Let $\triangle{ABC}$ be a scalene triangle with the longest side $AC$. (A ${\textit{scalene triangle}}$ has sides of different lengths.) Let $P$ and $Q$ be the points on the side $AC$ such that $AP=AB$ and $CQ=CB$. Thus we have a new triangle $\triangle{BPQ}$ inside $\triangle{ABC}$. Let $k_1$ be the circle circumscribed around the triangle $\triangle{BPQ}$ (that is, the circle passing through the vertices $B,P,$ and $Q$ of the triangle $\triangle{BPQ}$); and let $k_2$ be the circle inscribed in triangle $\triangle{ABC}$ (that is, the circle inside triangle $\triangle{ABC}$ that is tangent to the three sides $AB,BC$, and $CA$). Prove that the two circles $k_1$ and $k_2$ are concentric, that is, they have the same center.

Among 16 coins there are 8 heavy coins with weight of 11 g, and 8 light coins with weight of 10 g, but it's unknown what weight of any coin is. One of the coins is anniversary. How to know, is anniversary coin heavy or light, via three weighings on scales with two cups and without any weight?

Let \( f: \mathbb{R} \to \mathbb{R} \) be a continuous function. A chord is defined as a segment of integer length, parallel to the x-axis, whose endpoints lie on the graph of \( f \). It is known that the graph of \( f \) contains exactly \( N \) chords, one of which has length 2025. Find the minimum possible value of \( N \).

The number in each box below is the product of the numbers in the two boxes that touch it in the row above. For example, $30 = 6\times5$. What is the missing number in the top row? [asy] unitsize(0.8cm); draw((-1,0)--(1,0)--(1,-2)--(-1,-2)--cycle); draw((-2,0)--(0,0)--(0,2)--(-2,2)--cycle); draw((0,0)--(2,0)--(2,2)--(0,2)--cycle); draw((-3,2)--(-1,2)--(-1,4)--(-3,4)--cycle); draw((-1,2)--(1,2)--(1,4)--(-1,4)--cycle); draw((1,2)--(1,4)--(3,4)--(3,2)--cycle); label("600",(0,-1)); label("30",(-1,1)); label("6",(-2,3)); label("5",(0,3)); [/asy] $\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6$

Let $\triangle ABC$ be a triangle with circumcenter $O$ satisfying $AB=13$, $BC = 15$, and $AC = 14$. Suppose there is a point $P$ such that $PB \perp BC$ and $PA \perp AB$. Let $X$ be a point on $AC$ such that $BX \perp OP$. What is the ratio $AX/XC$? [i]Proposed by Thomas Lam[/i]

An integer is called [i]formidable[/i] if it can be written as a sum of distinct powers of $4$, and [i]successful [/i] if it can be written as a sum of distinct powers of $6$. Can $2005$ be written as a sum of a [i]formidable [/i] number and a [i]successful [/i] number? Prove your answer.

For a rectangle $ABCD$ which is not a square, there is $O$ such that $O$ is on the perpendicular bisector of $BD$ and $O$ is in the interior of $\triangle BCD$. Denote by $E$ and $F$ the second intersections of the circle centered at $O$ passing through $B, D$ and $AB, AD$. $BF$ and $DE$ meets at $G$, and $X, Y, Z$ are the foots of the perpendiculars from $G$ to $AB, BD, DA$. $L, M, N$ are the foots of the perpendiculars from $O$ to $CD, BD, BC$. $XY$ and $ML$ meets at $P$, $YZ$ and $MN$ meets at $Q$. Prove that $BP$ and $DQ$ are parallel.

Let $ABCD$ be a cyclic quadrilateral with $AB=BC$ and $AD = CD$. A point $M$ lies on the minor arc $CD$ of its circumcircle. The lines $BM$ and $CD$ meet at point $P$, the lines $AM$ and $BD$ meet at point $Q$. Prove that $PQ \parallel AC$.

