Let points $A_1$, $A_2$ and $A_3$ lie on the circle $\Gamma$ in a counter-clockwise order, and let $P$ be a point in the same plane. For $i \in \{1,2,3\}$, let $\tau_i$ denote the counter-clockwise rotation of the plane centred at $A_i$, where the angle of rotation is equial to the angle at vertex $A_i$ in $\triangle A_1A_2A_3$. Further, define $P_i$ to be the point $\tau_{i+2}(\tau_{i}(\tau_{i+1}(P)))$, where the indices are taken modulo $3$ (i.e., $\tau_4 = \tau_1$ and $\tau_5 = \tau_2$). Prove that the radius of the circumcircle of $\triangle P_1P_2P_3$ is at most the radius of $\Gamma$. [i]Proposed by Anant Mudgal[/i]

A circle concentric with the inscribed circle of $ABC$ intersects the sides of the triangle at six points forming a convex hexagon $A_1A_2B_1B_2C_1C_2$ (points $C_1$ and $C_2$ on the $AB$ side, $A_1$ and $A_2$ on $BC$, $B_1$ and $B_2$ on $AC$). Prove that if line $A_1B_1$ is parallel to the bisector of angle $B$, then line $A_2C_2$ is parallel to the bisector of angle $C$.

Find the slope of a line passing through the point $(0,\ 1)$ with which the area of the part bounded by the line and the parabola $y=x^2$ is $\frac{5\sqrt{5}}{6}.$

Given a set $ \mathcal{H}$ of points in the plane, $ P$ is called an "intersection point of $ \mathcal{H}$" if distinct points $ A,B,C,D$ exist in $ \mathcal{H}$ such that lines $ AB$ and $ CD$ are distinct and intersect in $ P$. Given a finite set $ \mathcal{A}_{0}$ of points in the plane, a sequence of sets is defined as follows: for any $ j\geq0$, $ \mathcal{A}_{j+1}$ is the union of $ \mathcal{A}_{j}$ and the intersection points of $ \mathcal{A}_{j}$. Prove that, if the union of all the sets in the sequence is finite, then $ \mathcal{A}_{i}=\mathcal{A}_{1}$ for any $ i\geq1$.

Define the [i]inverse [/i] of triangle $ABC$ with respect to a point $O$ in the following way: construct the circumcircle of $ABC$ and construct lines $AO$, $BO$, and $CO$. Let $A'$ be the other intersection of $AO$ and the circumcircle (if $AO$ is tangent, then let $A' = A$). Similarly define $B'$ and $C'$. Then $A'B'C'$ is the inverse of $ABC$ with respect to $O$. Compute the area of the inverse of the triangle given in the plane by $A(-6, -21)$, $B(-23, 10)$, $C(16, 23)$ with respect to $O(1, 3)$.

The sphere $ \omega $ passes through the vertex $S$ of the pyramid $SABC$ and intersects with the edges $SA,SB,SC$ at $A_1,B_1,C_1$ other than $S$. The sphere $ \Omega $ is the circumsphere of the pyramid $SABC$ and intersects with $ \omega $ circumferential, lies on a plane which parallel to the plane $(ABC)$. Points $A_2,B_2,C_2$ are symmetry points of the points $A_1,B_1,C_1$ respect to midpoints of the edges $SA,SB,SC$ respectively. Prove that the points $A$, $B$, $C$, $A_2$, $B_2$, and $C_2$ lie on a sphere.

Given two points $A$ and $B$ and a circle, find a point $X$ on the circle so that points $C$ and $D$ at which lines $AX$ and $BX$ intersect the circle are the endpoints of the chord $CD$ parallel to a given line $MN$.

The diagram below shows some small squares each with area $3$ enclosed inside a larger square. Squares that touch each other do so with the corner of one square coinciding with the midpoint of a side of the other square. Find integer $n$ such that the area of the shaded region inside the larger square but outside the smaller squares is $\sqrt{n}$. [asy] size(150); real r=1/(2sqrt(2)+1); path square=(0,1)--(r,1)--(r,1-r)--(0,1-r)--cycle; path square2=(0,.5)--(r/sqrt(2),.5+r/sqrt(2))--(r*sqrt(2),.5)--(r/sqrt(2),.5-r/sqrt(2))--cycle; defaultpen(linewidth(0.8)); filldraw(unitsquare,gray); filldraw(square2,white); filldraw(shift((0.5-r/sqrt(2),0.5-r/sqrt(2)))*square2,white); filldraw(shift(1-r*sqrt(2),0)*square2,white); filldraw(shift((0.5-r/sqrt(2),-0.5+r/sqrt(2)))*square2,white); filldraw(shift(0.5-r/sqrt(2)-r,-(0.5-r/sqrt(2)-r))*square,white); filldraw(shift(0.5-r/sqrt(2)-r,-(0.5+r/sqrt(2)))*square,white); filldraw(shift(0.5+r/sqrt(2),-(0.5+r/sqrt(2)))*square,white); filldraw(shift(0.5+r/sqrt(2),-(0.5-r/sqrt(2)-r))*square,white); filldraw(shift(0.5-r/2,-0.5+r/2)*square,white); [/asy]

Let $ABC$ be a triangle and $P$ be any point on the side $BC$. Let $I_1$,$I_2$ be the incenters of triangles $ABP$ and $ACP$, respectively. If $D$ is the point of tangency of the incircle of $ABC$ with the side $BC$, prove that $\angle I_1DI_2 = 90^o$.

