There is an Equilateral trapezoid $ ABCD. $ $ \bar{AB} =60, \bar{BC}=\bar{DA}= 36, \bar{CD}=108. $ $ M $ is the middle point of $ \bar {AB} $, and point $P$ on $ \bar{AM} $ follows that $ \bar {AP} $ =10. The foot of perpendicular dropped from $P$ to $ \bar {BD} $ is $E$. $ \bar{AC} \cap \bar{BD} $ is $ F $. Point $X$ is on $ \bar {AF} $ which follows $ \bar{MX}=\bar{ME} $ Find $ \bar{AX} \times \bar{AF} $

Let $ABCD$ be a convex quadrilateral. Let $E,F,G,H$ be points on segments $AB$, $BC$, $CD$, $DA$, respectively, and let $P$ be the intersection of $EG$ and $FH$. Given that quadrilaterals $HAEP$, $EBFP$, $FCGP$, $GDHP$ all have inscribed circles, prove that $ABCD$ also has an inscribed circle. [i]Evan O'Dorney.[/i]

Let $A$, $B$, $C$, $D$, $E$ be points such that $ABCD$ is a cyclic quadrilateral and $ABDE$ is a parallelogram. The diagonals $AC$ and $BD$ intersect at $S$ and the rays $AB$ and $DC$ intersect at $F$. Prove that $\sphericalangle{AFS}=\sphericalangle{ECD}$. [i](4th Middle European Mathematical Olympiad, Team Competition, Problem 6)[/i]

The convex quadrilateral $ABCD$ is has angle $A$ equal to $60^o$ , angle bisector of $A$ the diagonal $AC$ and $\angle ACD= 40^o$ and $\angle ACB = 120^o$. Inside the quadrilateral the point $P$ lies such that $\angle PDA = 40^o$ and $\angle PBA = 10^o$; a) Find the angle $\angle DPB$? b) Prove that $P$ lies on the diagonal $AC$.

Given a triangle $ABC$ and a point $P_0$ on the side $AB$. Construct points $P_i, Q_i, R_i $ as follows. $Q_i$ is the foot of the perpendicular from $P_i$ to $BC, R_i$ is the foot of the perpendicular from $Q_i$ to $AC$ and $P_i$ is the foot of the perpendicular from $R_{i-1}$ to $AB$. Show that the points $P_i$ converge to a point $P$ on $AB$ and show how to construct $P$.

Determine all integer numbers $n\ge 3$ such that the regular $n$-gon can be decomposed into isosceles triangles by non-intersecting diagonals.

Given six real numbers $a$, $b$, $c$, $x$, $y$, $z$ such that $0 < b-c < a < b+c$ and $ax + by + cz = 0$. What is the sign of the sum $ayz + bzx + cxy$ ?

Let $ a$, $ b$, $ c$ be the sides of a triangle, and $ S$ its area. Prove: \[ a^{2} \plus{} b^{2} \plus{} c^{2}\geq 4S \sqrt {3} \] In what case does equality hold?

The inscribed circle of the triangle $ABC$ touches the sides $BC, CA, AB$ at points $A_1, B_1, C_1$ respectively. The points $P_b, Q_b, R_b$ are the points of the segments $BC_1, C_1A_1, A_1B$, respectively, such that $BP_bQ_bR_b$ is parallelogram. In the same way, the points $P_c, Q_c, R_c$ are the points of the sections $CB_1, B_1A_1, A_1C$, respectively such that $CP_cQ_cR_c$ is a parallelogram. The intersection of the lines $P_bR_b$ and $P_cR_c$ is $T$. Show that $TQ_b = TQ_c$.

Two circles $\mathcal{C}_1$ and $\mathcal{C}_2$ intersect in $P$ and $Q$. A line through $P$ intersects $\mathcal{C}_1$ and $\mathcal{C}_2$ again at $A$ and $B$, respectively, and $X$ is the midpoint of $AB$. The line through $Q$ and $X$ intersects $C_1$ and $C_2$ again at $Y$ and $Z$, respectively. Prove that $X$ is the midpoint of $YZ$.

