Let $ABCD$ be a cyclic quadrilateral (In the same order) inscribed into the circle $\odot (O)$. Let $\overline{AC}$ $\cap$ $\overline{BD}$ $=$ $E$. A randome line $\ell$ through $E$ intersects $\overline{AB}$ at $P$ and $BC$ at $Q$. A circle $\omega$ touches $\ell$ at $E$ and passes through $D$. Given, $\omega$ $\cap$ $\odot (O)$ $=$ $R$. Prove, Points $B,Q,R,P$ are concyclic.

[b]p1.[/b] What is the units digit of the product of the first seven primes? [b]p2. [/b]In triangle $ABC$, $\angle BAC$ is a right angle and $\angle ACB$ measures $34$ degrees. Let $D$ be a point on segment $ BC$ for which $AC = CD$, and let the angle bisector of $\angle CBA$ intersect line $AD$ at $E$. What is the measure of $\angle BED$? [b]p3.[/b] Chad numbers five paper cards on one side with each of the numbers from $ 1$ through $5$. The cards are then turned over and placed in a box. Jordan takes the five cards out in random order and again numbers them from $ 1$ through $5$ on the other side. When Chad returns to look at the cards, he deduces with great difficulty that the probability that exactly two of the cards have the same number on both sides is $p$. What is $p$? [b]p4.[/b] Only one real value of $x$ satisfies the equation $kx^2 + (k + 5)x + 5 = 0$. What is the product of all possible values of $k$? [b]p5.[/b] On the Exeter Space Station, where there is no effective gravity, Chad has a geometric model consisting of $125$ wood cubes measuring $ 1$ centimeter on each edge arranged in a $5$ by $5$ by $5$ cube. An aspiring carpenter, he practices his trade by drawing the projection of the model from three views: front, top, and side. Then, he removes some of the original $125$ cubes and redraws the three projections of the model. He observes that his three drawings after removing some cubes are identical to the initial three. What is the maximum number of cubes that he could have removed? (Keep in mind that the cubes could be suspended without support.) [b]p6.[/b] Eric, Meena, and Cameron are studying the famous equation $E = mc^2$. To memorize this formula, they decide to play a game. Eric and Meena each randomly think of an integer between $1$ and $50$, inclusively, and substitute their numbers for $E$ and $m$ in the equation. Then, Cameron solves for the absolute value of $c$. What is the probability that Cameron’s result is a rational number? [b]p7.[/b] Let $CDE$ be a triangle with side lengths $EC = 3$, $CD = 4$, and $DE = 5$. Suppose that points $ A$ and $B$ are on the perimeter of the triangle such that line $AB$ divides the triangle into two polygons of equal area and perimeter. What are all the possible values of the length of segment $AB$? [b]p8.[/b] Chad and Jordan are raising bacteria as pets. They start out with one bacterium in a Petri dish. Every minute, each existing bacterium turns into $0, 1, 2$ or $3$ bacteria, with equal probability for each of the four outcomes. What is the probability that the colony of bacteria will eventually die out? [b]p9.[/b] Let $a = w + x$, $b = w + y$, $c = x + y$, $d = w + z$, $e = x + z$, and $f = y + z$. Given that $af = be = cd$ and $$(x - y)(x - z)(x - w) + (y - x)(y - z)(y - w) + (z - x)(z - y)(z - w) + (w - x)(w - y)(w - z) = 1,$$ what is $$2(a^2 + b^2 + c^2 + d^2 + e^2 + f^2) - ab - ac - ad - ae - bc - bd - bf - ce - cf - de - df - ef ?$$ [b]p10.[/b] If $a$ and $b$ are integers at least $2$ for which $a^b - 1$ strictly divides $b^a - 1$, what is the minimum possible value of $ab$? Note: If $x$ and $y$ are integers, we say that $x$ strictly divides $y$ if $x$ divides $y$ and $|x| \ne |y|$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Two acute triangles $ABC$ and $A_1B_1C_1$ are such that $B_1$ and $C_1$ lie on $BC$, and $A_1$ lies inside the triangle $ABC$. Let $S$ and $S_1$ be the areas of those triangles respectively. Prove that $\frac{S}{AB + AC}> \frac{S_1}{A_1B_1 + A_1C_1}$ (Nairi Sedrakyan, Ilya Bogdanov)

In a triangle $ABC$ with $ \angle A = 36^o$ and $AB = AC$, the bisector of the angle at $C$ meets the oposite side at $D$. Compute the angles of $\triangle BCD$. Express the length of side $BC$ in terms of the length $b$ of side $AC$ without using trigonometric functions.

