Let be given an isosceles triangle $ABC$ with the base $AB$. A point $P$ is chosen on the altitude $CD$ so that the incircles of $ABP$ and $PECF$ are congruent, where $E$ and $F$ are the intersections of $AP$ and $BP$ with the opposite sides of the triangle, respectively. Prove that the incircles of triangles $ADP$ and $BCP$ are also congruent.

[b]p1.[/b] It can be shown in Calculus that the area between the x-axis and the parabola $y=kx^2$ (к is a positive constant) on the $x$-interval $0 \le x \le a$ is $\frac{ka^3}{3}$ a) Find the area between the parabola $y=4x^2$ and the x-axis for $0 \le x \le 3$. b) Find the area between the parabola $y=5x^2$ and the x-axis for $-2 \le x \le 4$. c) A square $2$ by $2$ dartboard is situated in the $xy$-plane with its center at the origin and its sides parallel to the coordinate axes. Darts that are thrown land randomly on the dartboard. Find the probability that a dart will land at a point of the dartboard that is nearer to the point $(0, 1)$ than to the bottom edge of the dartboard. [b]p2.[/b] When two rows of a determinant are interchanged, the value of the determinant changes sign. There are also certain operations which can be performed on a determinant which leave its value unchanged. Two such operations are changing any row by adding a constant multiple of another row to it, and changing any column by adding a constant multiple of another column to it. Often these operations are used to generate lots of zeroes in a determinant in order to simplify computations. In fact, if we can generate zeroes everywhere below the main diagonal in a determinant, the value of the determinant is just the product of all the entries on that main diagonal. For example, given the determinant $\begin{vmatrix} 1 & 2 & 3 \\ 2 & 6 & 2 \\ 3 & 10 & 4 \end{vmatrix}$ we add $-2$ times the first row to the second row, then add $-2$ times the second row to the third row, giving the new determinant $\begin{vmatrix} 1 & 2 & 3 \\ 0 & 2 & -4 \\ 0 & 0 & 3 \end{vmatrix}$ , and the value is the product of the diagonal entries: $6$. a) Transform this determinant into another determinant with zeroes everywhere below the main diagonal, and find its value: $\begin{vmatrix} 1 & 3 & -1 \\ 4 & 7 & 2 \\ 3 & -6 & 5 \end{vmatrix}$ b) Do the same for this determinant: $\begin{vmatrix} 0 & 1 & 2 & 3 \\ 1 & 0 & 1 & 2 \\ 2 & 1 & 0 & 1 \\ 3 & 2 & 1 & 0 \end{vmatrix}$ [b]p3.[/b] In Pascal’s triangle, the entries at the ends of each row are both $1$, and otherwise each entry is the sum of the two entries diagonally above it: Row Number $0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1$ $1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 1 \,\,\,1$ $2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 1 \,\, 2 \,\,1$ $3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 1\,\, 3 \,\, 3 \,\, 1$ $4\,\,\,\,\,\,\,\,\,\,\,\,\,\,1 \,\,4 \,\, 6 \,\, 4 \,\, 1$ $...\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...$ This triangle gives the binomial coefficients in expansions like $( a + b)^3 = 1a^3 + 3a^2 b + 3 ab^2 + 1b^3$ . a) What is the sum of the numbers in row #$5$ of Pascal's triangle? b) What is the sum of the numbers in row #$n$ of Pascal's triangle? c) Show that in row #$6$ of Pascal's triangle, the sum of all the numbers is exactly twice the sum of the first, third, fifth, and seventh numbers in the row. d) Prove that in row #$n$ of Pascal's triangle, the sum of ail the numbers is exactly twice the sum of the numbers in the odd positions of that row. [b]p4.[/b] The product: of several terms is sometimes described using the symbol $\Pi$ which is capital pi, the Greek equivalent of $p$, for the word "product". For example the symbol $\prod^4_{k=1}(2k +1)$ means the product of numbers of the form $(2k + 1)$, for $k=1,2,3,4$. Thus it equals $945$. a) Evaluate as a reduced fraction $\prod_{k=1}^{10} \frac{k}{k + 2}$ b) Evaluate as a reduced fraction $\prod_{k=1}^{10} \frac{k^2 + 10k+ 17}{k^2+4k + 41}$ c) Evaluate as a reduced fraction $\prod_{k=1}^{\infty}\frac{k^3-1}{k^3+1}$ [b]p5.[/b] a) In right triangle $CAB$, the median $AF$, the angle bisector $AE$, and the altitude $AD$ divide the right angld $A$ into four equal angles. If $AB = 1$, find the area of triangle $AFE$. [img]https://cdn.artofproblemsolving.com/attachments/5/1/0d4a83e58a65c2546ce25d1081b99d45e30729.png[/img] b) If in any triangle, an angle is divided into four equal angles by the median, angle bisector, and altitude drawn from that angle, prove that the angle must be a right angle. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The following figures are composed of squares and circles. Which figure has a shaded region with largest area? [asy]/* AMC8 2003 #22 Problem */ size(3inch, 2inch); unitsize(1cm); pen outline = black+linewidth(1); filldraw((0,0)--(2,0)--(2,2)--(0,2)--cycle, mediumgrey, outline); filldraw(shift(3,0)*((0,0)--(2,0)--(2,2)--(0,2)--cycle), mediumgrey, outline); filldraw(Circle((7,1), 1), mediumgrey, black+linewidth(1)); filldraw(Circle((1,1), 1), white, outline); filldraw(Circle((3.5,.5), .5), white, outline); filldraw(Circle((4.5,.5), .5), white, outline); filldraw(Circle((3.5,1.5), .5), white, outline); filldraw(Circle((4.5,1.5), .5), white, outline); filldraw((6.3,.3)--(7.7,.3)--(7.7,1.7)--(6.3,1.7)--cycle, white, outline); label("A", (1, 2), N); label("B", (4, 2), N); label("C", (7, 2), N); draw((0,-.5)--(.5,-.5), BeginArrow); draw((1.5, -.5)--(2, -.5), EndArrow); label("2 cm", (1, -.5)); draw((3,-.5)--(3.5,-.5), BeginArrow); draw((4.5, -.5)--(5, -.5), EndArrow); label("2 cm", (4, -.5)); draw((6,-.5)--(6.5,-.5), BeginArrow); draw((7.5, -.5)--(8, -.5), EndArrow); label("2 cm", (7, -.5)); draw((6,1)--(6,-.5), linetype("4 4")); draw((8,1)--(8,-.5), linetype("4 4"));[/asy] $ \textbf{(A)}\ \text{A only}\qquad\textbf{(B)}\ \text{B only}\qquad\textbf{(C)}\ \text{C only}\qquad\textbf{(D)}\ \text{both A and B}\qquad\textbf{(E)}\ \text{all are equal}$

