Let $ABCD$ be a convex quadrilateral whose sides $AD$ and $BC$ are not parallel. Suppose that the circles with diameters $AB$ and $CD$ meet at points $E$ and $F$ inside the quadrilateral. Let $\omega_E$ be the circle through the feet of the perpendiculars from $E$ to the lines $AB,BC$ and $CD$. Let $\omega_F$ be the circle through the feet of the perpendiculars from $F$ to the lines $CD,DA$ and $AB$. Prove that the midpoint of the segment $EF$ lies on the line through the two intersections of $\omega_E$ and $\omega_F$. [i]Proposed by Carlos Yuzo Shine, Brazil[/i]

Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. Which of the following is closest to how much shorter Silvia's trip was, compared to Jerry's trip? $\textbf{(A)}\ 30 \%\qquad\textbf{(B)}\ 40 \%\qquad\textbf{(C)}\ 50 \%\qquad\textbf{(D)}\ 60 \%\qquad\textbf{(E)}\ 70 \%$

In a given group of people $\mathcal{F}$, each member has at least two acquaintances from $\mathcal{F}$. Moreover, for each cycle $A_{1} \leftrightarrow A_{2} \leftrightarrow ... \leftrightarrow A_{n} \leftrightarrow A_{1}$ in $\mathcal{F}$ (here '$X \leftrightarrow Y$' means that $X$ and $Y$ are acquaintances), each $A_i$ knows exactly two other members $A_j$ of the cycle. Prove that there exist $X, Y \in \mathcal{F}$ such that each of them has exactly two acquaintances in $\mathcal{F}$, and $X, Y$ have at least one common acquaintance in the group. [i]Authored by Mirko Petrusevski[/i]

Pyramid $OABCD$ has square base $ABCD,$ congruent edges $\overline{OA}, \overline{OB}, \overline{OC},$ and $\overline{OD},$ and $\angle AOB=45^\circ.$ Let $\theta$ be the measure of the dihedral angle formed by faces $OAB$ and $OBC.$ Given that $\cos \theta=m+\sqrt{n},$ where $m$ and $n$ are integers, find $m+n.$

In right angled triangle $ABC$ point $D$ is midpoint of hypotenuse, and $E$ and $F$ are points on shorter sides $AC$ and $BC$, respectively, such that $DE \perp DF$. Prove that $EF^2=AE^2+BF^2$

A student is asked to measure $30.0 {\text{g}}$ of methanol $(d=0.7914 \text{g/mL at 25}^{\circ}\text{C})$ but has only a graduated cylinder with which to measure it. What volume of methanol should the student use to obtain the required ${30.0 \text{g}}$? ${ \textbf{(A)}\ 23.7 \text{mL} \qquad\textbf{(B)}\ 30.0 \text{mL} \qquad\textbf{(C)}\ 32.4 \text{mL} \qquad\textbf{(D)}\ 37.9 \text{mL} }$

Prove that $2^{147} - 1$ is divisible by $343$.

Find the sum of the real roots of the polynomial \[ \prod_{k=1}^{100} \left( x^2-11x+k \right) = \left( x^2-11x+1 \right)\left( x^2-11x+2 \right)\dots\left(x^2-11x+100\right). \][i]Proposed by Evan Chen[/i]

Given is a square $ABCD$ with side length $1$. Points $K$, $L$, $M$, and $N$, distinct from the vertices of the square, lie on segments $AB$, $BC$, $CD$, and $DA$, respectively. Prove that the perimeter of at least one of the triangles $ANK$, $BKL$, $CLM$, $DMN$ is less than $2$.

Let $F_k(a,b)=(a+b)^k-a^k-b^k$ and let $S={1,2,3,4,5,6,7,8,9,10}$. For how many ordered pairs $(a,b)$ with $a,b\in S$ and $a\leq b$ is $\frac{F_5(a,b)}{F_3(a,b)}$ an integer?

$E$ and $F$ are interior points of convex quadrilateral $ABCD$ such that $AE = BE$, $CE = DE$, $\angle AEB = \angle CED$, $AF = DF$, $BF = CF$, $\angle AFD = \angle BFC$. Prove that $\angle AFD + \angle AEB = \pi$.

Determine a) the smallest number b) the biggest number $n \ge 3$ of non-negative integers $x_1, x_2, ... , x_n$, having the sum $2011$ and satisfying: $x_1 \le | x_2 - x_3 | , x_2 \le | x_3 - x_4 | , ... , x_{n-2} \le | x_{n-1} -x_n | , x_{n-1} \le | x_n - x_1 |$ and $x_n \le | x_1 - x_2 | $.

