In an isosceles trapezoid, the parallel bases have lengths $\log3$ and $\log192$, and the altitude to these bases has length $\log16$. The perimeter of the trapezoid can be written in the form $\log2^p3^q$, where $p$ and $q$ are positive integers. Find $p+q$.

Let $ABC$ be a scalene triangle and $l_0$ be a line that is tangent to the circumcircle of $ABC$ at point $A$. Let $l$ be a variable line which is parallel to line $l_0$. Let $l$ intersect segment $AB$ and $AC$ at the point $X$, $Y$ respectively. $BY$ and $CX$ intersects at point $T$ and the line $AT$ intersects the circumcirle of $ABC$ at $Z$. Prove that as $l$ varies, circumcircle of $XYZ$ passes through a fixed point.

The incircle $ \omega $ of a quadrilateral $ ABCD $ touches $ AB, BC, CD, DA $ at $ E, F, G, H $, respectively. Choose an arbitrary point $ X$ on the segment $ AC $ inside $ \omega $. The segments $ XB, XD $ meet $ \omega $ at $ I, J $ respectively. Prove that $ FJ, IG, AC $ are concurrent.

Let $ABCD$ be a convex quadrilateral with $\angle{BAD} = 2\angle{BCD}$ and $AB = AD$. Let $P$ be a point such that $ABCP$ is a parallelogram. Prove that $CP = DP$.

Let $n$ be a positive integer. E. Chen and E. Chen play a game on the $n^2$ points of an $n \times n$ lattice grid. They alternately mark points on the grid such that no player marks a point that is on or inside a non-degenerate triangle formed by three marked points. Each point can be marked only once. The game ends when no player can make a move, and the last player to make a move wins. Determine the number of values of $n$ between $1$ and $2013$ (inclusive) for which the first player can guarantee a win, regardless of the moves that the second player makes. [i]Ray Li[/i]

Let $ABCD$ be a quadrilateral with each interior angle less than $180^o$. Show that if $A, B, C, D$ do not lie on a circle, then $AB \cdot CD + AD\cdot BC > AC \cdot BD$

$AD$ and $BE$ are altitudes of an acute triangle $ABC$. From $D$, perpendiculars are dropped to $AB$ at $G$ and $AC$ at $K$. From $E$, perpendiculars are dropped to $AB$ at $F$ and $BC$ at $H$. Prove that $FG$ is parallel to $HK$ and $FK = GH$.

Given two circles ${\omega _1},{\omega _2}$, $S$ denotes all $\Delta ABC$ satisfies that ${\omega _1}$ is the circumcircle of $\Delta ABC$, ${\omega _2}$ is the $A$- excircle of $\Delta ABC$ , ${\omega _2}$ touches $BC,CA,AB$ at $D,E,F$. $S$ is not empty, prove that the centroid of $\Delta DEF$ is a fixed point.

Let $ABC$ be an isosceles triangle with base $AB$. We choose a point $P$ inside the triangle on altitude through $C$. The circle with diameter $CP$ intersects the straight line through $B$ and $P$ again at the point $D_P$ and the Straight through $A$ and $C$ one more time at point $E_P$. Prove that there is a point $F$ such that for any choice of $P$ the points $D_P , E_P$ and $F$ lie on a straight line. [i](Walther Janous)[/i]

Let $ABC$ be a triangle whose angles $\alpha=\angle CAB$ and $\beta=\angle CBA$ are greater than $45^{\circ}$. Above the side $AB$ a right isosceles triangle $ABR$ is constructed with $AB$ as the hypotenuse, such that $R$ is inside the triangle $ABC$. Analogously we construct above the sides $BC$ and $AC$ the right isosceles triangles $CBP$ and $ACQ$, right at $P$ and in $Q$, but with these outside the triangle $ABC$. Prove that $CQRP$ is a parallelogram.

A ray of light reflects from the rays of a given angle. A ray that enters the vertex of the angle is absorbed. Prove that there is a natural number $n$ such that any ray can reflect at most $n$ times

Let $E$ and $F$ be points on side $BC$ of a triangle $\vartriangle ABC$. Points $K$ and $L$ are chosen on segments $AB$ and $AC$, respectively, so that $EK \parallel AC$ and $FL \parallel AB$. The incircles of $\vartriangle BEK$ and $\vartriangle CFL$ touches segments $AB$ and $AC$ at $X$ and $Y$ , respectively. Lines $AC$ and $EX$ intersect at $M$, and lines $AB$ and $FY$ intersect at $N$. Given that $AX = AY$, prove that $MN \parallel BC$.