Convex pentagon $PQRST$ has $PQ = T P = 5$, $QR = RS = ST = 6$, and $\angle QRS = \angle RST = 90^o$. Given that points $U$ and $V$ exist such that $RU =UV = VS = 2$, find the area of pentagon $PQUVT$ . [i]Proposed by Kira Tang[/i]

[u]Round 1[/u] [b]p1. [/b]The Queen of Bees invented a new language for her hive. The alphabet has only $6$ letters: A, C, E, N, R, T; however, the alphabetic order is different than in English. A word is any sequence of $6$ different letters. In the dictionary for this language, the word TRANCE immediately follows NECTAR. What is the last word in the dictionary? [b]p2.[/b] Is it possible to solve the equation $\frac{1}{x}= \frac{1}{y} +\frac{1}{z}$ with $x,y,z$ integers (positive or negative) such that one of the numbers $x,y,z$ has one digit, another has two digits, and the remaining one has three digits? [b]p3.[/b] The $10,000$ dots in a $100\times 100$ square grid are all colored blue. Rekha can paint some of them red, but there must always be a blue dot on the line segment between any two red dots. What is the largest number of dots she can color red? The picture shows a possible coloring for a $5\times 7$ grid. [img]https://cdn.artofproblemsolving.com/attachments/0/6/795f5ab879938ed2a4c8844092b873fb8589f8.jpg[/img] [b]p4.[/b] Six flies rest on a table. You have a swatter with a checkerboard pattern, much larger than the table. Show that there is always a way to position and orient the swatter to kill at least five of the flies. Each fly is much smaller than a swatter square and is killed if any portion of a black square hits any part of the fly. [b]p5.[/b] Maryam writes all the numbers $1-81$ in the cells of a $9\times 9$ table. Tian calculates the product of the numbers in each of the nine rows, and Olga calculates the product of the numbers in every column. Could Tian's and Olga's lists of nine products be identical? [u]Round 2[/u] [b]p6.[/b] A set of points in the plane is epic if, for every way of coloring the points red or blue, it is possible to draw two lines such that each blue point is on a line, but none of the red points are. The figure shows a particular set of $4$ points and demonstrates that it is epic. What is the maximum possible size of an epic set? [img]https://cdn.artofproblemsolving.com/attachments/e/f/44fd1679c520bdc55c78603190409222d0b721.jpg[/img] [b]p7.[/b] Froggy Chess is a game played on a pond with lily pads. First Judit places a frog on a pad of her choice, then Magnus places a frog on a different pad of his choice. After that, they alternate turns, with Judit moving first. Each player, on his or her turn, selects either of the two frogs and another lily pad where that frog must jump. The jump must reduce the distance between the frogs (all distances between the lily pads are different), but both frogs cannot end up on the same lily pad. Whoever cannot make a move loses. The picture below shows the jumps permitted in a particular situation. Who wins the game if there are $2017$ lily pads? [img]https://cdn.artofproblemsolving.com/attachments/a/9/1a26e046a2a614a663f9d317363aac61654684.jpg[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A triangle $ABC$ is given. Let $C'$ and $C'_{a}$ be the touching points of sideline $AB$ with the incircle and with the excircle touching the side $BC$. Points $C'_{b}$, $C'_{c}$, $A'$, $A'_{a}$, $A'_{b}$, $A'_{c}$, $B'$, $B'_{a}$, $B'_{b}$, $B'_{c}$ are defined similarly. Consider the lengths of $12$ altitudes of triangles $A'B'C'$, $A'_{a}B'_{a}C'_{a}$, $A'_{b}B'_{b}C'_{b}$, $A'_{c}B'_{c}C'_{c}$. (a) (8-9) Find the maximal number of different lengths. (b) (10-11) Find all possible numbers of different lengths.

