Let $T_1$ be a triangle having $a, b, c$ as lengths of its sides and let $T_2$ be another triangle having $u, v,w$ as lengths of its sides. If $P,Q$ are the areas of the two triangles, prove that \[16PQ \leq a^2(-u^2 + v^2 + w^2) + b^2(u^2 - v^2 + w^2) + c^2(u^2 + v^2 - w^2).\] When does equality hold?

Let $I$ be the incenter of an acute-angled triangle $ABC$. Line $AI$ cuts the circumcircle of $BIC$ again at $E$. Let $D$ be the foot of the altitude from $A$ to $BC$, and let $J$ be the reflection of $I$ across $BC$. Show $D$, $J$ and $E$ are collinear.

Bill divides a $28 \times 30$ rectangular board into two smaller rectangular boards with a single straightcut, so that the side lengths of both boards are positive whole numbers. How many different pairs of rectangular boards, up to congruence and arrangement, can Bill possibly obtain? (For instance, a cut that is $1$ unit away from either of the edges with length $28$ will result in the same pair of boards: either way, one would end up with a $1 \times 28$ board and a $29 \times 28$ board.)

Let $ABCD$ be a quadtrilateral with no parallel sides. The diagonals intersect at $E$, and $P, Q$ are points on sides $AB, CD$ respectively such that $\frac{AP}{PB} = \frac{CQ}{QD}$. $PQ$ meet $AC$ and $BD$ at $R,S$. Prove that $(EAB),(ECD),(ERS)$ all meet a point other than $E$.

Find all prime numbers $ p,q$ less than 2005 and such that $ q|p^2 \plus{} 4$, $ p|q^2 \plus{} 4$.

The points $ A,B,C$ are on a circle $ O$. The tangent line at $ A$ and the secant $ BC$ intersect at $ P$, $ B$ lying between $ C$ and $ P$. If $ \overline{BC} \equal{} 20$ and $ \overline{PA} \equal{} 10\sqrt {3}$, then $ \overline{PB}$ equals: $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 10\sqrt {3} \qquad\textbf{(D)}\ 20 \qquad\textbf{(E)}\ 30$

In the triangle $ABC$, points $D, E, F$ are on the sides $BC, CA$ and $AB$ respectively such that $FE$ is parallel to $BC$ and $DF$ is parallel to $CA$, Let P be the intersection of $BE$ and $DF$, and $Q$ the intersection of $FE$ and $AD$. Prove that $PQ$ is parallel to $AB$.

Show that there exists some [b]positive[/b] integer $k$ with $$\gcd(2012,2020)=\gcd(2012+k,2020)$$$$=\gcd(2012,2020+k)=\gcd(2012+k,2020+k).$$

Let $k$ be a positive integer. $12k$ persons have participated in a party and everyone shake hands with $3k+6$ other persons. We know that the number of persons who shake hands with every two persons is a fixed number. Find $k.$

Let $ABCD$ be a convex quadrilateral with the property that $MA \cdot MC + MA \cdot CD = MB \cdot MD$, where $M$ is the intersection of the diagonals $AC$ and $BD$. The angle bisector of $\angle ACD$ is drawn intersecting ray $\overrightarrow{BA}$ at $K$. Prove that $BC = DK$ if and only if $AB \parallel CD$.

From a bag containing 5 pairs of socks, each pair a different color, a random sample of 4 single socks is drawn. Any complete pairs in the sample are discarded and replaced by a new pair draw from the bag. The process continues until the bag is empty or there are 4 socks of different colors held outside the bag. What is the probability of the latter alternative?

Find all values of $a$ such that the two polynomials \[x^2+ax-1\qquad\text{and}\qquad x^2-x+a\] share at least 1 root. [i]2018 CCA Math Bonanza Individual Round #7[/i]

Mustafa bought a big rug. The seller measured the rug with a ruler that was supposed to measure one meter. As it turned out to be $30$ meters long by $20$ meters wide, he charged Rs $120.000$ Rs. When Mustafa arrived home, he measured the rug again and realized that the seller had overcharged him by $9.408$ Rs. How many centimeters long is the ruler used by the seller?