Find the number of distinct permutations of $ITEST$. [i](.1 point)[/i]

If $p$ and $q$ are positive numbers with $p+q = 1$, knowing that any real numbers $x,y$ satisfy $(x-y)^2 \ge 0$, show that $\frac{x+y}{2} \ge \sqrt{xy}$, $\frac{x^2+y^2}{2} \ge \big(\frac{x+y}{2}\big)^2$, $\big(p+\frac{1}{p}\big)^2+\big(q+\frac{1}{q}\big)^2 \ge \frac{25}{2}$

For $1\leq i\leq 215$ let $a_i=\frac{1}{2^i}$ and $a_{216}=\frac{1}{2^{215}}$. Let $x_1,x_2,\ldots,x_{216}$ be positive real numbers such that \[ \sum\limits_{i=1}^{216} x_i=1 \text{\quad and \quad} \sum\limits_{1\leq i<j \leq 216} x_ix_j = \frac{107}{215}+ \sum\limits_{i=1}^{216} \frac{a_ix_i^2}{2(1-a_i)}.\] The maximum possible value of $x_2=\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Consider a triangle $ABC$ with $AB = 13, BC = 14, CA = 15$. A line perpendicular to $BC$ divides the interior of $\vartriangle BC$ into two regions of equal area. Suppose that the aforesaid perpendicular cuts $BC$ at $D$, and cuts $\vartriangle ABC$ again at $E$. If $L$ is the length of the line segment $DE$, find $L^2$.

How many natural number triples $(x,y,z)$ are there such that $xyz = 10^6$? $ \textbf{(A)}\ 568 \qquad\textbf{(B)}\ 784 \qquad\textbf{(C)}\ 812 \qquad\textbf{(D)}\ 816 \qquad\textbf{(E)}\ 824 $

Let $ABC$ be an acute triangle, and let $H_A$, $H_B$, and $H_C$ be the feet of the altitudes relative to vertices $A$, $B$, and $C$, respectively. Define $I_A$, $I_B$, and $I_C$ as the incenters of triangles $AH_B H_C$, $BH_C H_A$, and $CH_A H_B$, respectively. Let $T_A$, $T_B$, and $T_C$ be the intersection of the incircle of triangle $ABC$ with $BC$, $CA$, and $AB$, respectively. Prove that the triangles $I_A I_B I_C$ and $T_A T_B T_C$ are congruent.

Let $ S$ be the set of permutations of the sequence $ 1, 2, 3, 4, 5$ for which the first term is not $ 1$. A permutation is chosen randomly from $ S$. The probability that the second term is $ 2$, in lowest terms, is $ a/b$. What is $ a \plus{} b$? $ \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 11 \qquad \textbf{(D)}\ 16 \qquad \textbf{(E)}\ 19$

Laila took five math tests, each worth a maximum of 100 points. Laila's score on each test was an integer between 0 and 100, inclusive. Laila received the same score on the first four tests, and she received a higher score on the last test. Her average score on the five tests was 82. How many values are possible for Laila's score on the last test? $\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }9\qquad\textbf{(D) }10\qquad \textbf{(E) }18$

Let skew lines $PM, QN$ be given such that $PM\perp PQ\perp QN$. Let a plane $\sigma\perp PQ$ containing the midpoint $O$ of segment $PQ$ be given and in it a circle $k$ with center $O$ and given radius $r$. Consider all segments $XY$ with endpoint $X, Y$ on lines $PM, QN$, respectively, which contain a point of $k$. Show that segments $XY$ have the same length. Find the locus of all such points $X$.

There is a long-standing conjecture that there is no number with $2n + 1$ instances in Pascal’s triangle for $n \ge 2$. Assuming this is true, for how many $n \le 100, 000$ are there exactly $3$ instances of $n$ in Pascal’s triangle?

Let $ x$, $ y$, $ z$ be integers such that \[ \begin{array}{l} {x \minus{} 3y \plus{} 2z \equal{} 1} \\ {2x \plus{} y \minus{} 5z \equal{} 7} \end{array} \] Then $ z$ can be $\textbf{(A)}\ 3^{111} \qquad\textbf{(B)}\ 4^{111} \qquad\textbf{(C)}\ 5^{111} \qquad\textbf{(D)}\ 6^{111} \qquad\textbf{(E)}\ \text{None}$