[i]25 problems for 30 minutes.[/i] [b]p1.[/b] Chad, Ravi, Kevin, and Meena are four of the $551$ residents of Chadwick, Illinois. Expressing your answer to the nearest percent, how much of the population do they represent? [b]p2.[/b] Points $A$, $B$, and $C$ are on a line for which $AB = 625$ and $BC = 256$. What is the sum of all possible values of the length $AC$? [b]p3.[/b] An increasing arithmetic sequence has first term $2014$ and common difference $1337$. What is the least odd term of this sequence? [b]p4.[/b] How many non-congruent scalene triangles with integer side lengths have two sides with lengths $3$ and $4$? [b]p5.[/b] Let $a$ and $b$ be real numbers for which the function $f(x) = ax^2+bx+3$ satisfies $f(0)+2^0 = f(1)+2^1 = f(2) + 2^2$. What is $f(0)$? [b]p6.[/b] A pentomino is a set of five planar unit squares that are joined edge to edge. Two pentominoes are considered the same if and only if one can be rotated and translated to be identical to the other. We say that a pentomino is compact if it can fit within a $2$ by $3$ rectangle. How many distinct compact pentominoes exist? [b]p7.[/b] Consider a hexagon with interior angle measurements of $91$, $101$, $107$, $116$, $152$, and $153$ degrees. What is the average of the interior angles of this hexagon, in degrees? [b]p8.[/b] What is the smallest positive number that is either one larger than a perfect cube and one less than a perfect square, or vice versa? [b]p9.[/b] What is the first time after $4:56$ (a.m.) when the $24$-hour expression for the time has three consecutive digits that form an increasing arithmetic sequence with difference $1$? (For example, $23:41$ is one of those moments, while $23:12$ is not.) [b]p10.[/b] Chad has trouble counting. He wants to count from $1$ to $100$, but cannot pronounce the word "three," so he skips every number containing the digit three. If he tries to count up to $100$ anyway, how many numbers will he count? [b]p11.[/b] In square $ABCD$, point $E$ lies on side $BC$ and point $F$ lies on side $CD$ so that triangle $AEF$ is equilateral and inside the square. Point $M$ is the midpoint of segment $EF$, and $P$ is the point other than $E$ on $AE$ for which $PM = FM$. The extension of segment $PM$ meets segment $CD$ at $Q$. What is the measure of $\angle CQP$, in degrees? [b]p12.[/b] One apple is five cents cheaper than two bananas, and one banana is seven cents cheaper than three peaches. How much cheaper is one apple than six peaches, in cents? [b]p13.[/b] How many ordered pairs of integers $(a, b)$ exist for which |a| and |b| are at most $3$, and $a^3-a = b^3-b$? [b]p14.[/b] Five distinct boys and four distinct girls are going to have lunch together around a table. They decide to sit down one by one under the following conditions: no boy will sit down when more boys than girls are already seated, and no girl will sit down when more girls than boys are already seated. How many possible sequences of taking seats exist? [b]p15.[/b] Jordan is swimming laps in a pool. For each lap after the first, the time it takes her to complete is five seconds more than that of the previous lap. Given that she spends 10 minutes on the first six laps, how long does she spend on the next six laps, in minutes? [b]p16.[/b] Chad decides to go to trade school to ascertain his potential in carpentry. Chad is assigned to cut away all the vertices of a wooden regular tetrahedron with sides measuring four inches. Each vertex is cut away by a plane which passes through the three midpoints of the edges adjacent to that vertex. What is the surface area of the resultant solid, in square inches? Note: A tetrahedron is a solid with four triangular faces. In a regular tetrahedron, these faces are all equilateral triangles. [b]p17.[/b] Chad and Jordan independently choose two-digit positive integers. The two numbers are then multiplied together. What is the probability that the result has a units digit of zero? [b]p18.