Let $a$ and $b$ be parallel lines with $50$ distinct points marked on $a$ and $50$ distinct points marked on $b$. Find the greatest possible number of acute-angled triangles all of whose vertices are marked.

Point $D$ lies inside triangle $ABC$ such that $\angle DAC = \angle DCA = 30^{\circ}$ and $\angle DBA = 60^{\circ}$. Point $E$ is the midpoint of segment $BC$. Point $F$ lies on segment $AC$ with $AF = 2FC$. Prove that $DE \perp EF$.

Prove that if a triangle has integer side lengths and the area (in square units) equals the perimeter (in units), then the perimeter is not a prime number.

Let $\vartriangle ABC$ be a right-angled triangle with $\angle A = 90^o$ and circumcircle $\Gamma$. The inscribed circle is tangent to $BC$ in point $D$. Let $E$ be the midpoint of the arc $AB$ of $\Gamma$ not containing $C$ and let $F$ be the midpoint of the arc $AC$ of $\Gamma$ not containing $B$. (a) Prove that $\vartriangle ABC \sim \vartriangle DEF$. (b) Prove that $EF$ goes through the points of tangency of the incircle to $AB$ and $AC$.

Let $W$ be a regular octahedron and $O$ be its center. In a plane $P$ containing $O$ circles $k_1(O,r_1)$ and $k_2(O,r_2)$ are chosen so that $k_1 \subset P\cap W \subset k_2$. Prove that $\frac{r_1}{r_2}\le \frac{\sqrt3}{2}$

Let $ABC$ be a triangle with circumcircle $\Gamma$ and $AB\neq AC$. Let $D$ and $E$ lie on the arc $BC$ of $\Gamma$ not containing $A$ such that $\angle BAE=\angle DAC$. Let the incenters of $BAE$ and $CAD$ be $X$ and $Y$, respectively, and let the external tangents of the incircles of $BAE$ and $CAD$ intersect at $Z$. Prove that $Z$ lies on the common chord of $\Gamma$ and the circumcircle of $AXY$.

Let $AP,BQ$ and $CR$ be altitudes of an acute-angled triangle $ABC$. Show that for any point $X$ inside the triangle $PQR$ there exists a tetrahedron $ABCD$ such that $X$ is the point on the face $ABC$ at the greatest distance from $D$ (measured along the surface of the tetrahedron).

Let $ABC$ be an acute triangle with $|AB| \neq |AC|$ and the midpoints of segments $[AB]$ and $[AC]$ be $D$ resp. $E$. The circumcircles of the triangles $BCD$ and $BCE$ intersect the circumcircle of triangle $ADE$ in $P$ resp. $Q$ with $P \neq D$ and $Q \neq E$. Prove $|AP|=|AQ|$. [i](Notation: $|\cdot|$ denotes the length of a segment and $[\cdot]$ denotes the line segment.)[/i]

Let $ABC$ be a triangle. Circle $\Gamma$ passes through point $A$, meets segments $AB$ and $AC$ again at $D$ and $E$ respectively, and intersects segment $BC$ at $F$ and $G$ such that $F$ lies between $B$ and $G$. The tangent to circle $(BDF)$ at $F$ and the tangent to circle $(CEG)$ at $G$ meet at $T$. Suppose that points $A$ and $T$ are distinct. Prove that line $AT$ is parallel to $BC$.

Take two points $A\ (-1,\ 0),\ B\ (1,\ 0)$ on the $xy$-plane. Let $F$ be the figure by which the whole points $P$ on the plane satisfies $\frac{\pi}{4}\leq \angle{APB}\leq \pi$ and the figure formed by $A,\ B$. Answer the following questions: (1) Illustrate $F$. (2) Find the volume of the solid generated by a rotation of $F$ around the $x$-axis.

[b]a)[/b] Show that the sum of the squares of the minimum distances from a point that is situated on a sphere to the faces of the cube that circumscribe the sphere doesn't depend on the point. [b]b)[/b] Show that the sum of the cubes of the minimum distances from a point that is situated on a sphere to the faces of the cube that circumscribe the sphere doesn't depend on the point. [i]Alexandru Sergiu Alamă[/i]

In unequal triangle $ABC$ bisector $AK$ was drawn. Diameter $XY$ of its circumcircle is perpendicular to $AK$ (order of points on circumcircle is $B-X-A-Y-C$). A circle, passing on points $X$ and $Y$, intersect segments $BK$ and $CK$ in points $T$ and $Z$ respectively. Prove that if $KZ=KT$, then $XT \perp YZ$.