Let $D$ be a point on side $[AB]$ of triangle $ABC$ with $|AB|=|AC|$ such that $[CD]$ is an angle bisector and $m(\widehat{ABC})=40^\circ$. Let $F$ be a point on the extension of $[AB]$ after $B$ such that $|BC|=|AF|$. Let $E$ be the midpoint of $[CF]$. If $G$ is the intersection of lines $ED$ and $AC$, what is $m(\widehat{FBG})$? $ \textbf{(A)}\ 150^\circ \qquad\textbf{(B)}\ 135^\circ \qquad\textbf{(C)}\ 120^\circ \qquad\textbf{(D)}\ 105^\circ \qquad\textbf{(E)}\ \text{None of above} $

Two medians drawn from vertices A and B of triangle ABC are perpendicular. Prove that side AB is the shortest side of ABC.

Let $K$ and $L$ be the points on the semicircle with diameter $AB$. Denote intersection of $AK$ and $AL$ as $T$ and let $N$ be the point such that $N$ is on segment $AB$ and line $TN$ is perpendicular to $AB$. If $U$ is the intersection of perpendicular bisector of $AB$ an $KL$ and $V$ is a point on $KL$ such that angles $UAV$ and $UBV$ are equal. Prove that $NV$ is perpendicular to $KL$.

Point $ P$ lies on side $ AB$ of a convex quadrilateral $ ABCD$. Let $ \omega$ be the incircle of triangle $ CPD$, and let $ I$ be its incenter. Suppose that $ \omega$ is tangent to the incircles of triangles $ APD$ and $ BPC$ at points $ K$ and $ L$, respectively. Let lines $ AC$ and $ BD$ meet at $ E$, and let lines $ AK$ and $ BL$ meet at $ F$. Prove that points $ E$, $ I$, and $ F$ are collinear. [i]Author: Waldemar Pompe, Poland[/i]

Consider the $2\times3$ rectangle below. We fill in the small squares with the numbers $1,2,3,4,5,6$ (one per square). Define a [i]tasty[/i] filling to be one such that each row is [b]not[/b] in numerical order from left to right and each column is [b]not[/b] in numerical order from top to bottom. If the probability that a randomly selected filling is tasty is $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, what is $m+n$? \begin{tabular}{|c|c|c|c|} \hline & & \\ \hline & & \\ \hline \end{tabular} [i]2016 CCA Math Bonanza Lightning #4.2[/i]

On side, $BC, AB$ of a parallelogram $ABCD$ lie points $M,N$ respectively such that $|AM| =|CN|$. Let $P$ be the intersection of $AM$ and $CN$. Prove that the angle bisector of $\angle APC$ passes through $D$.