Consider an acute non-isosceles triangle. In a single step it is allowed to cut any one of the available triangles into two triangles along its median. Is it possible that after a finite number of cuttings all triangles will be isosceles? [i]Proposed by E. Bakaev[/i]

In the regular hexagonal prism $ABCDEFA_1B_1C_1D_1E_1F_1$, We construct $, Q$, the projections of point $A$ on the lines $A_1B$ and $A_1C$ repsectilvely. We construct $R,S$, the projections of point $D_1$ on the lines $A_1D$ and $C_1D$ respectively. a) Determine the measure of the angle between the planes $(AQP)$ and $(D_1RS)$. b) Show that $\angle AQP = \angle D_1RS$.

[b]p17.[/b] Let the roots of the polynomial $f(x) = 3x^3 + 2x^2 + x + 8 = 0$ be $p, q$, and $r$. What is the sum $\frac{1}{p} +\frac{1}{q} +\frac{1}{r}$ ? [b]p18.[/b] Two students are playing a game. They take a deck of five cards numbered $1$ through $5$, shuffle them, and then place them in a stack facedown, turning over the top card next to the stack. They then take turns either drawing the card at the top of the stack into their hand, showing the drawn card to the other player, or drawing the card that is faceup, replacing it with the card on the top of the pile. This is repeated until all cards are drawn, and the player with the largest sum for their cards wins. What is the probability that the player who goes second wins, assuming optimal play? [b]p19.[/b] Compute the sum of all primes $p$ such that $2^p + p^2$ is also prime. [b]p20.[/b] In how many ways can one color the $8$ vertices of an octagon each red, black, and white, such that no two adjacent sides are the same color? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given triangle $ ABC$ and $ AB\equal{}c$, $ AC\equal{}b$ and $ BC\equal{}a$ satisfying $ a \geq b \geq c$, $ BE$ and $ CF$ are two interior angle bisectors. $ P$ is a point inside triangle $ AEF$. $ R$ and $ Q$ are the projections of $ P$ on sides $ AB$ and $ AC$. Prove that $ PR \plus{} PQ \plus{} RQ < b$.

A line parallel to the side $AC$ of triangle $ABC$ cuts the side $AB$ at $K$ and the side $BC$ at $M$. $O$ is the intersection point of $AM$ and $CK$. If $AK = AO$ and $KM = MC$, prove that $AM = KB$.