In the below figure, there is a regular hexagon and three squares whose sides are equal to $4$ cm. Let $M,N$, and $P$ be the centers of the squares. The perimeter of the triangle $MNP$ can be written in the form $a + b\sqrt3$ (cm), where $a, b$ are integers. Compute the value of $a + b$. [img]https://cdn.artofproblemsolving.com/attachments/e/8/5996e994d4bbed8d3b3269d3e38fc2ec5d2f0b.png[/img]

[u]Round 1[/u] [b]p1.[/b] Suppose that gold satisfies the relation $p = v + v^2$, where $p$ is the price and $v$ is the volume. How many pieces of gold with volume $1$ can be bought for the price of a piece with volume $2$? [b]p2.[/b] Find the smallest prime number with each digit greater or equal to $8$. [b]p3.[/b] What fraction of regular hexagon $ZUMING$ is covered by both quadrilateral $ZUMI$ and quadrilateral$ MING$? [u]Round 2[/u] [b]p4.[/b] The two smallest positive integers expressible as the sum of two (not necessarily positive) perfect cubes are $1 = 1^3 +0^3$ and $2 = 1^3 +1^3$. Find the next smallest positive integer expressible in this form. [b]p5.[/b] In how many ways can the numbers $1, 2, 3,$ and $4$ be written in a row such that no two adjacent numbers differ by exactly $1$? [b]p6.[/b] A real number is placed in each cell of a grid with $3$ rows and $4$ columns. The average of the numbers in each column is $2016$, and the average of the numbers in each row is a constant $x$. Compute $x$. [u]Round 3[/u] [b]p7.[/b] Fardin is walking from his home to his oce at a speed of $1$ meter per second, expecting to arrive exactly on time. When he is halfway there, he realizes that he forgot to bring his pocketwatch, so he runs back to his house at $2$ meters per second. If he now decides to travel from his home to his office at $x$ meters per second, find the minimum $x$ that will allow him to be on time. [b]p8.[/b] In triangle $ABC$, the angle bisector of $\angle B$ intersects the perpendicular bisector of $AB$ at point $P$ on segment $AC$. Given that $\angle C = 60^o$, determine the measure of $\angle CPB$ in degrees. [b]p9.[/b] Katie colors each of the cells of a $6\times 6$ grid either black or white. From top to bottom, the number of black squares in each row are $1$, $2$, $3$, $4$, $5$, and $6$, respectively. From left to right, the number of black squares in each column are $6$, $5$, $4$, $3$, $2$, and $1$, respectively. In how many ways could Katie have colored the grid? [u]Round 4[/u] [b]p10.[/b] Lily stands at the origin of a number line. Each second, she either moves $2$ units to the right or $1$ unit to the left. At how many different places could she be after $2016$ seconds? [b]p11.[/b] There are $125$ politicians standing in a row. Each either always tells the truth or always lies. Furthermore, each politician (except the leftmost politician) claims that at least half of the people to his left always lie. Find the number of politicians that always lie. [b]p12.[/b] Two concentric circles with radii $2$ and $5$ are drawn on the plane. What is the side length of the largest square whose area is contained entirely by the region between the two circles? PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2934055p26256296]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For a positive integer $n$, a cubic polynomial $p(x)$ is said to be [i]$n$-good[/i] if there exist $n$ distinct integers $a_1, a_2, \ldots, a_n$ such that all the roots of the polynomial $p(x) + a_i = 0$ are integers for $1 \le i \le n$. Given a positive integer $n$ prove that there exists an $n$-good cubic polynomial.

Let $ABCD$ be a circumscribed quadrilateral. Let $g$ be a line through $A$ which meets the segment $BC$ in $M$ and the line $CD$ in $N$. Denote by $I_1$, $I_2$ and $I_3$ the incenters of $\triangle ABM$, $\triangle MNC$ and $\triangle NDA$, respectively. Prove that the orthocenter of $\triangle I_1I_2I_3$ lies on $g$. [i]Proposed by Nikolay Beluhov, Bulgaria[/i]

a) Each of the side of the convex hexagon is longer than $1$. Does it necessary have a diagonal longer than $2$? b) Each of the main diagonals of the convex hexagon is longer than $2$. Does it necessary have a side longer than $1$?

