Let $ABCD$ be a parallelogram. Let $X \notin AC $ lie inside $ABCD$ so that $\angle AXB = \angle CXD = 90^ {\circ}$ and $\Omega$ denote the circumcircle of $AXC$. Consider a diameter $EF$ of $\Omega$ and assume neither $E, \ X, \ B$ nor $F, \ X, \ D$ are collinear. Let $K \neq X$ be an intersection point of circumcircles of $BXE$ and $DXF$ and $L \neq X$ be such point on $\Omega$ so that $\angle KXL = 90^{\circ}$. Prove that $AB = KL$.

Let $\triangle ABC$ be a triangle and $\omega$ its circumcircle. The exterior angle bisector of $\angle BAC$ intersects $\omega$ at point $D$. Let $X$ be the foot of the altitude from $C$ to $AD$ and let $F$ be the intersection of the internal angle bisector of $\angle BAC$ and $BC$. Show that $BX$ bisects segment $AF$.

[b]p1.[/b] Velociraptor $A$ is located at $x = 10$ on the number line and runs at $4$ units per second. Velociraptor $B$ is located at $x = -10$ on the number line and runs at $3$ units per second. If the velociraptors run towards each other, at what point do they meet? [b]p2.[/b] Let $n$ be a positive integer. There are $n$ non-overlapping circles in a plane with radii $1, 2, ... , n$. The total area that they enclose is at least $100$. Find the minimum possible value of $n$. [b]p3.[/b] How many integers between $1$ and $50$, inclusive, are divisible by $4$ but not $6$? [b]p4.[/b] Let $a \star b = 1 + \frac{b}{a}$. Evaluate $((((((1 \star 1) \star 1) \star 1) \star 1) \star 1) \star 1) \star 1$. [b]p5.[/b] In acute triangle $ABC$, $D$ and $E$ are points inside triangle $ABC$ such that $DE \parallel BC$, $B$ is closer to $D$ than it is to $E$, $\angle AED = 80^o$ , $\angle ABD = 10^o$ , and $\angle CBD = 40^o$. Find the measure of $\angle BAE$, in degrees. [b]p6. [/b]Al is at $(0, 0)$. He wants to get to $(4, 4)$, but there is a building in the shape of a square with vertices at $(1, 1)$, $(1, 2)$, $(2, 2)$, and $(2, 1)$. Al cannot walk inside the building. If Al is not restricted to staying on grid lines, what is the shortest distance he can walk to get to his destination? [b]p7. [/b]Point $A = (1, 211)$ and point $B = (b, 2011)$ for some integer $b$. For how many values of $b$ is the slope of $AB$ an integer? [b]p8.[/b] A palindrome is a number that reads the same forwards and backwards. For example, $1$, $11$ and $141$ are all palindromes. How many palindromes between $1$ and 1000 are divisible by $11$? [b]p9.[/b] Suppose $x, y, z$ are real numbers that satisfy: $$x + y - z = 5$$ $$y + z - x = 7$$ $$z + x - y = 9$$ Find $x^2 + y^2 + z^2$. [b]p10.[/b] In triangle $ABC$, $AB = 3$ and $AC = 4$. The bisector of angle $A$ meets $BC$ at $D$. The line through $D$ perpendicular to $AD$ intersects lines $AB$ and $AC$ at $F$ and $E$, respectively. Compute $EC - FB$. (See the following diagram.) [img]https://cdn.artofproblemsolving.com/attachments/2/7/e26fbaeb7d1f39cb8d5611c6a466add881ba0d.png[/img] [b]p11.[/b] Bob has a six-sided die with a number written on each face such that the sums of the numbers written on each pair of opposite faces are equal to each other. Suppose that the numbers $109$, $131$, and $135$ are written on three faces which share a corner. Determine the maximum possible sum of the numbers on the three remaining faces, given that all three are positive primes less than $200$. [b]p12.[/b] Let $d$ be a number chosen at random from the set $\{142, 143, ..., 198\}$. What is the probability that the area of a rectangle with perimeter $400$ and diagonal length $d$ is an integer? [b]p13.[/b] There are $3$ congruent circles such that each circle passes through the centers of the other two. Suppose that $A, B$, and $C$ are points on the circles such that each circle has exactly one of $A, B$, or $C$ on it and triangle $ABC$ is equilateral. Find the ratio of the maximum possible area of $ABC$ to the minimum possible area of $ABC$. (See the following diagram.) [img]https://cdn.artofproblemsolving.com/attachments/4/c/162554fcc6aa21ce3df3ce6a446357f0516f5d.png[/img] [b]p14.[/b] Let $k$ and $m$ be constants such that for all triples $(a, b, c)$ of positive real numbers, $$\sqrt{ \frac{4}{a^2}+\frac{36}{b^2}+\frac{9}{c^2}+\frac{k}{ab} }=\left| \frac{2}{a}+\frac{6}{b}+\frac{3}{c}\right|$$ if and only if $am^2 + bm + c = 0$. Find $k$. [b]p15.[/b] A bored student named Abraham is writing $n$ numbers $a_1, a_2, ..., a_n$. The value of each number is either $1, 2$, or $3$; that is, $a_i$ is $1, 2$ or $3$ for $1 \le i \le n$. Abraham notices that the ordered triples $$(a_1, a_2, a_3), (a_2, a_3, a_4), ..., (a_{n-2}, a_{n-1}, a_n), (a_{n-1}, a_n, a_1), (a_n, a_1, a_2)$$ are distinct from each other. What is the maximum possible value of $n$? Give the answer n, along with an example of such a sequence. Write your answer as an ordered pair. (For example, if the answer were $5$, you might write $(5, 12311)$.) PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be a triangle with $AB=AC$, and let $M$ be the midpoint of $BC$. Let $P$ be a point such that $PB<PC$ and $PA$ is parallel to $BC$. Let $X$ and $Y$ be points on the lines $PB$ and $PC$, respectively, so that $B$ lies on the segment $PX$, $C$ lies on the segment $PY$, and $\angle PXM=\angle PYM$. Prove that the quadrilateral $APXY$ is cyclic.