A trapezoid $ABCD$ with bases $AB$ and $CD$ is such that the circumcircle of the triangle $BCD$ intersects the line $AD$ in a point $E$, distinct from $A$ and $D$. Prove that the circumcircle oF the triangle $ABE$ is tangent to the line $BC$.

Prove the following theorem: If there are $n$ pairs of different points $P_i$, $i = 1, 2, ..., n$, $n > 2$ in three dimensions space, such that each of them is at a smaller distance from one and the same point $Q$ than any other $P_i$, then $n < 15$.

Let $A',\,B',\,C'$ be points, in which excircles touch corresponding sides of triangle $ABC$. Circumcircles of triangles $A'B'C,\,AB'C',\,A'BC'$ intersect a circumcircle of $ABC$ in points $C_1\ne C,\,A_1\ne A,\,B_1\ne B$ respectively. Prove that a triangle $A_1B_1C_1$ is similar to a triangle, formed by points, in which incircle of $ABC$ touches its sides.

Given three points $P$, $A$, $B$ and a circle such that the lines $PA$ and $PB$ are tangent to the circle at the points $A$ and $B$, respectively. A line through the point $P$ intersects that circle at two points $C$ and $D$. Through the point $B$, draw a line parallel to $PA$; let this line intersect the lines $AC$ and $AD$ at the points $E$ and $F$, respectively. Prove that $BE = BF$.

The sides of rectangle $ABCD$ have lengths 10 and 11. An equilateral triangle is drawn so that no point of the triangle lies outside $ABCD.$ The maximum possible area of such a triangle can be written in the form $p\sqrt{q}-r,$ where $p, q,$ and $r$ are positive integers, and $q$ is not divisible by the square of any prime number. Find $p+q+r.$

[b]p1.[/b] Compute $17^2 + 17 \cdot 7 + 7^2$. [b]p2.[/b] You have $\$1.17$ in the minimum number of quarters, dimes, nickels, and pennies required to make exact change for all amounts up to $\$1.17$. How many coins do you have? [b]p3.[/b] Suppose that there is a $40\%$ chance it will rain today, and a $20\%$ chance it will rain today and tomorrow. If the chance it will rain tomorrow is independent of whether or not it rained today, what is the probability that it will rain tomorrow? (Express your answer as a percentage.) [b]p4.[/b] A number is called boxy if the number of its factors is a perfect square. Find the largest boxy number less than $200$. [b]p5.[/b] Alice, Bob, Carl, and Dave are either lying or telling the truth. If the four of them make the following statements, who has the coin? [i]Alice: I have the coin. Bob: Carl has the coin. Carl: Exactly one of us is telling the truth. Dave: The person who has the coin is male.[/i] [b]p6.[/b] Vicky has a bag holding some blue and some red marbles. Originally $\frac23$ of the marbles are red. After Vicky adds $25$ blue marbles, $\frac34$ of the marbles are blue. How many marbles were originally in the bag? [b]p7.[/b] Given pentagon $ABCDE$ with $BC = CD = DE = 4$, $\angle BCD = 90^o$ and $\angle CDE = 135^o$, what is the length of $BE$? [b]p8.[/b] A Berkeley student decides to take a train to San Jose, stopping at Stanford along the way. The distance from Berkeley to Stanford is double the distance from Stanford to San Jose. From Berkeley to Stanford, the train's average speed is $15$ meters per second. From Stanford to San Jose, the train's average speed is $20$ meters per second. What is the train's average speed for the entire trip? [b]p9.[/b] Find the area of the convex quadrilateral with vertices at the points $(-1, 5)$, $(3, 8)$, $(3,-1)$, and $(-1,-2)$. [b]p10.[/b] In an arithmetic sequence $a_1$, $a_2$, $a_3$, $...$ , twice the sum of the first term and the third term is equal to the fourth term. Find $a_4/a_1$. [b]p11.[/b] Alice, Bob, Clara, David, Eve, Fred, Greg, Harriet, and Isaac are on a committee. They need to split into three subcommittees of three people each. If no subcommittee can be all male or all female, how many ways are there to do this? [b]p12.[/b] Usually, spaceships have $6$ wheels. However, there are more advanced spaceships that have $9$ wheels. Aliens invade Earth with normal spaceships, advanced spaceships, and, surprisingly, bicycles (which have $2$ wheels). There are $10$ vehicles and $49$ wheels in total. How many bicycles are there? [b]p13.[/b] If you roll three regular six-sided dice, what is the probability that the three numbers showing will form an arithmetic sequence? (The order of the dice does matter, but we count both $(1,3, 2)$ and $(1, 2, 3)$ as arithmetic sequences.) [b]p14.[/b] Given regular hexagon $ABCDEF$ with center $O$ and side length $6$, what is the area of pentagon $ABODE$? [b]p15.[/b] Sophia, Emma, and Olivia are eating dinner together. The only dishes they know how to make are apple pie, hamburgers, hotdogs, cheese pizza, and ice cream. If Sophia doesn't eat dessert, Emma is vegetarian, and Olivia is allergic to apples, how many dierent options are there for dinner if each person must have at least one dish that they can eat? [b]p16.[/b] Consider the graph of $f(x) = x^3 + x + 2014$. A line intersects this cubic at three points, two of which have $x$-coordinates $20$ and $14$. Find the $x$-coordinate of the third intersection point. [b]p17.[/b] A frustum can be formed from a right circular cone by cutting of the tip of the cone with a cut perpendicular to the height. What is the surface area of such a frustum with lower radius $8$, upper radius $4$, and height $3$? [b]p18.[/b] A quadrilateral $ABCD$ is dened by the points $A = (2,-1)$, $B = (3, 6)$, $C = (6, 10)$ and $D = (5,-2)$. Let $\ell$ be the line that intersects and is perpendicular to the shorter diagonal at its midpoint. What is the slope of $\ell$? [b]p19.[/b] Consider the sequence $1$, $1$, $2$, $2$, $3$, $3$, $3$, $5$, $5$, $5$, $5$, $5$, $...$ where the elements are Fibonacci numbers and the Fibonacci number $F_n$ appears $F_n$ times. Find the $2014$th element of this sequence. (The Fibonacci numbers are defined as $F_1 = F_2 = 1$ and for $n > 2$, $F_n = F_{n-1}+F_{n-2}$.) [b]p20.[/b] Call a positive integer top-heavy if at least half of its digits are in the set $\{7, 8, 9\}$. How many three digit top-heavy numbers exist? (No number can have a leading zero.) PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