[b]p1.[/b] In the following $3$ by $3$ grid, $a, b, c$ are numbers such that the sum of each row is listed at the right and the sum of each column is written below it: [center][img]https://cdn.artofproblemsolving.com/attachments/d/9/4f6fd2bc959c25e49add58e6e09a7b7eed9346.png[/img][/center] What is $n$? [b]p2.[/b] Suppose in your sock drawer of $14$ socks there are 5 different colors and $3$ different lengths present. One day, you decide you want to wear two socks that have both different colors and different lengths. Given only this information, what is the maximum number of choices you might have? [b]p3.[/b] The population of Arveymuddica is $2014$, which is divided into some number of equal groups. During an election, each person votes for one of two candidates, and the person who was voted for by $2/3$ or more of the group wins. When neither candidate gets $2/3$ of the vote, no one wins the group. The person who wins the most groups wins the election. What should the size of the groups be if we want to minimize the minimum total number of votes required to win an election? [b]p4.[/b] A farmer learns that he will die at the end of the year (day $365$, where today is day $0$) and that he has a number of sheep. He decides that his utility is given by ab where a is the money he makes by selling his sheep (which always have a fixed price) and $b$ is the number of days he has left to enjoy the profit; i.e., $365-k$ where $k$ is the day. If every day his sheep breed and multiply their numbers by $103/101$ (yes, there are small, fractional sheep), on which day should he sell them all? [b]p5.[/b] Line segments $\overline{AB}$ and $\overline{AC}$ are tangent to a convex arc $BC$ and $\angle BAC = \frac{\pi}{3}$ . If $\overline{AB} = \overline{AC} = 3\sqrt3$, find the length of arc $BC$. [b]p6.[/b] Suppose that you start with the number $8$ and always have two legal moves: $\bullet$ Square the number $\bullet$ Add one if the number is divisible by $8$ or multiply by $4$ otherwise How many sequences of $4$ moves are there that return to a multiple of $8$? [b]p7.[/b] A robot is shuffling a $9$ card deck. Being very well machined, it does every shuffle in exactly the same way: it splits the deck into two piles, one containing the $5$ cards from the bottom of the deck and the other with the $4$ cards from the top. It then interleaves the cards from the two piles, starting with a card from the bottom of the larger pile at the bottom of the new deck, and then alternating cards from the two piles while maintaining the relative order of each pile. The top card of the new deck will be the top card of the bottom pile. The robot repeats this shuffling procedure a total of n times, and notices that the cards are in the same order as they were when it started shuffling. What is the smallest possible value of $n$? [b]p8.[/b] A secant line incident to a circle at points $A$ and $C$ intersects the circle's diameter at point $B$ with a $45^o$ angle. If the length of $AB$ is $1$ and the length of $BC$ is $7$, then what is the circle's radius? [b]p9.[/b] If a complex number $z$ satisfies $z + 1/z = 1$, then what is $z^{96} + 1/z^{96}$? [b]p10.[/b] Let $a, b$ be two acute angles where $\tan a = 5 \tan b$. Find the maximum possible value of $\sin (a - b)$. [b]p11.[/b] A pyramid, represented by $SABCD$ has parallelogram $ABCD$ as base ($A$ is across from $C$) and vertex $S$. Let the midpoint of edge $SC$ be $P$. Consider plane $AMPN$ where$ M$ is on edge $SB$ and $N$ is on edge $SD$. Find the minimum value $r_1$ and maximum value $r_2$ of $\frac{V_1}{V_2}$ where $V_1$ is the volume of pyramid $SAMPN$ and $V_2$ is the volume of pyramid $SABCD$. Express your answer as an ordered pair $(r_1, r_2)$. [b]p12.[/b] A $5 \times 5$ grid is missing one of its main diagonals. In how many ways can we place $5$ pieces on the grid such that no two pieces share a row or column? [b]p13.[/b] There are $20$ cities in a country, some of which have highways connecting them. Each highway goes from one city to another, both ways. There is no way to start in a city, drive along the highways of the country such that you travel through each city exactly once, and return to the same city you started in. What is the maximum number of roads this country could have? [b]p14.[/b] Find the area of the cyclic quadrilateral with side lengths given by the solutions to $$x^4-10x^3+34x^2- 45x + 19 = 0.$$ [b]p15.[/b] Suppose that we know $u_{0,m} = m^2 + m$ and $u_{1,m} = m^2 + 3m$ for all integers $m$, and that $$u_{n-1,m} + u_{n+1,m} = u_{n,m-1} + u_{n,m+1}$$ Find $u_{30,-5}$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Carl has $5$ cubes each having side length $1$, and Kate has $5$ cubes each having side length $2$. What is the total volume of the $10$ cubes? $\textbf{(A) }24 \qquad \textbf{(B) }25 \qquad \textbf{(C) } 28\qquad \textbf{(D) } 40\qquad \textbf{(E) } 45$

What is the largest number of squares on $9 \times 9$ square board that can be cut along their both diagonals so that the board does not fall apart into several pieces?

