 6c = 1200$. [b]p9.[/b] Define $$f(x) = \begin{cases} 2x \,\,\,\, ,\,\,\,\, 0 \le x < \frac12 \\ 2 - 2x \,\,\,\, , \,\,\,\, \frac12 \le x \le 1 \end{cases}$$ Michael picks a real number $0 \le x \le 1$. Michael applies $f$ repeatedly to $x$ until he reaches $x$ again. Find the number of real numbers $x$ for which Michael applies $f$ exactly $12$ times. [b]p10.[/b] In $\vartriangle ABC$, let point $H$ be the intersection of its altitudes and let $M$ be the midpoint of side $BC$. Given that $BC = 4$, $MA = 3$, and $\angle HMA = 60^o$, find the circumradius of $\vartriangle ABC$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Show that the edges of a connected simple (no loops and no multiple edges) finite graph can be oriented so that the number of edges leaving each vertex is even if and only if the total number of edges is even

On two sides of an angle, points $A, B$ are chosen. The midpoint $M$ of the segment $AB$ belongs to two lines such that one of them meets the sides of the angle at points $A_1, B_1$, and the other at points $A_2, B_2$. The lines $A_1B_2$ and $A_2B_1$ meet $AB$ at points $P$ and $Q$. Prove that $M$ is the midpoint of $PQ$.