Given is an acute triangle $ABC$ with $AB<AC$ with altitudes $BD$ and $CE$. Let the tangents to the circumcircle at $B$ and $C$ meet at $Y$. Let $\omega_1$ be the circle through $A$ tangent to $DE$ at $E$; define $\omega_2$ similarly, and let their intersection point be $X$. Prove that $A, X, Y$ are colinear. $\textit{Proposed by Nikola Velov}$

Let $f:[0,1]\rightarrow\mathbb{R}$ be a continuous function. Prove that $$\int_0^{\pi/2}f(\sin(2x))\sin x\, dx = \int_0^{\pi/2} f(\cos^2 x)\cos x\, dx.$$

For how many natural numbers $n$ between $1$ and $2014$ (both inclusive) is $\frac{8n}{9999-n}$ an integer?

p1. A fraction is called Toba-$n$ if the fraction has a numerator of $1$ and the denominator of $n$. If $A$ is the sum of all the fractions of Toba-$101$, Toba-$102$, Toba-$103$, to Toba-$200$, show that $\frac{7}{12} <A <\frac56$. p2. If $a, b$, and $c$ satisfy the system of equations $$ \frac{ab}{a+b}=\frac12$$ $$\frac{bc}{b+c}=\frac13 $$ $$ \frac{ac}{a+c}=\frac17 $$ Determine the value of $(a- c)^b$. p3. Given triangle $ABC$. If point $M$ is located at the midpoint of $AC$, point $N$ is located at the midpoint of $BC$, and the point $P$ is any point on $AB$. Determine the area of ​​the quadrilateral $PMCN$. [img]https://cdn.artofproblemsolving.com/attachments/4/d/175e2d55f889b9dd2d8f89b8bae6c986d87911.png[/img] p4. Given the rule of motion of a particle on a flat plane $xy$ as following: $N: (m, n)\to (m + 1, n + 1)$ $T: (m, n)\to (m + 1, n - 1)$, where $m$ and $n$ are integers. How many different tracks are there from $(0, 3)$ to $(7, 2)$ by using the above rules ? p5. Andra and Dedi played “SUPER-AS”. The rules of this game as following. Players take turns picking marbles from a can containing $30$ marbles. For each take, the player can take the least a minimum of $ 1$ and a maximum of $6$ marbles. The player who picks up the the last marbels is declared the winner. If Andra starts the game by taking $3$ marbles first, determine how many marbles should be taken by Dedi and what is the next strategy to take so that Dedi can be the winner.

$ABCD$ is a parallelogram. The excircle of $ABC$ opposite $A$ has center $E$ and touches the line $AB$ at $X$. The excircle of $ADC$ opposite $A$ has center $F$ and touches the line $AD$ at $Y$. The line $FC$ meets the line$ AB$ at $W$, and the line $EC$ meets the line $AD$ at $Z$. Show that $WX = YZ$.

Let $n$ points be given inside a rectangle $R$ such that no two of them lie on a line parallel to one of the sides of $R$. The rectangle $R$ is to be dissected into smaller rectangles with sides parallel to the sides of $R$ in such a way that none of these rectangles contains any of the given points in its interior. Prove that we have to dissect $R$ into at least $n + 1$ smaller rectangles. [i]Proposed by Serbia[/i]

Given that $\log_{10}2 \approx 0.30103$, find the smallest positive integer $n$ such that the decimal representation of $2^{10n}$ does not begin with the digit $1$.