In triangle $ABC$,let $M$ be the midpoint of the side $BC$,and $N$ is the midpoint of the segment $AM$,the circle going through $N$ and tangent the line $AC$ at $A$ intersects the segment $AB$ again in $P$. prove that the circumcircle of triangle $BPM$ is tangent the line $AM$

Let $ABCD$ be a convex quadrilateral whose diagonals $AC$ and $BD$ intersect at point $O$. Let a line through $O$ intersect segment $AB$ at $M$ and segment $CD$ at $N$. Prove that the segment $MN$ is not longer than at least one of the segments $AC$ and $BD$.

The $n$ points $P_1,P_2, \ldots, P_n$ are placed inside or on the boundary of a disk of radius 1 in such a way that the minimum distance $D_n$ between any two of these points has its largest possible value $D_n.$ Calculate $D_n$ for $n = 2$ to 7. and justify your answer.

Given $n$ lines on the plane, they divide the plane onto several bounded or bounded polygonal regions. Define the rank of a region as the number of vertices on its boundary (a vertex is a point which belongs to at least two lines). Prove that the sum of squares of ranks of all regions does not exceed $10n^2$. (D. Fomin)

Let $ABC$ with $AB<AC<BC$ be an acute angled triangle and $c$ its circumcircle. Let $D$ be the point diametrically opposite to $A$. Point $K$ is on $BD$ such that $KB=KC$. The circle $(K, KC)$ intersects $AC$ at point $E$. Prove that the circle $(BKE)$ is tangent to $c$.

Let $B$ be a point on a circle $S_1$, and let $A$ be a point distinct from $B$ on the tangent at $B$ to $S_1$. Let $C$ be a point not on $S_1$ such that the line segment $AC$ meets $S_1$ at two distinct points. Let $S_2$ be the circle touching $AC$ at $C$ and touching $S_1$ at a point $D$ on the opposite side of $AC$ from $B$. Prove that the circumcentre of triangle $BCD$ lies on the circumcircle of triangle $ABC$.