The twelve-sided figure shown has been drawn on $1 \text{ cm}\times 1 \text{ cm}$ graph paper. What is the area of the figure in $\text{cm}^2$? [asy] unitsize(8mm); for (int i=0; i<7; ++i) { draw((i,0)--(i,7),gray); draw((0,i+1)--(7,i+1),gray); } draw((1,3)--(2,4)--(2,5)--(3,6)--(4,5)--(5,5)--(6,4)--(5,3)--(5,2)--(4,1)--(3,2)--(2,2)--cycle,black+2bp); [/asy] $\textbf{(A) } 12 \qquad \textbf{(B) } 12.5 \qquad \textbf{(C) } 13 \qquad \textbf{(D) } 13.5 \qquad \textbf{(E) } 14$

A net for a polyhedron is cut along an edge to give two [b]pieces[/b]. For example, we may cut a cube net along the red edge to form two pieces as shown. [asy] size(5.5cm); draw((1,0)--(1,4)--(2,4)--(2,0)--cycle); draw((1,1)--(2,1)); draw((1,2)--(2,2)); draw((1,3)--(2,3)); draw((0,1)--(3,1)--(3,2)--(0,2)--cycle); draw((2,1)--(2,2),red+linewidth(1.5)); draw((3.5,2)--(5,2)); filldraw((4.25,2.2)--(5,2)--(4.25,1.8)--cycle,black); draw((6,1.5)--(10,1.5)--(10,2.5)--(6,2.5)--cycle); draw((7,1.5)--(7,2.5)); draw((8,1.5)--(8,2.5)); draw((9,1.5)--(9,2.5)); draw((7,2.5)--(7,3.5)--(8,3.5)--(8,2.5)--cycle); draw((11,1.5)--(11,2.5)--(12,2.5)--(12,1.5)--cycle); [/asy] Are there two distinct polyhedra for which this process may result in the same two pairs of pieces? If you think the answer is no, prove that no pair of polyhedra can result in the same two pairs of pieces. If you think the answer is yes, provide an example; a clear example will suffice as a proof.

A point $M$ is chosen inside the square $ABCD$ in such a way that $\angle MAC = \angle MCD = x$ . Find $\angle ABM$.

In the complex plane consider the unit circle with the origin as its center. Furthermore, consider inscribed regular 17-gon with one of its vertices being $1+0i.$ How many of its vertices lie in the (open) unit disc centered in $\sqrt{3/2}(1+i)$?

It is known that the graph of the function $y = f(x)$ after a rotation of $45^o$ around a certain point turns into the graph of the function $y = x^3 + ax^2 + 19x + 97$. At what $a$ is this possible?

$H$ is the orthocenter of an acute triangle $ABC$, and let $M$ be the midpoint of $BC$. Suppose $(AH)$ meets $AB$ and $AC$ at $D,E$ respectively. $AH$ meets $DE$ at $P$, and the line through $H$ perpendicular to $AH$ meets $DM$ at $Q$. Prove that $P,Q,B$ are collinear.

Let ABC be a triangle with $\angle A = 90^o$, $\angle B = 75^o$, and $AB = 2$. Points $P$ and $Q$ of the sides $AC$ and $BC$ respectively, are such that $\angle APB = \angle CPQ$ and $\angle BQA = \angle CQP$. Calculate the lenght of $QA$.

Let $ABC$ be a triangle with incenter $I$. The points $D$ and $E$ lie on the segments $CA$ and $BC$ respectively, such that $CD = CE$. Let $F$ be a point on the segment $CD$. Prove that the quadrilateral $ABEF$ is circumscribable if and only if the quadrilateral $DIEF$ is cyclic. [i]Proposed by Dorlir Ahmeti, Albania[/i]

Let $ ABCDE$ be a convex pentagon such that \[ \angle BAC \equal{} \angle CAD \equal{} \angle DAE\qquad \text{and}\qquad \angle ABC \equal{} \angle ACD \equal{} \angle ADE. \]The diagonals $BD$ and $CE$ meet at $P$. Prove that the line $AP$ bisects the side $CD$. [i]Proposed by Zuming Feng, USA[/i]

In triangle $ABC$, the incircle touches sides $BC,CA,AB$ at $D,E,F$ respectively. Assume there exists a point $X$ on the line $EF$ such that \[\angle{XBC} = \angle{XCB} = 45^{\circ}.\] Let $M$ be the midpoint of the arc $BC$ on the circumcircle of $ABC$ not containing $A$. Prove that the line $MD$ passes through $E$ or $F$. United Kingdom

Find the smallest positive integer whose cube ends in 888.

