In a triangle $ABC, P$ and $P'$ are points on side $BC, Q$ on side $CA$, and $R $ on side $AB$, such that $\frac{AR}{RB}=\frac{BP}{PC}=\frac{CQ}{QA}=\frac{CP'}{P'B}$ . Let $G$ be the centroid of triangle $ABC$ and $K$ be the intersection point of $AP'$ and $RQ$. Prove that points $P,G,K$ are collinear.

Given a family $C$ of circles of the same radius $R$, which completely covers the plane (that is, every point in the plane belongs to at least one circle of the family), prove that there exist two circles of the family such that the distance between their centers is less than or equal to $R\sqrt3$ .

A cuboctahedron is the convex hull of (smallest convex set containing) the $12$ points $(\pm 1, \pm 1, 0), (\pm 1, 0, \pm 1), (0, \pm 1, \pm 1)$. Find the cosine of the solid angle of one of the triangular faces, as viewed from the origin. (Take a figure and consider the set of points on the unit sphere centered on the origin such that the ray from the origin through the point intersects the figure. The area of that set is the solid angle of the figure as viewed from the origin.)

Determine if there exists a convex polyhedron such that (1) it has 12 edges, 6 faces and 8 vertices; (2) it has 4 faces with each pair of them sharing a common edge of the polyhedron.

In a triangle $ABC$, let $D$ and $E$ be the feet of the angle bisectors of angles $A$ and $B$, respectively. A rhombus is inscribed into the quadrilateral $AEDB$ (all vertices of the rhombus lie on different sides of $AEDB$). Let $\varphi$ be the non-obtuse angle of the rhombus. Prove that $\varphi \le \max \{ \angle BAC, \angle ABC \}$.

Let $A_1$ and $C_1$ be the feet of altitudes $AH$ and $CH$ of an acute-angled triangle $ABC$. Points $A_2$ and $C_2$ are the reflections of $A_1$ and $C_1$ about $AC$. Prove that the distance between the circumcenters of triangles $C_2HA_1$ and $C_1HA_2$ equals $AC$.

For any two points $A (x_1 , y_1)$ and $B (x_2, y_2)$, the distance $r (A, B)$ between them is determined by the equality $r(A, B) = max\{| x_1- x_2 | , | y_1 - y_2 |\}$. Prove that the triangle inequality $r(A, C) + r(C, B) \ge r(A, B)$. holds for the distance introduced in this way . Let $A$ and $B$ be two points of the plane . Find the locus of points $C$ for which a) $r(A, C) + r(C, B) = r(A, B)$ b) $r(A, C) = r(C, B).$

Let $ABC$ be a triangle and let $D$ be a point on the segment $BC$ such that $AD=BC$. \\ Suppose $\angle{CAD}=x^{\circ}, \angle{ABC}=y^{\circ}$ and $\angle{ACB}=z^{\circ}$ and $x,y,z$ are in an arithmetic progression in that order where the first term and the common difference are positive integers. Find the largest possible value of $\angle{ABC}$ in degrees.

Points $E$ and $C$ are chosen on a semicircle with diameter $AB$ and center $O$ such that $OE \perp AB$ and the intersection point $D$ of $AC$ and $OE$ is inside the semicircle. Find all values of $\angle{CAB}$ for which the quadrilateral $OBCD$ is tangent.

How many noncongruent integer-sided triangles with positive area and perimeter less than $15$ are neither equilateral, isosceles, nor right triangles? $\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }7$

Points $A$ and $B$ are chosen on a circle $k$. Let AP and $BQ$ be segments of the same length tangent to $k$, drawn on different sides of line $AB$. Prove that the line $AB$ bisects the segment $PQ$.

In the Cartesian plane, a line segment is called [i]tame[/i] if it lies parallel to one of the coordinate axes and its distance to this axis is an integer. Otherwise it is called [i]wild[/i]. Let $m$ and $n$ be odd positive integers. The rectangle with vertices $(0,0),(m,0),(m,n)$ and $(0,n)$ is partitioned into finitely many triangles. Let $M$ be the set of these triangles. Assume that (1) Each triangle from $M$ has at least one tame side. (2) For each tame side of a triangle from $M$, the corresponding altitude has length $1$. (3) Each wild side of a triangle from $M$ is a common side of exactly two triangles from $M$. Show that at least two triangles from $M$ have two tame sides each.

