In $\vartriangle ABC, PQ // BC$ where $P$ and $Q$ are points on $AB$ and $AC$ respectively. The lines $PC$ and $QB$ intersect at $G$. It is also given $EF//BC$, where $G \in EF, E \in AB$ and $F\in AC$ with $PQ = a$ and $EF = b$. Find value of $BC$.

Let n > 3 be a given integer. Find the largest integer d (in terms of n) such that for any set S of n integers, there are four distinct (but not necessarily disjoint) nonempty subsets, the sum of the elements of each of which is divisible by d.

Let $k$, $a$, and $b$, be fixed integers such that $0 \le a < k$, $0 \le b < k+1$, and $a$, $b$ are not both zero. The sequence $\{T_n\}_{n \ge k}$ satisfies $T_n = T_{n-1}+T_{n-2} \pmod{n}$, $0 \le T_n < n$, $T_k = a$, and $T_{k+1} = b$. Let the decimal expression of $T_n$ form a sequence $x=\overline{0.T_kT_{k+1} \dots}$. For instance, when $k = 66, a = 5, b = 20$, we get $T_{66}=5$, $T_{67}=20$, $T_{68}=25$, $T_{69}=45$, $T_{70}=0$, $T_{71}=45, \dots$, and thus $x=0.522545045 \dots$. Prove that $x$ is irrational.

For a complex number $z \neq 3$,$4$, let $F(z)$ denote the real part of $\tfrac{1}{(3-z)(4-z)}$. If \[ \int_0^1 F \left( \frac{\cos 2 \pi t + i \sin 2 \pi t}{5} \right) \; dt = \frac mn \] for relatively prime positive integers $m$ and $n$, find $100m+n$. [i]Proposed by Evan Chen[/i]

Let $B_1,C_1$ be the midpoints of sides $AC,AB$ of a triangle $ABC$ respectively. The tangents to the circumcircle at $B$ and $C$ meet the rays $CC_1,BB_1$ at points $K$ and $L$ respectively. Prove that $\angle BAK = \angle CAL$.

A circle is divided by $2019$ points into equal parts. Two players delete these points in turns. A player loses, if after his turn it is possible to draw a diameter of the circle such that there are no undeleted points on one side of it. Which player has a winning strategy?

Let $ABC$ be a triangle with $b\neq c$. Points $D$ is the midpoint of $BC$ and let $E$ be the foot of angle $A$ bisector. In the exterior of the triangle we construct the similar triangles $AMB$ and $ANC$ . Prove: a) $MN\bot AD \Longleftrightarrow MA \bot AB$ b) $MN\bot AE \Longleftrightarrow M,A,N$ are colinear.

The triangle $ABC$ is inscribed in the circle $S$ and $AB <AC$. The line containing $A$ and is perpendicular to $BC$ meets $S$ in $P$ ($P \ne A$). Point $X$ is on the segment $AC$ and the line $BX$ intersects $S$ in $Q$ ($Q \ne B$). Show that $BX = CX$ if, and only if, $PQ$ is a diameter of $S$.

Find all function $ f: \mathbb R\to \mathbb R$ satisfying the condition: \[ f(f(x \minus{} y)) \equal{} f(x)\cdot f(y) \minus{} f(x) \plus{} f(y) \minus{} xy \]

Bag A has three chips labeled 1, 3, and 5. Bag B has three chips labeled 2, 4, and 6. If one chip is drawn from each bag, how many different values are possible for the sum of the two numbers on the chips? $ \textbf{(A)} 4 \qquad\textbf{(B)} 5 \qquad\textbf{(C)} 6 \qquad\textbf{(D)} 7 \qquad\textbf{(E)} 9 $

Show that there exists a real constant $C>1$ with the following property: For any positive integer $n$, there are at least $C^n$ positive integers with exactly $n$ decimal digits, which are divisible by the product of their digits. (In particular, these $n$ digits are all non-zero.) [i]Proposed by Jean-Marie De Koninck and Florian Luca[/i]

Annie drew a rectangle and partitioned it into $n$ rows and $k$ columns with horizontal and vertical lines. Annie knows the area of the resulting $n \cdot k$ little rectangles while Benny does not. Annie reveals the area of some of these small rectangles to Benny. Given $n$ and $k$ at least how many of the small rectangle’s areas did Annie have to reveal, if from the given information Benny can determine the areas of all the $n \cdot k$ little rectangles? For example in the case $n = 3$ and $k = 4$ revealing the areas of the $10$ small rectangles if enough information to find the areas of the remaining two little rectangles. [img]https://cdn.artofproblemsolving.com/attachments/b/1/c4b6e0ab6ba50068ced09d2a6fe51e24dd096a.png[/img]

In the plane of a given triangle $A_1A_2A_3$ determine (with proof) a straight line $l$ such that the sum of the distances from $A_1, A_2$, and $A_3$ to $l$ is the least possible.