[/b] For art class, Jordan needs to cut a circle out of the coordinate grid. She would like to find a circle passing through at least $16$ lattice points so that her cut is accurate. What is the smallest possible radius of her circle? Note: A lattice point is defined as one whose coordinates are both integers. For example, $(5, 8)$ is a lattice point whereas $(3.5, 5)$ is not. [b]p19.[/b] Chad's ant Arctica is on one of the eight corners of Chad's toolbox, which measures two decimeters in width, three decimeters in length, and four decimeters in height. One day, Arctica wanted to go to the opposite corner of this box. Assuming she can only crawl on the surface of the toolbox, what is the shortest distance she has to crawl to accomplish this task, in decimeters? (You may assume that the toolbox is oating in the Exeter Space Station, so that Arctica can crawl on all six faces.) [b]p20.[/b] Jordan is counting numbers for fun. She starts with the number $1$, and then counts onward, skipping any number that is a divisor of the product of all previous numbers she has said. For example, she starts by counting $1$, $2$, $3$, $4$, $5$, but skips 6, a divisor of $1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 = 120$. What is the $20^{th}$ number she counts? [b]p21.[/b] Chad and Jordan are having a race in the lake shown below. The lake has a diameter of four kilometers and there is a circular island in the middle of the lake with a diameter of two kilometers. They start at one point on the edge of the lake and finish at the diametrically opposite point. Jordan makes the trip only by swimming in the water, while Chad swims to the island, runs across it, and then continues swimming. They both take the fastest possible route and, amazingly, they tie! Chad swims at two kilometers an hour and runs at five kilometers an hour. At what speed does Jordan swim? [img]https://cdn.artofproblemsolving.com/attachments/f/6/22b3b0bba97d25ab7aabc67d30821d0b12efc0.png[/img] [b]p22.[/b] Cameron has stolen Chad's barrel of oil and is driving it around on a truck on the coordinate grid on his truck. Cameron is a bad truck driver, so he can only move the truck forward one kilometer at a $4$ $EMC^2$ $2014$ Problems time along one of the gridlines. In fact, Cameron is so bad at driving the truck that between every two one-kilometer movements, he has to turn exactly $90$ degrees. After $50$ one-kilometer movements, given that Cameron's first one-kilometer movement was westward, how many points he could be on? [b]p23.[/b] Let $a$, $b$, and $c$ be distinct nonzero base ten digits. Assume there exist integers $x$ and $y$ for which $\overline{abc} \cdot \overline{cb} = 100x^2 + 1$ and $\overline{acb} \cdot \overline{bc} = 100y^2 + 1$. What is the minimum value of the number $\overline{abbc}$? Note: The notation $\overline{pqr}$ designates the number whose hundreds digit is $p$, tens digit is $q$, and units digit is $r$, not the product $p \cdot q \cdot r$. [b]p24.[/b] Let $r_1, r_2, r_3, r_4$ and $r_5$ be the five roots of the equation $x^5-4x^4+3x^2-2x+1 = 0$. What is the product of $(r_1 +r_2 +r_3 +r_4)$, $(r_1 +r_2 +r_3 +r_5)$, $(r_1 +r_2 +r_4 +r_5)$, $(r_1 +r_3 +r_4 +r_5)$, and $(r_2 +r_3 +r_4 +r_5)$? [b]p25.[/b] Chad needs seven apples to make an apple strudel for Jordan. He is currently at 0 on the metric number line. Every minute, he randomly moves one meter in either the positive or the negative direction with equal probability. Arctica's parents are located at $+4$ and $-4$ on the number line. They will bite Chad for kidnapping Arctica if he walks onto those numbers. Also, there is one apple located at each integer between $-3$ and $3$, inclusive. Whenever Chad lands on an integer with an unpicked apple, he picks it. What is the probability that Chad picks all the apples without getting bitten by Arctica's parents? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The circle inscribed in the triangle $ABC$ with center $I$ touches the side $BC$ at the point $D$. The line $DI$ intersects the side $AC$ at the point $M$. The tangent from $M$ to the inscribed circle, different from $AC$, intersects the side $AB$ at the point $N$. The line $NI$ intersects the side $BC$ at the point $P$. Prove that $AB = BP$.