We are given a non-isosceles triangle $ABC$ with incenter $I$. Show that the circumcircle $k$ of the triangle $AIB$ does not touch the lines $CA$ and $CB$. Let $P$ be the second point of intersection of $k$ with $CA$ and let $Q$ be the second point of intersection of $k$ with $CB$. Show that the four points $A$, $B$, $P$ and $Q$ (not necessarily in this order) are the vertices of a trapezoid.

Does there exist a rectangle which can be dissected into $15$ congruent polygons which are not rectangles? Can a square be dissected into $15$ congruent polygons which are not rectangles?

Three circles, with radii of $1, 1$, and $2$, are externally tangent to each other. The minimum possible area of a quadrilateral that contains and is tangent to all three circles can be written as $a + b\sqrt{c}$ where $c$ is not divisible by any perfect square larger than $1$. Find $a + b + c$

Let $O$ be the circumcenter of triangle $ABC$. Define $O_{A0} = O_{B0} = O_{C0} = O$. Recursively, define $O_{An}$ to be the circumcenter of $\vartriangle BO_{A(n-1)}C$. Similarly define $O_{Bn}, O_{Cn}$. Find all $n \ge 1$ so that for any triangle $ABC$ such that $O_{An}, O_{Bn}, O_{Cn}$ all exist, it is true that $AO_{An}, BO_{Bn}, CO_{Cn}$ are concurrent. (Li4)

Does there exist a convex pentagon such that for any of its inner angles, the angle bisector contains one of the diagonals?

Let $ABC$ be a triangle with $\angle{C} = 90^{\circ}$, and let $H$ be the foot of the altitude from $C$. A point $D$ is chosen inside the triangle $CBH$ so that $CH$ bisects $AD$. Let $P$ be the intersection point of the lines $BD$ and $CH$. Let $\omega$ be the semicircle with diameter $BD$ that meets the segment $CB$ at an interior point. A line through $P$ is tangent to $\omega$ at $Q$. Prove that the lines $CQ$ and $AD$ meet on $\omega$.

The heights of a triangular pyramid intersect at one point. Prove that all flat angles at any vertex of the surface are either acute, or right, or obtuse.

For what values of $k{}$ can a regular octagon with side-length $k{}$ be cut into $1 \times 2{}$ dominoes and rhombuses with side-length 1 and a $45^\circ{}$ angle?

The point $O$ inside the convex polygon makes isosceles triangle with all the pairs of its vertices. Prove that $O$ is the centre of the circumscribed circle. [u]other formulation:[/u] $P$ is a convex polygon and $X$ is an interior point such that for every pair of vertices $A, B$, the triangle $XAB$ is isosceles. Prove that all the vertices of $P$ lie on a circle with center $X$.

A square, a regular pentagon, and a regular hexagon are all inscribed in the same circle. The $15$ vertices of these polygons divide the circle into at most $15$ arcs. Let $M$ be the degree measure of the longest of these arcs. Find the minimum possible value for $M$.

Triangle $ABC$ has circumcircle $\Omega$ and circumcenter $O$. A circle $\Gamma$ with center $A$ intersects the segment $BC$ at points $D$ and $E$, such that $B$, $D$, $E$, and $C$ are all different and lie on line $BC$ in this order. Let $F$ and $G$ be the points of intersection of $\Gamma$ and $\Omega$, such that $A$, $F$, $B$, $C$, and $G$ lie on $\Omega$ in this order. Let $K$ be the second point of intersection of the circumcircle of triangle $BDF$ and the segment $AB$. Let $L$ be the second point of intersection of the circumcircle of triangle $CGE$ and the segment $CA$. Suppose that the lines $FK$ and $GL$ are different and intersect at the point $X$. Prove that $X$ lies on the line $AO$. [i]Proposed by Greece[/i]

For any given $k$ points in a plane, we define the diameter of the points as the maximum distance between any two points among the given points. Suppose $n$ points are there in a plane with diameter $d$. Show that we can always find a circle with radius $\frac{\sqrt{3}}{2}d$ such that all points lie inside the circle.

