Consider a pyramid whose base is an equilateral triangle $BCD$ and whose other faces are triangles isosceles, right at the common vertex $A$. An ant leaves the vertex $B$ arrives at a point $P$ of the $CD$ edge, from there goes to a point $Q$ of the edge $AC$ and returns to point $B$. If the path you made is minimal, how much is the angle $PQA$ ?

A hexagon has all its angles equal, and the lengths of four consecutive sides are $5$, $3$, $6$ and $7$, respectively. Find the lengths of the remaining two edges.

Two non-intersecting circles, not lying inside each other, are drawn in the plane. Two lines pass through a point P which lies outside each circle. The first line intersects the first circle at A and A′ and the second circle at B and B′; here A and B are closer to P than A′ and B′, respectively, and P lies on segment AB. Analogously, the second line intersects the first circle at C and C′ and the second circle at D and D′. Prove that the points A, B, C, D are concyclic if and only if the points A′, B′, C′, D′ are concyclic.

[b]p1.[/b] A permutation on $\{1, 2,..., n\}$ is an ordered arrangement of the numbers. For example, $32154$ is a permutation of $\{1, 2, 3, 4, 5\}$. Does there exist a permutation $a_1a_2... a_n$ of $\{1, 2,..., n\}$ such that $i+a_i$ is a perfect square for every $1 \le i \le n$ when a) $n = 6$ ? b) $n = 13$ ? c) $n = 86$ ? Justify your answers. [b]p2.[/b] Circle $C$ and circle $D$ are tangent at point $P$. Line $L$ is tangent to $C$ at point $Q$ and to $D$ at point $R$ where $Q$ and $R$ are distinct from $P$. Circle $E$ is tangent to $C, D$, and $L$, and lies inside triangle $PQR$. $C$ and $D$ both have radius $8$. Find the radius of $E$, and justify your answer. [img]https://cdn.artofproblemsolving.com/attachments/f/b/4b98367ea64e965369345247fead3456d3d18a.png[/img] [b]p3.[/b] (a) Prove that $\sin 3x = 4 \cos^2 x \sin x - \sin x$ for all real $x$. (b) Prove that $$(4 \cos^2 9^o - 1)(4 \cos^2 27^o - 1)(4 cos^2 81^o - 1)(4 cos^2 243^o - 1)$$ is an integer. [b]p4.[/b] Consider a $3\times 3\times 3$ stack of small cubes making up a large cube (as with the small cubes in a Rubik's cube). An ant crawls on the surface of the large cube to go from one corner of the large cube to the opposite corner. The ant walks only along the edges of the small cubes and covers exactly nine of these edges. How many different paths can the ant take to reach its goal? [b]p5.[/b] Let $m$ and $n$ be positive integers, and consider the rectangular array of points $(i, j)$ with $1 \le i \le m$, $1 \le j \le n$. For what pairs m; n of positive integers does there exist a polygon for which the $mn$ points $(i, j)$ are its vertices, such that each edge is either horizontal or vertical? The figure below depicts such a polygon with $m = 10$, $n = 22$. Thus $10$, $22$ is one such pair. [img]https://cdn.artofproblemsolving.com/attachments/4/5/c76c0fe197a8d1ebef543df8e39114fe9d2078.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A triangle $ABC$, its circumcircle, and its incenter $I$ are drawn on the plane. Construct the circumcenter of $ABC$ using only a ruler.

Let the line $\ell$ intersects the sides $AC,BC$ of the triangle $ABC$ respectively at the points $E$ and $F$. Prove that the line $\ell$ is passing through the incenter of the triangle $ABC$ if and only if the following equality is true: $$BC\cdot\frac{AE}{CE}+AC\cdot\frac{BF}{CF}=AB.$$ [i]H. Lesov[/i]

Let $B$ and $C$ be points on a semicircle with diameter $AD$ such that $B$ is closer to $A$ than $C$. Diagonals $AC$ and $BD$ intersect at point $E$. Let $P$ and $Q$ be points such that $\overline{PE} \perp \overline{BD}$ and $\overline{PB} \perp \overline{AD}$, while $\overline{QE} \perp \overline{AC}$ and $\overline{QC} \perp \overline{AD}$. If $BQ$ and $CP$ intersect at point $T$, prove that $\overline{TE} \perp \overline{BC}$. [i]Proposed by Andrew Wen[/i]

The quadrilateral $ ABCD$ is cyclic. Points $ E$ and $ F$ are chosen at the diagonals $ AC$ and $ BD$ in such a way that $ AF\bot CD$ and $ DE\bot AB.$ Prove that $ EF \parallel BC.$

Let $ABCD$ be a square and let point $F$ be any point on side $BC$. Let the line perpendicular to $DF$, that passes through $B$, intersect line $DC$ at $Q$. What is value of $\angle FQC$?