Let $ABC$ be a triangle with $m (\angle C) = 90^\circ$ and the points $D \in [AC], E\in [BC]$. Inside the triangle we construct the semicircles $C_1, C_2, C_3, C_4$ of diameters $[AC], [BC], [CD], [CE]$ and let $\{C, K\} = C_1 \cap C_2, \{C, M\} =C_3 \cap C_4, \{C, L\} = C_2 \cap C_3, \{C, N\} =C_1 \cap C_4$. Show that points $K, L, M, N$ are concyclic.

Chords $ AB$ and $ CD$ of a circle intersect at a point $ E$ inside the circle. Let $ M$ be an interior point of the segment $ EB$. The tangent line at $ E$ to the circle through $ D$, $ E$, and $ M$ intersects the lines $ BC$ and $ AC$ at $ F$ and $ G$, respectively. If \[ \frac {AM}{AB} \equal{} t, \] find $\frac {EG}{EF}$ in terms of $ t$.

[b]p1.[/b] If Clark wants to divide $100$ pizzas among $25$ people so that each person receives the same number of pizzas, how many pizzas should each person receive? [b]p2.[/b] In a group of $3$ people, every pair of people shakes hands once. How many handshakes occur? [b]p3.[/b] Dylan and Joey have $14$ costumes in total. Dylan gives Joey $4$ costumes, and Joey now has the number of costumes that Dylan had before giving Joey any costumes. How many costumes does Dylan have now? [b]p4.[/b] At Banjo Borger, a burger costs $7$ dollars, a soda costs $2$ dollars, and a cookie costs $3$ dollars. Alex, Connor, and Tony each spent $11$ dollars on their order, but none of them got the same order. If Connor bought the most cookies, how many cookies did Connor buy? [b]p5.[/b] Joey, James, and Austin stand on a large, flat field. If the distance from Joey to James is $30$ and the distance from Austin to James is $18$, what is the minimal possible distance from Joey to Austin? [b]p6.[/b] If the first and third terms of a five-term arithmetic sequence are $3$ and $8$, respectively, what is the sum of all $5$ terms in the sequence? [b]p7.[/b] What is the area of the $S$-shaped figure below, which has constant vertical height $5$ and width $10$? [img]https://cdn.artofproblemsolving.com/attachments/3/c/5bbe638472c8ea8289b63d128cd6b449440244.png[/img] [b]p8.[/b] If the side length of square $A$ is $4$, what is the perimeter of square $B$, formed by connecting the midpoints of the sides of $A$? [b]p9.[/b] The Chan Shun Auditorium at UC Berkeley has room number $2050$. The number of seats in the auditorium is a factor of the room number, and there are between $150$ and $431$ seats, inclusive. What is the sum of all of the possible numbers of seats in Chan Shun Auditorium? [b]p10.[/b] Krishna has a positive integer $x$. He notices that $x^2$ has the same last digit as $x$. If Krishna knows that $x$ is a prime number less than $50$, how many possible values of $x$ are there? [b]p11.[/b] Jing Jing the Kangaroo starts on the number $1$. If she is at a positive integer $n$, she can either jump to $2n$ or to the sum of the digits of $n$. What is the smallest positive integer she cannot reach no matter how she jumps? [b]p12.[/b] Sylvia is $3$ units directly east of Druv and runs twice as fast as Druv. When a whistle blows, Druv runs directly north, and Sylvia runs along a straight line. If they meet at a point a distance $d$ units away from Druv's original location, what is the value of $d$? [b]p13.[/b] If $x$ is a real number such that $\sqrt{x} + \sqrt{10} = \sqrt{x + 20}$, compute $x$. [b]p14.[/b] Compute the number of rearrangments of the letters in $LATEX$ such that the letter $T$ comes before the letter $E$ and the letter $E$ comes before the letter $X$. For example, $TLEAX$ is a valid rearrangment, but $LAETX$ is not. [b]p15.[/b] How many integers $n$ greater than $2$ are there such that the degree measure of each interior angle of a regular $n$-gon is an even integer? [b]p16.[/b] Students are being assigned to faculty mentors in the Berkeley Math Department. If there are $7$ students and $3$ mentors and each student has exactly one mentor, in how many ways can students be assigned to mentors given that each mentor has at least one student? [b]p17.[/b] Karthik has a paper square of side length $2$. He folds the square along a crease that connects the midpoints of two opposite sides (as shown in the left diagram, where the dotted line indicates the fold). He takes the resulting rectangle and folds it such that one of its vertices lands on the vertex that is diagonally opposite. Find the area of Karthik's final figure. [img]https://cdn.artofproblemsolving.com/attachments/1/e/01aa386f6616cafeed5f95ababb27bf24657f6.png[/img] [b]p18.[/b] Sally is inside a pen consisting of points $(a, b)$ such that $0 \le a, b \le 4$. If she is currently on the point $(x, y)$, she can move to either $(x, y + 1)$, $(x, y - 1)$, or $(x + 1, y)$. Given that she cannot revisit any point she has visited before, find the number of ways she can reach $(4, 4)$ from $(0, 0)$. [b]p19.[/b] An ant sits on the circumference of the circular base of a party hat (a cone without a circular base for the ant to walk on) of radius $2$ and height $\sqrt{5}$. If the ant wants to reach a point diametrically opposite of its current location on the hat, what is the minimum possible distance the ant needs to travel? [img]https://cdn.artofproblemsolving.com/attachments/3/4/6a7810b9862fd47106c3c275c96337ef6d23c2.png[/img] [b]p20.[/b] If $$f(x) = \frac{2^{19}x + 2^{20}}{ x^2 + 2^{20}x + 2^{20}}.$$ find the value of $f(1) + f(2) + f(4) + f(8) + ... + f(220)$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABCD$ be a rhombus with angle $\angle A = 60^\circ$. Let $E$ be a point, different from $D$, on the line $AD$. The lines $CE$ and $AB$ intersect at $F$. The lines $DF$ and $BE$ intersect at $M$. Determine the angle $\angle BMD$ as a function of the position of $E$ on $AD.$