On the extension of the hypotenuse $AB$ of the right-angled triangle $ABC$, the point $D$ after the point B is marked so that $DC = 2BC$. Let the point $H$ be the foot of the altitude dropped from the vertex $C$. If the distance from the point $H$ to the side $BC$ is equal to the length of the segment $HA$, prove that $\angle BDC = 18$.

Let $ABC$ be a triangle with $AB < AC$ and incircle $(I)$ tangent to $BC$ at $D$. Take $K$ on $AD$ such that $CD = CK$. Suppose that $AD$ cuts $(I)$ at $G$ and $BG$ cuts $CK$ at $L$. Prove that K is the midpoint of $CL$.

In a triangle with sides $a,b,c,t_a,t_b,t_c$ are the corresponding medians and $D$ the diameter of the circumcircle. Prove that \[\frac{a^2+b^2}{t_c}+\frac{b^2+c^2}{t_a}+\frac{c^2+a^2}{t_b}\le 6D\]

$\triangle{ABC}$ can cover a convex polygon $M$.Prove that there exsit a triangle which is congruent to $\triangle{ABC}$ such that it can also cover $M$ and has one side line paralel to or superpose one side line of $M$.

Let $n_1, n_2$ be positive integers. Consider in a plane $E$ two disjoint sets of points $M_1$ and $M_2$ consisting of $2n_1$ and $2n_2$ points, respectively, and such that no three points of the union $M_1 \cup M_2$ are collinear. Prove that there exists a straightline $g$ with the following property: Each of the two half-planes determined by $g$ on $E$ ($g$ not being included in either) contains exactly half of the points of $M_1$ and exactly half of the points of $M_2.$

Consider the sequences of six positive integers $a_1,a_2,a_3,a_4,a_5,a_6$ with the properties that $a_1=1$, and if for some $j > 1$, $a_j = m > 1$, then $m-1$ appears in the sequence $a_1,a_2,\dots,a_{j-1}$. Such sequences include $1,1,2,1,3,2$ and $1,2,3,1,4,1$ but not $1,2,2,4,3,2$. How many such sequences of six positive integers are there?

Let $ABCD$ be a convex quadrilateral with perpendicular diagonals. If $AB = 20, BC = 70$ and $CD = 90$, then what is the value of $DA$?

The longest diagonal of a convex hexagon is $2$. Is there necessarily a side or diagonal in this hexagon whose length does not exceed $1$?

Let $ABC$ be a triangle such that midpoints of three altitudes are collinear. If the largest side of triangle is $10$, what is the largest possible area of the triangle? $ \textbf{(A)}\ 20 \qquad\textbf{(B)}\ 25 \qquad\textbf{(C)}\ 30 \qquad\textbf{(D)}\ 40 \qquad\textbf{(E)}\ 50 $

Given triangle $ ABC$ with medians $ AE$, $ BF$, $ CD$; $ FH$ parallel and equal to $ AE$; $ BH$ and $ HE$ are drawn; $ FE$ extended meets $ BH$ in $ G$. Which one of the following statements is not necessarily correct? $ \textbf{(A)}\ AEHF \text{ is a parallelogram} \qquad \textbf{(B)}\ HE\equal{}HG \\ \textbf{(C)}\ BH\equal{}DC \qquad \textbf{(D)}\ FG\equal{}\frac{3}{4}AB \qquad \textbf{(E)}\ FG\text{ is a median of triangle }BFH$