Triangle $ABC$ is isosceles ($AB=AC$). From $A$, we draw a line $\ell$ parallel to $BC$. $P,Q$ are on perpendicular bisectors of $AB,AC$ such that $PQ\perp BC$. $M,N$ are points on $\ell$ such that angles $\angle APM$ and $\angle AQN$ are $\frac\pi2$. Prove that \[\frac{1}{AM}+\frac1{AN}\leq\frac2{AB}\]

A cone is constructed with a semicircular piece of paper, with radius 10. Find the height of the cone.

Let $ABC$ be a right triangle, right at $B$, and let $M$ be the midpoint of the side $BC$. Let $P$ be the point in bisector of the angle $ \angle BAC$ such that $PM$ is perpendicular to $BC (P$ is outside the triangle $ABC$). Determine the triangle area $ABC$ if $PM = 1$ and $MC = 5$.

Consider equilateral triangle $ABC$ and suppose that there exist three distinct points $X, Y,Z$ lie inside triangle $ABC$ such that i) $AX = BY = CZ$ ii) The triplets of points $(A,X,Z), (B,Y,X), (C,Z,Y )$ are collinear in that order. Prove that $XY Z$ is an equilateral triangle.

In trapezoid $ABCD$, $BC \parallel AD$, $AB = 13$, $BC = 15$, $CD = 14$, and $DA = 30$. Find the area of $ABCD$.

In an acute triangle $ ABC$ segments $ BE$ and $ CF$ are altitudes. Two circles passing through the point $ A$ and $ F$ and tangent to the line $ BC$ at the points $ P$ and $ Q$ so that $ B$ lies between $ C$ and $ Q$. Prove that lines $ PE$ and $ QF$ intersect on the circumcircle of triangle $ AEF$. [i]Proposed by Davood Vakili, Iran[/i]

Let us consider a triangle $\Delta{PQR}$ in the co-ordinate plane. Show for every function $f: \mathbb{R}^2\to \mathbb{R}\;,f(X)=ax+by+c$ where $X\equiv (x,y) \text{ and } a,b,c\in\mathbb{R}$ and every point $A$ on $\Delta PQR$ or inside the triangle we have the inequality: \begin{align*} & f(A)\le \text{max}\{f(P),f(Q),f(R)\} \end{align*}

[b]p1.[/b] Find $20 \cdot 18 + 20 + 18 + 1$. [b]p2.[/b] Suzie’s Ice Cream has $10$ flavors of ice cream, $5$ types of cones, and $5$ toppings to choose from. An ice cream cone consists of one flavor, one cone, and one topping. How many ways are there for Sebastian to order an ice cream cone from Suzie’s? [b]p3.[/b] Let $a = 7$ and $b = 77$. Find $\frac{(2ab)^2}{(a+b)^2-(a-b)^2}$ . [b]p4.[/b] Sebastian invests $100,000$ dollars. On the first day, the value of his investment falls by $20$ percent. On the second day, it increases by $25$ percent. On the third day, it falls by $25$ percent. On the fourth day, it increases by $60$ percent. How many dollars is his investment worth by the end of the fourth day? [b]p5.[/b] Square $ABCD$ has side length $5$. Points $K,L,M,N$ are on segments $AB$,$BC$,$CD$,$DA$ respectively,such that $MC = CL = 2$ and $NA = AK = 1$. The area of trapezoid $KLMN$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $m + n$. [b]p6.[/b] Suppose that $p$ and $q$ are prime numbers. If $p + q = 30$, find the sum of all possible values of $pq$. [b]p7.[/b] Tori receives a $15 - 20 - 25$ right triangle. She cuts the triangle into two pieces along the altitude to the side of length $25$. What is the difference between the areas of the two pieces? [b]p8.[/b] The factorial of a positive integer $n$, denoted $n!$, is the product of all the positive integers less than or equal to $n$. For example, $1! = 1$ and $5! = 120$. Let $m!$ and $n!$ be the smallest and largest factorial ending in exactly $3$ zeroes, respectively. Find $m + n$. [b]p9.[/b] Sam is late to class, which is located at point $B$. He begins his walk at point $A$ and is only allowed to walk on the grid lines. He wants to get to his destination quickly; how many paths are there that minimize his walking distance? [img]https://cdn.artofproblemsolving.com/attachments/a/5/764e64ac315c950367357a1a8658b08abd635b.png[/img] [b]p10.[/b] Mr. Iyer owns a set of $6$ antique marbles, where $1$ is red, $2$ are yellow, and $3$ are blue. Unfortunately, he has randomly lost two of the marbles. His granddaughter starts drawing the remaining $4$ out of a bag without replacement. She draws a yellow marble, then the red marble. Suppose that the probability that the next marble she draws is blue is equal to $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positiveintegers. What is $m + n$? [b]p11.[/b] If $a$ is a positive integer, what is the largest integer that will always be a factor of $(a^3+1)(a^3+2)(a^3+3)$? [b]p12.[/b] What is the largest prime number that is a factor of $160,401$? [b]p13.[/b] For how many integers $m$ does the equation $x^2 + mx + 2018 = 0$ have no real solutions in $x$? [b]p14.[/b] What is the largest palindrome that can be expressed as the product of two two-digit numbers? A palindrome is a positive integer that has the same value when its digits are reversed. An example of a palindrome is $7887887$. [b]p15.[/b] In circle $\omega$ inscribe quadrilateral $ADBC$ such that $AB \perp CD$. Let $E$ be the intersection of diagonals $AB$ and $CD$, and suppose that $EC = 3$, $ED = 4$, and $EB = 2$. If the radius of $\omega$ is $r$, then $r^2 =\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Determine $m + n$. [b]p16.[/b] Suppose that $a, b, c$ are nonzero real numbers such that $2a^2 + 5b^2 + 45c^2 = 4ab + 6bc + 12ca$. Find the value of $\frac{9(a + b + c)^3}{5abc}$ . [b]p17.[/b] Call a positive integer n spicy if there exist n distinct integers $k_1, k_2, ... , k_n$ such that the following two conditions hold: $\bullet$ $|k_1| + |k_2| +... + |k_n| = n2$, $\bullet$ $k_1 + k_2 + ...+ k_n = 0$. Determine the number of spicy integers less than $10^6$. [b]p18.[/b] Consider the system of equations $$|x^2 - y^2 - 4x + 4y| = 4$$ $$|x^2 + y^2 - 4x - 4y| = 4.$$ Find the sum of all $x$ and $y$ that satisfy the system. [b]p19.[/b] Determine the number of $8$ letter sequences, consisting only of the letters $W,Q,N$, in which none of the sequences $WW$, $QQQ$, or $NNNN$ appear. For example, $WQQNNNQQ$ is a valid sequence, while $WWWQNQNQ$ is not. [b]p20.[/b] Triangle $\vartriangle ABC$ has $AB = 7$, $CA = 8$, and $BC = 9$. Let the reflections of $A,B,C$ over the orthocenter H be $A'$,$B'$,$C'$. The area of the intersection of triangles $ABC$ and $A'B'C'$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ , where $b$ is squarefree and $a$ and $c$ are relatively prime. determine $a+b+c$. (The orthocenter of a triangle is the intersection of its three altitudes.) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given a regular polygon with apothem $ A $ and circumradius $ R $. Find for a regular polygon of equal perimeter and with double number of sides, the apothem $ a $ and the circumcircle $ r $ in terms of $A,R$