In a certain triangle, the bisectors of the two interior angles were extended to the intersection with the circumscribed circle and two equal chords were obtained. Is it true that the triangle is isosceles?

Let $ABC$ be a right triangle with $A = 90^{\circ}$ and $D \in (AC)$. Denote by $E$ the reflection of $A$ in the line $BD$ and $F$ the intersection point of $CE$ with the perpendicular in $D$ to $BC$. Prove that $AF, DE$ and $BC$ are concurrent.

Let $\triangle ABC$ be a triangle and let $X$ be a point in the interior of the triangle. The second intersection points of the lines $XA,XB$ and $XC$ with the circumcircle of $\triangle ABC$ are $P,Q$ and $R$. Let $U$ be a point on the ray $XP$ (these are the points on the line $XP$ such that $P$ and $U$ lie on the same side of $X$). The line through $U$ parallel to $AB$ intersects $BQ$ in $V$ . The line through $U$ parallel to $AC$ intersects $CR$ in $W$. Prove that $Q, R, V$ , and $W$ lie on a circle.

Let $B$ and $C$ be points on a semicircle with diameter $AD$ such that $B$ is closer to $A$ than $C$. Diagonals $AC$ and $BD$ intersect at point $E$. Let $P$ and $Q$ be points such that $\overline{PE} \perp \overline{BD}$ and $\overline{PB} \perp \overline{AD}$, while $\overline{QE} \perp \overline{AC}$ and $\overline{QC} \perp \overline{AD}$. If $BQ$ and $CP$ intersect at point $T$, prove that $\overline{TE} \perp \overline{BC}$. [i]Proposed by Andrew Wen[/i]

Let $ m,n$ be two natural numbers with $ m > 1$ and $ 2^{2m \plus{} 1} \minus{} n^2\geq 0$. Prove that: \[ 2^{2m \plus{} 1} \minus{} n^2\geq 7 .\]

A biased coin has a $ \dfrac{6 + 2\sqrt{3}}{12} $ chance of landing heads, and a $ \dfrac{6 - 2\sqrt{3}}{12} $ chance of landing tails. What is the probability that the number of times the coin lands heads after being flipped 100 times is a multiple of 4? The answer can be expressed as $ \dfrac{1}{4} + \dfrac{1 + a^b}{c \cdot d^e} $ where $ a, b, c, d, e $ are positive integers. Find the minimal possible value of $ a + b + c + d + e $.

Given an integer $n\geq 2$, determine the maximum value the sum $x_1+\cdots+x_n$ may achieve, as the $x_i$ run through the positive integers, subject to $x_1\leq x_2\leq \cdots \leq x_n$ and $x_1+\cdots+x_n=x_1 x_2\cdots x_n$.

Which of the following does not divide the number of ordered pairs $(x,y)$ of integers satisfying the equation $x^3 - 13y^3 = 1453$? $ \textbf{a)}\ 2 \qquad\textbf{b)}\ 3 \qquad\textbf{c)}\ 5 \qquad\textbf{d)}\ 7 \qquad\textbf{e)}\ \text{None of above} $

Find all functions $f:R \rightarrow R$ such that $$(x+y^2)f(yf(x))=xyf(y^2+f(x))$$, where $x,y \in \mathbb{R}$

Let $\triangle ABC$ be a triangle such that $AB=40$ and $AC=30.$ Points $X$ and $Y$ are on the segment $AB$ and $BC,$ respectively such that $AX:BX=3:2$ and $BY:CY=1:4.$ Given that $XY=12,$ the area of $\triangle ABC$ can be written as $a\sqrt{b}$ where $a$ and $b$ are positive integers and $b$ is squarefree. Compute $a+b.$

Given positive real numbers $p,q$ with $p+q=1$. Prove that for all positive integers $m,n$ the following inequality holds $(1-p^m)^n+(1-q^n)^m \geq 1$.

[b]p1.[/b] Let $p_b(m)$ be the sum of digits of $m$ when $m$ is written in base $b$. (So, for example, $p_2(5) = 2$). Let $f(0) = 2007^{2007}$, and for $n \ge 0$ let $f(n + 1) = p_7(f(n))$. What is $f(10^{10000})$? [b]p2.[/b] Compute: $$\sum^{\infty}_{n=1}\frac{(-1)^{n+1}4n}{n^4 - 8n^2 + 4}.$$ [b]p3.[/b] $ABCDEFGH$ is an octagon whose eight interior angles all have the same measure. The lengths of the eight sides of this octagon are, in some order, $$2, 2\sqrt2, 4, 4\sqrt2, 6, 7, 7, \,\,\, and \,\,\, 8.$$ Find the area of $ABCDEFGH$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].