On a cartesian plane a parabola $y = x^2$ is drawn. For a given $k > 0$ we consider all trapezoids inscribed into this parabola with bases parallel to the x-axis, and the product of the lengths of their bases is exactly $k$. Prove that the lateral sides of all such trapezoids share a common point.

Several straight lines such that no two are parallel, cut the plane into several regions. A point $A$ is marked inside of one region. Prove that a point, separated from $A$ by each of these lines, exists if and only if $A$ belongs to an unbounded region.

The distances from a point $ P$ inside an equilateral triangle to the vertices of the triangle are $ 3,4$, and $ 5$. Find the area of the triangle.

Every side of a right triangle is an integer when measured in cm, and the difference between the hypotenuse and one of the legs is $75$ cm. What is the smallest possible value of its perimeter?

[u]Round 5[/u] [b]p13.[/b] Simplify $\frac11+\frac13+\frac16+\frac{1}{10}+\frac{1}{15}+\frac{1}{21}$. [b]p14.[/b] Given that $x + y = 7$ and $x^2 + y^2 = 29$, what is the sum of the reciprocals of $x$ and $y$? [b]p15.[/b] Consider a rectangle $ABCD$ with side lengths $AB = 3$ and $BC = 4$. If circles are inscribeδ in triangles $ABC$ and $BCD$, how far are the centers of the circles from each other? [u]Round 6[/u] [b]p16.[/b] Evaluate $\frac{2!}{1!} +\frac{3!}{2!} +\frac{4!}{3!} + ... +\frac{99!}{98!}+\frac{100!}{99!}$ . [b]p17.[/b] Let $ABCD$ be a square of side length $2$. A semicircle is drawn with diameter $\overline{AC}$ that passes through point $B$. Find the area of the region inside the semicircle but outside the square. [b]p18.[/b] For how many positive integer values of $k$ is $\frac{37k - 30}{k}$ a positive integer? [u]Round 7[/u] [b]p19.[/b] Two parallel planar slices across a sphere of radius $25$ create cross sections of area $576\pi$ and $225\pi$. What is the maximum possible distance between the two slices? [b]p20.[/b] How many positive integers cannot be expressed in the form $3\ell + 4m + 5t$, where $\ell$, $m$, and $t$ are nonnegative integers? [b]p21.[/b] In April, a fool is someone who is fooled by a classmate. In a class of $30$ students, $14$ people were fooled by someone else and $29$ people fooled someone else. What is the largest positive integer $n$ for which we can guarantee that at least one person was fooled by at least $n$ other people? [u]Round 8[/u] [b]p22.[/b] Let $$S = 4 + \dfrac{12}{4 +\dfrac{ 12}{4 +\dfrac{ 12}{4+ ...}}}.$$ Evaluate $4 +\frac{ 12}{S}.$ [b]p23.[/b] Jonathan is buying bananagram sets for $\$11$ each and flip-flops for $\$17$ each. If he spends $\$227$ on purchases for bananagram sets and flip-flops, what is the total number of bananagram sets and flip-flops he bought? [b]p24.[/b] Alan has a $3 \times 3$ array of squares. He starts removing the squares one at a time such that each time he removes one square, all remaining squares share a side with at least two other remaining squares. What is the maximum number of squares Alan can remove? PS. You should use hide for answers. Rounds 1-4 are [url=https://artofproblemsolving.com/community/c3h2952214p26434209]here[/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3134133p28400917]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In triangle $ABC$, $AB>AC.$ The bisector of $\angle BAC$ meets $BC$ at $D.$ $P$ is on line $DA,$ such that $A$ lies between $P$ and $D$. $PQ$ is tangent to $\odot(ABD)$ at $Q.$ $PR$ is tangent to $\odot(ACD)$ at $R.$ $CQ$ meets $BR$ at $K.$ The line parallel to $BC$ and passing through $K$ meets $QD,AD,RD$ at $E,L,F,$ respectively. Prove that $EL=KF.$