.Let $CH$ be an altitude of right-angled triangle $ABC$ ($\angle C = 90^o$), $HA_1$, $HB_1$ be the bisectors of angles $CHB$, $AHC$ respectively, and $E, F$ be the midpoints of $HB_1$ and $HA_1$ respectively. Prove that the lines $AE$ and $BF$ meet on the bisector of angle $ACB$.

[u]Round 1[/u] [b]p1.[/b] Ravi has a bag with $100$ slips of paper in it. Each slip has one of the numbers $3, 5$, or $7$ written on it. Given that half of the slips have the number $3$ written on them, and the average of the values on all the slips is $4.4$, how many slips have $7$ written on them? [b]p2.[/b] In triangle $ABC$, point $D$ lies on side $AB$ such that $AB \perp CD$. It is given that $\frac{CD}{BD}=\frac12$, $AC = 29$, and $AD = 20$. Find the area of triangle $BCD$. [b]p3.[/b] Compute $(123 + 4)(123 + 5) - 123\cdot 132$. [u]Round 2[/u] [b]p4. [/b] David is evaluating the terms in the sequence $a_n = (n + 1)^3 - n^3$ for $n = 1, 2, 3,....$ (that is, $a_1 = 2^3 - 1^3$ , $a_2 = 3^3 - 2^3$, $a_3 = 4^3 - 3^3$, and so on). Find the first composite number in the sequence. (An positive integer is composite if it has a divisor other than 1 and itself.) [b]p5.[/b] Find the sum of all positive integers strictly less than $100$ that are not divisible by $3$. [b]p6.[/b] In how many ways can Alex draw the diagram below without lifting his pencil or retracing a line? (Two drawings are different if the order in which he draws the edges is different, or the direction in which he draws an edge is different). [img]https://cdn.artofproblemsolving.com/attachments/9/6/9d29c23b3ca64e787e717ceff22d45851ae503.png[/img] [u]Round 3[/u] [b]p7.[/b] Fresh Mann is a $9$th grader at Euclid High School. Fresh Mann thinks that the word vertices is the plural of the word vertice. Indeed, vertices is the plural of the word vertex. Using all the letters in the word vertice, he can make $m$ $7$-letter sequences. Using all the letters in the word vertex, he can make $n$ $6$-letter sequences. Find $m - n$. [b]p8.[/b] Fresh Mann is given the following expression in his Algebra $1$ class: $101 - 102 = 1$. Fresh Mann is allowed to move some of the digits in this (incorrect) equation to make it into a correct equation. What is the minimal number of digits Fresh Mann needs to move? [b]p9.[/b] Fresh Mann said, “The function $f(x) = ax^2+bx+c$ passes through $6$ points. Their $x$-coordinates are consecutive positive integers, and their y-coordinates are $34$, $55$, $84$, $119$, $160$, and $207$, respectively.” Sophy Moore replied, “You’ve made an error in your list,” and replaced one of Fresh Mann’s numbers with the correct y-coordinate. Find the corrected value. [u]Round 4[/u] [b]p10.[/b] An assassin is trying to find his target’s hotel room number, which is a three-digit positive integer. He knows the following clues about the number: (a) The sum of any two digits of the number is divisible by the remaining digit. (b) The number is divisible by $3$, but if the first digit is removed, the remaining two-digit number is not. (c) The middle digit is the only digit that is a perfect square. Given these clues, what is a possible value for the room number? [b]p11.[/b] Find a positive real number $r$ that satisfies $$\frac{4 + r^3}{9 + r^6}=\frac{1}{5 - r^3}- \frac{1}{9 + r^6}.$$ [b]p12.[/b] Find the largest integer $n$ such that there exist integers $x$ and $y$ between $1$ and $20$ inclusive with $$\left|\frac{21}{19} -\frac{x}{y} \right|<\frac{1}{n}.$$ PS. You had better use hide for answers. Last rounds have been posted [url=https://artofproblemsolving.com/community/c4h2784267p24464980]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].