In a acute triangle $\triangle ABC$, denote $D, E$ as the foot of the perpendicular from $B$ to $AC$ and $C$ to $AB$. Denote the reflection of $E$ with respect to $AC, BC$ as $S, T$. The circumcircle of $\triangle CST$ hits $AC$ at point $X (\not= C)$. Denote the circumcenter of $\triangle CST$ as $O$. Prove that $XO \perp DE$.

p1. A quadratic equation has the natural roots $a$ and $ b$. Another quadratic equation has roots $ b$ and $c$ with $a\ne c$. If $a$, $ b$, and $c$ are prime numbers less than $15$, how many triplets $(a,b,c)$ that might meet these conditions are there (provided that the coefficient of the quadratic term is equal to $ 1$)? p2. In Indonesia, was formerly known the "Archipelago Fraction''. The [i]Archipelago Fraction[/i] is a fraction $\frac{a}{b}$ such that $a$ and $ b$ are natural numbers with $a < b$. Find the sum of all Archipelago Fractions starting from a fraction with $b = 2$ to $b = 1000$. p3. Look at the following picture. The letters $a, b, c, d$, and $e$ in the box will replaced with numbers from $1, 2, 3, 4, 5, 6, 7, 8$, or $9$, provided that $a,b, c, d$, and $e$ must be different. If it is known that $ae = bd$, how many arrangements are there? [img]https://cdn.artofproblemsolving.com/attachments/f/2/d676a57553c1097a15a0774c3413b0b7abc45f.png[/img] p4. Given a triangle $ABC$ with $A$ as the vertex and $BC$ as the base. Point $P$ lies on the side $CA$. From point $A$ a line parallel to $PB$ is drawn and intersects extension of the base at point $D$. Point $E$ lies on the base so that $CE : ED = 2 :3$. If $F$ is the midpoint between $E$ and $C$, and the area of ​​triangle ABC is equal with $35$ cm$^2$, what is the area of ​​triangle $PEF$? p5. Each side of a cube is written as a natural number. At the vertex of each angle is given a value that is the product of three numbers on three sides that intersect at the vertex. If the sum of all the numbers at the points of the angle is equal to $1001$, find the sum of all the numbers written on the sides of the cube.

If an arc of $ 45^\circ$ on circle $ A$ has the same length as an arc of $ 30^\circ$ on circle $ B$, then the ratio of the area of circle $ A$ to the area of circle $ B$ is $ \textbf{(A)}\ \frac {4}{9} \qquad \textbf{(B)}\ \frac {2}{3} \qquad \textbf{(C)}\ \frac {5}{6} \qquad \textbf{(D)}\ \frac {3}{2} \qquad \textbf{(E)}\ \frac {9}{4}$

In triangle $ABC$ it holds $|BC|= \frac{1}{2}(|AB|+|AC|)$. Let $M$ and $N$ be midpoints of $AB$ and $AC$, and let $I$ be the incenter of $ABC$. Prove that $A, M, I, N$ are concyclic.