In triangle $ABC, \angle A$ is smaller than $\angle C$. Point $D$ lies on the (extended) line $BC$ (with $B$ between $C$ and $D$) such that $|BD| = |AB|$. Point $E$ lies on the bisector of $\angle ABC$ such that $\angle BAE = \angle ACB$. Line segment $BE$ intersects line segment $AC$ in point $F$. Point $G$ lies on line segment $AD$ such that $EG$ and $BC$ are parallel. Prove that $|AG| =|BF|$. [asy] unitsize (1.5 cm); real angleindegrees(pair A, pair B, pair C) { real a, b, c; a = abs(B - C); b = abs(C - A); c = abs(A - B); return(aCos((a^2 + c^2 - b^2)/(2*a*c))); }; pair A, B, C, D, E, F, G; B = (0,0); A = 2*dir(190); D = 2*dir(310); C = 1.5*dir(310 - 180); E = extension(B, incenter(A,B,C), A, rotate(angleindegrees(A,C,B),A)*(B)); F = extension(B,E,A,C); G = extension(E, E + D - B, A, D); filldraw(anglemark(A,C,B,8),gray(0.8)); filldraw(anglemark(B,A,E,8),gray(0.8)); draw(C--A--B); draw(E--A--D); draw(interp(C,D,-0.1)--interp(C,D,1.1)); draw(interp(E,B,-0.2)--interp(E,B,1.2)); draw(E--G); dot("$A$", A, SW); dot("$B$", B, NE); dot("$C$", C, NE); dot("$D$", D, NE); dot("$E$", E, N); dot("$F$", F, N); dot("$G$", G, SW); [/asy]

Points $K$ and $L$ are chosen on the sides $AB$ and $AC$ of an equilateral triangle $ABC$ such that $BK = AL$. Segments $BL$ and $CK$ intersect at $P$. Determine the ratio $\frac{AK}{KB}$ for which the segments $AP$ and $CK$ are perpendicular.

Let $ABC$ be a triangle and $D$, $E$, $F$ be the midpoints of arcs $BC$, $CA$, $AB$ on the circumcircle. Line $\ell_a$ passes through the feet of the perpendiculars from $A$ to $DB$ and $DC$. Line $m_a$ passes through the feet of the perpendiculars from $D$ to $AB$ and $AC$. Let $A_1$ denote the intersection of lines $\ell_a$ and $m_a$. Define points $B_1$ and $C_1$ similarly. Prove that triangle $DEF$ and $A_1B_1C_1$ are similar to each other.

The point $P$ lies inside the triangle $ABC$. A line is drawn through $P$ parallel to each side of the triangle. The lines divide $AB$ into three parts length $c, c', c"$ (in that order), and $BC$ into three parts length $a, a', a"$ (in that order), and $CA$ into three parts length $b, b', b"$ (in that order). Show that $abc = a'b'c' = a"b"c"$.

In triangle $ABC$, $AB=\frac{20}{11} AC$. The angle bisector of $\angle A$ intersects $BC$ at point $D$, and point $M$ is the midpoint of $AD$. Let $P$ be the point of the intersection of $AC$ and $BM$. The ratio of $CP$ to $PA$ can be expressed in the form $\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

A triangle with integer sides is called Heronian if its area is an integer. Does there exist a Heronian triangle whose sides are the arithmetic, geometric and harmonic means of two positive integers?

In an acute triangle $ABC$, the incircle $\omega$ touches $BC$ at $D$. Let $I_a$ be the excenter of $ABC$ opposite to $A$, and let $M$ be the midpoint of $DI_a$. Prove that the circumcircle of triangle $BMC$ is tangent to $\omega$. [i]Patrik Bak (Slovakia)[/i]

We divide up the plane into disjoint regions using a circle, a rectangle and a triangle. What is the greatest number of regions that we can get?

A satellite is considered accessible from the point $A{}$ of the planet's surface if it is located relative to the tangent plane drawn at point $A{}$, strictly on the other side than the planet. What is the smallest number of satellites that need to be launched over a spherical planet so that at some point the signals of at least two satellites are available from each point on the planet's surface? [i]Proposed by S. Volchenkov[/i]