If possible, construct an equilateral triangle whose three vertices are on three given circles.

A right triangle $ABC$ has the right angle at vertex $A$. Circle $c$ passes through vertices $A$ and $B$ of the triangle $ABC$ and intersects the sides $AC$ and $BC$ correspondingly at points $D$ and $E$. The line segment $CD$ has the same length as the diameter of the circle $c$. Prove that the triangle $ABE$ is isosceles.

Let $ABC$ be a given triangle. Find the locus $\mathbf M$ of all vertices $Z$ such that triangle $XYZ$ is equilateral where $X$ is any point of segment $AB$ and $Y\neq X$ lies on ray $AC.$

If you have $65$ points in a plane, we will make the lines that passes by any two points in this plane and we obtain exactly $2015$ distinct lines, prove that least $4$ points are collinears!!

Given a triangle $ABC$ satisfying $AC+BC=3\cdot AB$. The incircle of triangle $ABC$ has center $I$ and touches the sides $BC$ and $CA$ at the points $D$ and $E$, respectively. Let $K$ and $L$ be the reflections of the points $D$ and $E$ with respect to $I$. Prove that the points $A$, $B$, $K$, $L$ lie on one circle. [i]Proposed by Dimitris Kontogiannis, Greece[/i]

The four points of a circle are in the following order: $A, B, C, D$. Extensions of chord $AB$ beyond point $B$ and of chord $CD$ beyond point $C$ intersect at point $E$, with $\angle AED= 60^o$. If $\angle ABD =3 \angle BAC$ , prove that $AD$ is the diameter of the circle.