[i]25 problems for 30 minutes[/i] [b]p1.[/b] Compute the sum $2019 + 201 + 20 + 2$. [b]p2.[/b] The sequence $100, 102, 104,..., 996$ and $998$ is the sequence of all three-digit even numbers. How many three digit even numbers are there? [b]p3.[/b] Find the units digit of $25\times 37\times 113\times 22$. [b]p4.[/b] Samuel has a number in his head. He adds $4$ to the number and then divides the result by $2$. After doing this, he ends up with the same number he had originally. What is his original number? [b]p5.[/b] According to Shay's Magazine, every third president is terrible (so the third, sixth, ninth president and so on were all terrible presidents). If there have been $44$ presidents, how many terrible presidents have there been in total? [b]p6.[/b] In the game Tic-Tac-Toe, a player wins by getting three of his or her pieces in the same row, column, or diagonal of a $3\times 3$ square. How many configurations of $3$ pieces are winning? Rotations and reflections are considered distinct. [b]p7.[/b] Eddie is a sad man. Eddie is cursed to break his arm $4$ times every $20$ years. How many times would he break his arm by the time he reaches age $100$? [b]p8. [/b]The figure below is made from $5$ congruent squares. If the figure has perimeter $24$, what is its area? [img]https://cdn.artofproblemsolving.com/attachments/1/9/6295b26b1b09cacf0c32bf9d3ba3ce76ddb658.png[/img] [b]p9.[/b] Sancho Panza loves eating nachos. If he eats $3$ nachos during the first minute, $4$ nachos during the second, $5$ nachos during the third, how many nachos will he have eaten in total after $15$ minutes? [b]p10.[/b] If the day after the day after the day before Wednesday was two days ago, then what day will it be tomorrow? [b]p11.[/b] Neetin the Rabbit and Poonam the Meerkat are in a race. Poonam can run at $10$ miles per hour, while Neetin can only hop at $2$ miles per hour. If Neetin starts the race $2$ miles ahead of Poonam, how many minutes will it take for Poonam to catch up with him? [b]p12.[/b] Dylan has a closet with t-shirts: $3$ gray, $4$ blue, $2$ orange, $7$ pink, and $2$ black. Dylan picks one shirt at random from his closet. What is the probability that Dylan picks a pink or a gray t-shirt? [b]p13.[/b] Serena's brain is $200\%$ the size of Eric's brain, and Eric's brain is $200\%$ the size of Carlson's. The size of Carlson's brain is what percent the size of Serena's? [b]p14.[/b] Find the sum of the coecients of $(2x + 1)^3$ when it is fully expanded. [b]p15. [/b]Antonio loves to cook. However, his pans are weird. Specifically, the pans are rectangular prisms without a top. What is the surface area of the outside of one of Antonio's pans if their volume is $210$, and their length and width are $6$ and $5$, respectively? [b]p16.[/b] A lattice point is a point on the coordinate plane with $2$ integer coordinates. For example, $(3, 4)$ is a lattice point since $3$ and $4$ are both integers, but $(1.5, 2)$ is not since $1.5$ is not an integer. How many lattice points are on the graph of the equation $x^2 + y^2 = 625$? [b]p17.[/b] Jonny has a beaker containing $60$ liters of $50\%$ saltwater ($50\%$ salt and $50\%$ water). Jonny then spills the beaker and $45$ liters pour out. If Jonny adds $45$ liters of pure water back into the beaker, what percent of the new mixture is salt? [b]p18.[/b] There are exactly 25 prime numbers in the set of positive integers between $1$ and $100$, inclusive. If two not necessarily distinct integers are randomly chosen from the set of positive integers from $1$ to $100$, inclusive, what is the probability that at least one of them is prime? [b]p19.[/b] How many consecutive zeroes are at the end of $12!$ when it is expressed in base $6$? [b]p20.[/b] Consider the following figure. How many triangles with vertices and edges from the following figure contain exactly $1$ black triangle? [img]https://cdn.artofproblemsolving.com/attachments/f/2/a1c400ff7d06b583c1906adf8848370e480895.png[/img] [b]p21.[/b] After Akshay got kicked o the school bus for rowdy behavior, he worked out a way to get home from school with his dad. School ends at $2:18$ pm, but since Akshay walks slowly he doesn't get to the front door until $2:30$. His dad doesn't like to waste time, so he leaves home everyday such that he reaches the high school at exactly $2:30$ pm, instantly picks up Akshay and turns around, then drives home. They usually get home at $3:30$ pm. However, one day Akshay left school early at exactly $2:00$ pm because he was expelled. Trying to delay telling his dad for as long as possible, Akshay starts jogging home. His dad left home at the regular time, saw Akshay on the way, picked him up and turned around instantly. They then drove home while Akshay's dad yelled at him for being a disgrace. They reached home at $3:10$ pm. How long had Akshay been walking before his dad picked him up? [b]p22.[/b] In quadrilateral $ABCD$, diagonals $AC$ and $BD$ intersect at $O$. Then $\angle BOC = \angle BCD$, $\angle COD =\angle BAD$, $AB = 4$, $DC = 6$, and $BD = 5$. What is the length of $BO$? [b]p23.[/b] A standard six-sided die is rolled. The number that comes up first determines the number of additional times the die will be rolled (so if the first number is $3$, then the die will be rolled $3$ more times). Each time the die is rolled, its value is recorded. What is the expected value of the sum of all the rolls? [b]p24.[/b] Dora has a peculiar calculator that can only perform $2$ operations: either adding $1$ to the current number or squaring the current number. Each minute, Dora randomly chooses an operation to apply to her number. She starts with $0$. What is the expected number of minutes it takes Dora's number to become greater than or equal to $10$? [b]p25.[/b] Let $\vartriangle ABC$ be such that $AB = 2$, $BC = 1$, and $\angle ACB = 90^o$. Let points $D$ and $E$ be such that $\vartriangle ADE$ is equilateral, $D$ is on segment $\overline{BC}$, and $D$ and $E$ are not on the same side of $\overline{AC}$. Segment $\overline{BE}$ intersects the circumcircle of $\vartriangle ADE$ at a second point $F$. If $BE =\sqrt{6}$, find the length of $\overline{BF}$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $p,q$ be positive integers. Consider a rectangle $ABCD$ with lengths of sides $p$ and $q$ that consists of $pq$ unital squares. How many of these squares are crossed by diagonal $AC$?