The bisector of angle $A$ in parallelogram $ABCD$ intersects side $BC$ at $M$ and the bisector of $\angle AMC$ passes through point $D$. Find angles of the parallelogram if it is known that $\angle MDC = 45^o$. [img]https://cdn.artofproblemsolving.com/attachments/e/7/7cfb22f0c26fe39aa3da3898e181ae013a0586.png[/img]

Let $I$ be the incenter of a triangle $ABC$, $D$ be an arbitrary point of segment $AC$, and $A_{1}, A_{2}$ be the common points of the perpendicular from $D$ to the bisector $CI$ with $BC$ and $AI$ respectively. Define similarly the points $C_{1}$, $C_{2}$. Prove that $B$, $A_{1}$, $A_{2}$, $I$, $C_{1},$ $C_{2}$ are concyclic. Proposed by:D.Shvetsov

How many ordered pairs $(a, b)$ of positive integers satisfy the equation $$a\cdot b + 63 = 20\cdot \text{lcm}(a, b) + 12\cdot\text{gcd}(a,b),$$ where $\text{gcd}(a,b)$ denotes the greatest common divisor of $a$ and $b$, and $\text{lcm}(a,b)$ denotes their least common multiple? $\textbf{(A)}\ 0\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 8$

Adrian has $2n$ cards numbered from $ 1$ to $2n$. He gets rid of $n$ cards that are consecutively numbered. The sum of the numbers of the remaining papers is $1615$. Find all the possible values of $n$.

Determine how many primes are there in the sequence \[101, 10101, 1010101 ....\]

Find all real parameters $p$ for which the equation $x^3 -2p(p+1)x^2+(p^4 +4p^3 -1)x-3p^3 = 0$ has three distinct real roots which are sides of a right triangle.

Given an acute angles triangle $ABC$, and $H$ is its orthocentre. The external bisector of the angle $\angle BHC$ meets the sides $AB$ and $AC$ at the points $D$ and $E$ respectively. The internal bisector of the angle $\angle BAC$ meets the circumcircle of the triangle $ADE$ again at the point $K$. Prove that $HK$ is through the midpoint of the side $BC$.

Jasper and Rose are playing a game. Twenty-six $32$-ounce jugs are in a line, labeled Quart $\text{A}$ through Quart $\text{Z}$ from left to right. All twenty-six jugs are initially full. Jasper and Rose take turns making one of the following two moves: [list] [*] remove a positive integer number of ounces (possibly all) from the leftmost nonempty jug, or [*] remove an [i]equal[/i] positive integer number of ounces from the two leftmost nonempty jugs, possibly emptying them. Neither player may remove more ounces from a jug than it currently contains. [/list] Jasper plays first. A player’s score is the number of ounces they take from Quart $\text{Z}.$ If both players play to maximize their score, compute the maximum score that Jasper can guarantee.

Tamika selects two different numbers at random from the set $ \{ 8,9,10\} $ and adds them. Carlos takes two different numbers at random from the set $ \{ 3,5,6\} $ and multiplies them. What is the probability that Tamika's result is greater than Carlos' result? $ \text{(A)}\ \frac{4}{9}\qquad\text{(B)}\ \frac{5}{9}\qquad\text{(C)}\ \frac{1}{2}\qquad\text{(D)}\ \frac{1}{3}\qquad\text{(E)}\ \frac{2}{3} $

A group of children is sitting around a round table . At first, each child has an even number of candies. Each turn, each child gives half of his candies to the child sitting at his right. If, after a turn, a child has an odd number of candies, the teacher gives him\her an extra candy. Show that after a finite number of rounds all children will have the same number of candies.

Tanya and Serezha take turns putting chips in empty squares of a chessboard. Tanya starts with a chip in an arbitrary square. At every next move, Serezha must put a chip in the column where Tanya put her last chip, while Tanya must put a chip in the row where Serezha put his last chip. The player who cannot make a move loses. Which of the players has a winning strategy? [i]Proposed by A. Golovanov[/i]

From a $8\times 7$ rectangle divided into unit squares, we cut the corner, which consists of the first row and the first column. (that is, the corner has $14$ unit squares). For the following, when we say corner we reffer to the above definition, along with rotations and symmetry. Consider an infinite lattice of unit squares. We will color the squares with $k$ colors, such that for any corner, the squares in that corner are coloured differently (that means that there are no squares coloured with the same colour). Find out the minimum of $k$.