[b]p1.[/b] What is the smallest positive integer divisible by $20$, $12$, and $13$? [b]p2.[/b] Two circles of radius $5$ are placed in the plane such that their centers are $7$ units apart. What is the largest possible distance between a point on one circle and a point on the other? [b]p3.[/b] In a magic square, all the numbers in the rows, columns, and diagonals sum to the same value. How many $2\times 2$ magic squares containing the integers $\{1, 2, 3, 4\}$ are there? [b]p4.[/b] Ethan's sock drawer contains two pairs of white socks and one pair of red socks. Ethan picks two socks at random. What is the probability that he picks two white socks? [b]p5.[/b] The sum of the time on a digital clock is the sum of the digits displayed on the screen. For example, the sum of the time at $10:23$ would be $6$. Assuming the clock is a $12$ hour clock, what is the greatest possible positive difference between the sum of the time at some time and the sum of the time one minute later? [b]p6.[/b] Given the expression $1 \div 2 \div 3 \div 4$, what is the largest possible resulting value if one were to place parentheses $()$ somewhere in the expression? [b]p7.[/b] At a convention, there are many astronomers, astrophysicists, and cosmologists. At $first$, all the astronomers and astrophysicists arrive. At this point, $\frac35$ of the people in the room are astronomers. Then, all the cosmologists come, so now, $30\%$ of the people in the room are astrophysicists. What fraction of the scientists are cosmologists? [b]p8.[/b] At $10:00$ AM, a minuteman starts walking down a $1200$-step stationary escalator at $40$ steps per minute. Halfway down, the escalator starts moving up at a constant speed, while the minuteman continues to walk in the same direction and at the same pace that he was going before. At $10:55$ AM, the minuteman arrives back at the top. At what speed is the escalator going up, in steps per minute? [b]p9.[/b] Given that $x_1 = 57$, $x_2 = 68$, and $x_3 = 32$, let $x_n = x_{n-1} -x_{n-2} +x_{n-3}$ for $n \ge 4$. Find $x_{2013}$. [b]p10.[/b] Two squares are put side by side such that one vertex of the larger one coincides with a vertex of the smaller one. The smallest rectangle that contains both squares is drawn. If the area of the rectangle is $60$ and the area of the smaller square is $24$, what is the length of the diagonal of the rectangle? [b]p11.[/b] On a dield trip, $2$ professors, $4$ girls, and $4$ boys are walking to the forest to gather data on butterflies. They must walk in a line with following restrictions: one adult must be the first person in the line and one adult must be the last person in the line, the boys must be in alphabetical order from front to back, and the girls must also be in alphabetical order from front to back. How many such possible lines are there, if each person has a distinct name? [b]p12.[/b] Flatland is the rectangle with vertices $A, B, C$, and $D$, which are located at $(0, 0)$, $(0, 5)$, $(5, 5)$, and $(5, 0)$, respectively. The citizens put an exact map of Flatland on the rectangular region with vertices $(1, 2)$, $(1, 3)$, $(2, 3)$, and $(2, 2)$ in such a way so that the location of $A$ on the map lies on the point $(1, 2)$ of Flatland, the location of $B$ on the map lies on the point $(1, 3)$ of Flatland, the location of C on the map lies on the point $(2, 3)$ of Flatland, and the location of D on the map lies on the point $(2, 2)$ of Flatland. Which point on the coordinate plane is thesame point on the map as where it actually is on Flatland? [b]p13.[/b] $S$ is a collection of integers such that any integer $x$ that is present in $S$ is present exactly $x$ times. Given that all the integers from $1$ through $22$ inclusive are present in $S$ and no others are, what is the average value of the elements in $S$? [b]p14.[/b] In rectangle $PQRS$ with $PQ < QR$, the angle bisector of $\angle SPQ$ intersects $\overline{SQ}$ at point $T$ and $\overline{QR }$ at $U$. If $PT : TU = 3 : 1$, what is the ratio of the area of triangle $PTS$ to the area of rectangle $PQRS$? [b]p15.[/b] For a function $f(x) = Ax^2 + Bx + C$, $f(A) = f(B)$ and $A + 6 = B$. Find all possible values of $B$. [b]p16.[/b] Let $\alpha$ be the sum of the integers relatively prime to $98$ and less than $98$ and $\beta$ be the sum of the integers not relatively prime to $98$ and less than $98$. What is the value of $\frac{\alpha}{\beta}$ ? [b]p17.[/b] What is the value of the series $\frac{1}{3} + \frac{3}{9} + \frac{6}{27} + \frac{10}{81} + \frac{15}{243} + ...$? [b]p18.[/b] A bug starts at $(0, 0)$ and moves along lattice points restricted to $(i, j)$, where $0 \le i, j \le 2$. Given that the bug moves $1$ unit each second, how many different paths can the bug take such that it ends at $(2, 2)$ after $8$ seconds? [b]p19.[/b] Let $f(n)$ be the sum of the digits of $n$. How many different values of $n < 2013$ are there such that $f(f(f(n))) \ne f(f(n))$ and $f(f(f(n))) < 10$? [b]p20.[/b] Let $A$ and $B$ be points such that $\overline{AB} = 14$ and let $\omega_1$ and $\omega_2$ be circles centered at $A$ and $B$ with radii $13$ and $15$, respectively. Let $C$ be a point on $\omega_1$ and $D$ be a point on $\omega_2$ such that $\overline{CD}$ is a common external tangent to $\omega_1$ and $\omega_2$. Let $P$ be the intersection point of the two circles that is closer to $\overline{CD}$. If $M$ is the midpoint of $\overline{CD}$, what is the length of segment $\overline{PM}$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be a triangle and let $AD,BE,CF$ be its altitudes . $FA_{1},DB_{1},EC_{1}$ are perpendicular segments to $BC,AC,AB$ respectively. Prove that : $ABC$~$A_{1}B_{1}C_{1}$