Find all numbers $n \geq 4$ such that there exists a convex polyhedron with exactly $n$ faces, whose all faces are right-angled triangles. (Note that the angle between any pair of adjacent faces in a convex polyhedron is less than $180^\circ$.) [i]Proposed by Hesam Rajabzadeh[/i]

Bugs Bunny plays a game in the Euclidean plane. At the $n$-th minute $(n \geq 1)$, Bugs Bunny hops a distance of $F_n$ in the North, South, East, or West direction, where $F_n$ is the $n$-th Fibonacci number (defined by $F_1 = F_2 =1$ and $F_n = F_{n-1} + F_{n-2}$ for $n \geq 3$). If the first two hops were perpendicular, prove that Bugs Bunny can never return to where he started. [i]Proposed by Dylan Toh[/i]

Let $n$ be a positive integer and let $a_1,\cdots ,a_n$ be real numbers satisfying $0\le a_i\le 1$ for $i=1,\cdots ,n.$ Prove the inequality \[(1-{a_i}^n)(1-{a_2}^n)\cdots (1-{a_n}^n)\le (1-a_1a_2\cdots a_n)^n.\]

Let $n$ be a positive integer. Given is a subset $A$ of $\{0,1,...,5^n\}$ with $4n+2$ elements. Prove that there exist three elements $a<b<c$ from $A$ such that $c+2a>3b$. [i]Proposed by Dominik Burek and Tomasz Ciesla, Poland[/i]

Two circles $ k_1 $ and $ k_2 $ are given with centers $ P $ and $ R $ respectively, touching externally at point $ A $. Let $ p $ be their common tangent line which does not pass trough $ A $ and touch $ k_1 $ at $ B $ and $ k_2 $ at $ C $. $ PR $ cuts $ BC $ at point $ E $ and $ k_2 $ at $ A $ and $ D $. If $ AB=2AC $ find $ \frac{BC}{DE} $.

A rectangular piece of paper with vertices $ABCD$ is being cut by a pair of scissors. The pair of scissors starts at vertex $A$, and then cuts along the angle bisector of $DAB$ until it reaches another edge of the paper. One of the two resulting pieces of paper has $4$ times the area of the other piece. What is the ratio of the longer side of the original paper to the shorter side?

Amy has a $2 \times 10$ puzzle grid which she can use $1 \times 1$ and $1 \times 2$ (1 vertical, 2 horizontal) tiles to cover. How many ways can she exactly cover the grid without any tiles overlapping and without rotating the tiles?

Define $K(n,0)=\varnothing $ and, for all nonnegative integers m and n, $K(n,m+1)=\left\{ \left. k \right|\text{ }1\le k\le n\text{ and }K(k,m)\cap K(n-k,m)=\varnothing \right\}$. Find the number of elements of $K(2004,2004)$.

Let $a$, $b$, and $c$ be positive real numbers satisfying $ab+bc+ca=abc$. Determine the minimum value of $$a^abc + b^bca + c^cab.$$

Let $ABC$ be an arbitrary triangle. Let the excircle tangent to side $a$ be tangent to lines $AB,BC$ and $CA$ at points $C_a,A_a,$ and $B_a,$ respectively. Similarly, let the excircle tangent to side $b$ be tangent to lines $AB,BC,$ and $CA$ at points $C_b,A_b,$ and $B_b,$ respectively. Finally, let the excircle tangent to side $c$ be tangent to lines $AB,BC,$ and $CA$ at points $C_c,A_c,$ and $B_c,$ respectively. Let $A'$ be the intersection of lines $A_bC_b$ and $A_cB_c.$ Similarly, let $B'$ be the intersection of lines $B_aC_a$ and $A_cB_c,$ and let $C$ be the intersection of lines $B_aC_a$ and $A_bC_b.$ Finally, let the incircle be tangent to sides $a,b,$ and $c$ at points $T_a,T_b,$ and $T_c,$ respectively. a) Prove that lines $A'A_a,B'B_b,$ and $C'C_c$ are concurrent. b) Prove that lines $A'T_a, B'T_b,$ and $C'T_c$ are also concurrent, and their point of intersection is on the line defined by the orthocentre and the incentre of triangle $ABC.$ [i]Proposed by Viktor Csaplár, Bátorkeszi and Dániel Hegedűs, Gyöngyös[/i]