[b]p1.[/b] Nicky is studying biology and has a tank of $17$ lizards. In one day, he can either remove $5$ lizards or add $2$ lizards to his tank. What is the minimum number of days necessary for Nicky to get rid of all of the lizards from his tank? [b]p2.[/b] What is the maximum number of spheres with radius $1$ that can fit into a sphere with radius $2$? [b]p3.[/b] A positive integer $x$ is sunny if $3x$ has more digits than $x$. If all sunny numbers are written in increasing order, what is the $50$th number written? [b]p4.[/b] Quadrilateral $ABCD$ satisfies $AB = 4$, $BC = 5$, $DA = 4$, $\angle DAB = 60^o$, and $\angle ABC = 150^o$. Find the area of $ABCD$. [b]p5. [/b]Totoro wants to cut a $3$ meter long bar of mixed metals into two parts with equal monetary value. The left meter is bronze, worth $10$ zoty per meter, the middle meter is silver, worth $25$ zoty per meter, and the right meter is gold, worth $40$ zoty per meter. How far, in meters, from the left should Totoro make the cut? [b]p6.[/b] If the numbers $x_1, x_2, x_3, x_4$, and $x5$ are a permutation of the numbers $1, 2, 3, 4$, and $5$, compute the maximum possible value of $$|x_1 - x_2| + |x_2 - x_3| + |x_3 - x_4| + |x_4 - x_5|.$$ [b]p7.[/b] In a $3 \times 4$ grid of $12$ squares, find the number of paths from the top left corner to the bottom right corner that satisfy the following two properties: $\bullet$ The path passes through each square exactly once. $\bullet$ Consecutive squares share a side. Two paths are considered distinct if and only if the order in which the twelve squares are visited is different. For instance, in the diagram below, the two paths drawn are considered the same. [img]https://cdn.artofproblemsolving.com/attachments/7/a/bb3471bbde1a8f58a61d9dd69c8527cfece05a.png[/img] [b]p8.[/b] Scott, Demi, and Alex are writing a computer program that is $25$ ines long. Since they are working together on one computer, only one person may type at a time. To encourage collaboration, no person can type two lines in a row, and everyone must type something. If Scott takes $10$ seconds to type one line, Demi takes $15$ seconds, and Alex takes $20$ seconds, at least how long, in seconds, will it take them to finish the program? [b]p9.[/b] A hand of four cards of the form $(c, c, c + 1, c + 1)$ is called a tractor. Vinjai has a deck consisting of four of each of the numbers $7$, $8$, $9$ and $10$. If Vinjai shuffles and draws four cards from his deck, compute the probability that they form a tractor. [b]p10. [/b]The parabola $y = 2x^2$ is the wall of a fortress. Totoro is located at $(0, 4)$ and fires a cannonball in a straight line at the closest point on the wall. Compute the y-coordinate of the point on the wall that the cannonball hits. [b]p11. [/b]How many ways are there to color the squares of a $10$ by $10$ grid with black and white such that in each row and each column there are exactly two black squares and between the two black squares in a given row or column there are exactly [b]4[/b] white squares? Two configurations that are the same under rotations or reflections are considered different. [b]p12.[/b] In rectangle $ABCD$, points $E$ and $F$ are on sides $AB$ and $CD$, respectively, such that $AE = CF > AD$ and $\angle CED = 90^o$. Lines $AF, BF, CE$ and $DE$ enclose a rectangle whose area is $24\%$ of the area of $ABCD$. Compute $\frac{BF}{CE}$ . [b]p13.[/b] Link cuts trees in order to complete a quest. He must cut $3$ Fenwick trees, $3$ Splay trees and $3$ KD trees. If he must also cut 3 trees of the same type in a row at some point during his quest, in how many ways can he cut the trees and complete the quest? (Trees of the same type are indistinguishable.) [b]p14.[/b] Find all ordered pairs (a, b) of positive integers such that $\sqrt{64a + b^2} + 8 = 8\sqrt{a} + b$. [b]p15.[/b] Let $ABCDE$ be a convex pentagon such that $\angle ABC = \angle BCD = 108^o$, $\angle CDE = 168^o$ and $AB =BC = CD = DE$. Find the measure of $\angle AEB$ PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].