Point $P$ is chosen inside triangle $ABC$ so that $\angle APC+\angle ABC=180^o$ and $BC=AP.$ On the side $AB$, a point $K$ is chosen such that $AK = KB + PC$. Prove that $CK \perp AB$.

Let $(C)$ and $(L)$ be a circle and a line. $P_{1},\dots,P_{2n+1}$ are odd number of points on $(L)$. $A_{1}$ is an arbitrary point on $(C)$. $A_{k+1}$ is the intersection point of $A_{k}P_{k}$ and $(C)$ ($1\leq k\leq 2n+1$). Prove that $A_{1}A_{2n+2}$ passes through a constant point while $A_{1}$ varies on $(C)$.

A set of lines in the plane is in [i]general position[/i] if no two are parallel and no three pass through the same point. A set of lines in general position cuts the plane into regions, some of which have finite area; we call these its [i]finite regions[/i]. Prove that for all sufficiently large $n$, in any set of $n$ lines in general position it is possible to colour at least $\sqrt{n}$ lines blue in such a way that none of its finite regions has a completely blue boundary. [i]Note[/i]: Results with $\sqrt{n}$ replaced by $c\sqrt{n}$ will be awarded points depending on the value of the constant $c$.

Show there do not exist four points in the Euclidean plane such that the pairwise distances between the points are all odd integers.

2. In a square $ABCD$, let $P$ be a point in the side $CD$, different from $C$ and $D$. In the triangle $ABP$, the altitudes $AQ$ and $BR$ are drawn, and let $S$ be the intersection point of lines $CQ$ and $DR$. Show that $\angle ASB=90$.

The two points $P(8, 2)$ and $Q(5, 11)$ are considered in the plane. A mobile moves from $P$ to $Q$ according to a path that has to fulfill the following conditions: The moving part of $ P$ and arrives at a point on the $x$-axis, along which it travels a segment of length $1$, then it departs from this axis and goes towards a point on the $y$ axis, on which travels a segment of length $2$, separates from the $y$ axis finally and goes towards the point $Q$. Among all the possible paths, determine the one with the minimum length, thus like this same length.

Two parallel sides of a quadrilateral have integer lengths. Prove that this quadrilateral can be cut into congruent triangles. (A Shapovalov)

Let $ABCD$ be a cyclic quadrilateral. The internal angle bisector of $\angle BAD$ and line $BC$ intersect at $E.$ $M$ is the midpoint of segment $AE.$ The exterior angle bisector of $\angle BCD$ and line $AD$ intersect at $F.$ The lines $MF$ and $AB$ intersect at $G.$ Prove that if $AB=2AD,$ then $MF=2MG.$

Let $I$ be the incenter of fixed triangle $ABC$, and $D$ be an arbitrary point on $BC$. The perpendicular bisector of $AD$ meets $BI,CI$ at $F$ and $E$ respectively. Find the locus of orthocenters of $\triangle IEF$ as $D$ varies.

In a rectangle $ABCD$, let $M$ and $N$ be the midpoints of sides $BC$ and $CD$, respectively, such that $AM$ is perpendicular to $MN$. Given that the length of $AN$ is $60$, the area of rectangle $ABCD$ is $m \sqrt{n}$ for positive integers $m$ and $n$ such that $n$ is not divisible by the square of any prime. Compute $100m+n$. [i]Proposed by Yannick Yao[/i]

Let $k_1$ and $k_2$ be two circles with centers $O_1$ and $O_2$ and equal radius $r$ such that $O_1O_2 = r$. Let $A$ and $B$ be two points lying on the circle $k_1$ and being symmetric to each other with respect to the line $O_1O_2$. Let $P$ be an arbitrary point on $k_2$. Prove that \[PA^2 + PB^2 \geq 2r^2.\]