For every $n\geq 3$, determine all the configurations of $n$ distinct points $X_1,X_2,\ldots,X_n$ in the plane, with the property that for any pair of distinct points $X_i$, $X_j$ there exists a permutation $\sigma$ of the integers $\{1,\ldots,n\}$, such that $\textrm{d}(X_i,X_k) = \textrm{d}(X_j,X_{\sigma(k)})$ for all $1\leq k \leq n$. (We write $\textrm{d}(X,Y)$ to denote the distance between points $X$ and $Y$.) [i](United Kingdom) Luke Betts[/i]

Prove that there exists a unique point $M$ on the side $AD$ of a convex quadrilateral $ABCD$ such that \[\sqrt{S_{ABM}}+\sqrt{S_{CDM}} = \sqrt{S_{ABCD}}\] if and only if $AB\parallel CD$.

The base of a triangle is $ 15$ inches. Two lines are drawn parallel to the base, terminating in the other two sides, and dividing the triangle into three equal areas. The length of the parallel closer to the base is: $ \textbf{(A)}\ 5\sqrt{6}\text{ inches} \qquad\textbf{(B)}\ 10\text{ inches} \qquad\textbf{(C)}\ 4\sqrt{3}\text{ inches} \qquad\textbf{(D)}\ 7.5\text{ inches}\\ \textbf{(E)}\ \text{none of these}$

Two players, Red and Blue, play in alternating turns on a 10x10 board. Blue goes first. In his turn, a player picks a row or column (not chosen by any player yet) and color all its squares with his own color. If any of these squares was already colored, the new color substitutes the old one. The game ends after 20 turns, when all rows and column were chosen. Red wins if the number of red squares in the board exceeds at least by 10 the number of blue squares; otherwise Blue wins. Determine which player has a winning strategy and describe this strategy.

Let $I$ and $H$ be the incentre and orthocentre of triangle $ABC$ respectively. Let $P,Q$ be the midpoints of $AB,AC$. The rays $PI,QI$ intersect $AC,AB$ at $R,S$ respectively. Suppose that $T$ is the circumcentre of triangle $BHC$. Let $RS$ intersect $BC$ at $K$. Prove that $A,I$ and $T$ are collinear if and only if $[BKS]=[CKR]$. [i]Shen Wunxuan[/i]

In the coordinate plane, a point is called a $\text{lattice point}$ if both of its coordinates are integers. Let $A$ be the point $(12,84)$. Find the number of right angled triangles $ABC$ in the coordinate plane $B$ and $C$ are lattice points, having a right angle at vertex $A$ and whose incenter is at the origin $(0,0)$.

Given a circle of radius $K$. Two other circles, the sum of the radii of which are also equal to $K$, tangent to the circle from the inside. Prove that the line connecting the points of tangency passes through one of the common points of these circles.

Let $ABC$ be a triangle with sides $AB = 34$, $BC = 15$, $AC = 35$ and let $\Gamma$ be the circle of smallest possible radius passing through $A$ tangent to $BC$. Let the second intersections of $\Gamma$ and sides $AB$, $AC$ be the points $X, Y$ . Let the ray $XY$ intersect the circumcircle of the triangle $ABC$ at $Z$. If $AZ =\frac{p}{q}$ for relatively prime integers $p$ and $q$, find $p + q$.