The point $ I $ is the center of the inscribed circle of the triangle $ ABC $. The points $ B_1 $ and $ C_1 $ are the midpoints of the sides $ AC $ and $ AB $, respectively. It is known that $ \angle BIC_1 + \angle CIB_1 = 180^\circ $. Prove the equality $ AB + AC = 3BC $

Two circles with centers $O_1$ and $O_2$ meet at points $A$ and $B$. The bisector of angle $O_1AO_2$ meets the circles for the second time at points $C $and $D$. Prove that the distances from the circumcenter of triangle $CBD$ to $O_1$ and to $O_2$ are equal.

Let $ABCD$ be a quadrilateral and $P$ a point in the plane of the quadrilateral. Let $M,N$ be on the sides $AC,BD$ respectively such that $PM \parallel BC, PN \parallel AD$. $AC$ meets $BD$ at $E$. Prove that the orthocenter of triangles $EBC, EAD, EMN$ are collinear if and only if $P$ is on the line $AB$. Đỗ Thanh Sơn PS. Instead of the word [b]collinear[/b], it was written [b]concurrent[/b], probably a typo.

Let $D$ and $E$ be points on sides $AB$ and $AC$ respectively of a triangle $ABC$ such that $DE$ is parallel to $BC$ and tangent to the incircle of $ABC$. Prove that \[DE\le\frac{1}{8}(AB+BC+CA) \]

A tetrahedron $ABCD$ with acute-angled faces is inscribed in a sphere with center $O$. A line passing through $O$ perpendicular to plane $ABC$ crosses the sphere at point $D'$ that lies on the opposide side of plane $ABC$ than point $D$. Line $DD'$ crosses plane $ABC$ in point $P$ that lies inside the triangle $ABC$. Prove, that if $\angle APB=2\angle ACB$, then $\angle ADD'=\angle BDD'$.

Let $ABC$ be a triangle with $\angle A=90^{\circ}$ and $AB=AC$. Let $D$ and $E$ be points on the segment $BC$ such that $BD:DE:EC = 1:2:\sqrt{3}$. Prove that $\angle DAE= 45^{\circ}$

The (Fibonacci) sequence $f_n$ is defined by $f_1=f_2=1$ and $f_{n+2}=f_{n+1}+f_n$ for $n\ge1$. Prove that the area of the triangle with the sides $\sqrt{f_{2n+1}},\sqrt{f_{2n+2}},$ and $\sqrt{f_{2n+3}}$ is equal to $\frac12$.

Let $P$ and $Q$ be two distinct points in the plane. Let us denote by $m(PQ)$ the segment bisector of $PQ$. Let $S$ be a finite subset of the plane, with more than one element, that satisfies the following properties: (i) If $P$ and $Q$ are in $S$, then $m(PQ)$ intersects $S$. (ii) If $P_1Q_1, P_2Q_2, P_3Q_3$ are three diferent segments such that its endpoints are points of $S$, then, there is non point in $S$ such that it intersects the three lines $m(P_1Q_1)$, $m(P_2Q_2)$, and $m(P_3Q_3)$. Find the number of points that $S$ may contain.

Let $H$ be the orthocenter of a triangle $ABC$, and $M$, $N$ be the midpoints of segments $BC$, $AH$ respectively. The perpendicular from $N$ to $MH$ meets $BC$ at point $A^{\prime}$. Points $B^{\prime}$ and $C^{\prime}$ are defined similarly. Prove that $A^{\prime}$, $B^{\prime}$, $C^{\prime}$ are collinear. Proposed by: F.Ivlev

Six congruent isosceles triangles have been put together as described in the picture below. Prove that points M, F, C lie on one line. [img]https://i.imgur.com/1LU5Zmb.png[/img]

In acute-angled triangle $ABC,$ $E,F$ are on the side $BC,$ such that $\angle BAE=\angle CAF,$ and let $M,N$ be the projections of $F$ onto $AB,AC,$ respectively. The line $AE$ intersects $ \odot (ABC) $ at $D$(different from point $A$). Prove that $S_{AMDN}=S_{\triangle ABC}.$

In the triangle $[ABC], D$ is the midpoint of $[AB]$ and $E$ is the trisection point of $[BC]$ closer to $C$. If $\angle ADC= \angle BAE$ , find the measue of $\angle BAC$ .