Triangle $ABC$ with $\angle BAC=60^\circ$ is given. The circumcircle of $ABC$ is $\Omega$, and the orthocenter of $ABC$ is $H$. Let $S$ denote the midpoint of the arc $BC$ of $\Omega$ which doesn't contain $A$. Point $P$ was chosen on $\Omega$ so that $\angle HPS=90^\circ$. Prove that there exists a circle that goes through $P$ and $S$ and is tangent to lines $AB$, $AC$.

Given is an acute triangle $ ABC$ and the points $ A_1,B_1,C_1$, that are the feet of its altitudes from $ A,B,C$ respectively. A circle passes through $ A_1$ and $ B_1$ and touches the smaller arc $ AB$ of the circumcircle of $ ABC$ in point $ C_2$. Points $ A_2$ and $ B_2$ are defined analogously. Prove that the lines $ A_1A_2$, $ B_1B_2$, $ C_1C_2$ have a common point, which lies on the Euler line of $ ABC$.

There are $n$ different points $A_1, \ldots , A_n$ in the plain and each point $A_i$ it is assigned a real number $\lambda_i$ distinct from zero in such way that $(\overline{A_i A_j})^2 = \lambda_i + \lambda_j$ for all the $i$,$j$ with $i\neq{}j$} Show that: (1) $n \leq 4$ (2) If $n=4$, then $\frac{1}{\lambda_1} + \frac{1}{\lambda_2} + \frac{1}{\lambda_3}+ \frac{1}{\lambda_4} = 0$

A semicircle has diameter $AB$ and center $S$,with a point $M$ on the circumference.$U,V$ are the incircles of sectors $ASM$ and $BSM$.Prove that circles $U,V$ can be seperated by a line perpendicular to $AB$.

In acute-angled $\bigtriangleup ABC$, $H$ is the orthocenter, $O$ is the circumcenter and $I$ is the incenter. Given that $\angle C > \angle B > \angle A$, prove that $I$ lies within $\bigtriangleup BOH$.

Point \( A_1 \) inside the acute-angled triangle \( ABC \) is such that \[ \angle ACB = 2\angle A_1BC \quad \text{and} \quad \angle ABC = 2\angle A_1CB. \] Point \( A_2 \) is chosen so that points \( A \) and \( A_2 \) lie on opposite sides of line \( BC \), \( AA_2 \perp BC \), and the perpendicular bisector of \( AA_2 \) is tangent to the circumcircle of \( \triangle ABC \). Define points \( B_1, B_2, C_1, C_2 \) analogously. Prove that the circumcircles of \( \triangle AA_1A_2 \), \( \triangle BB_1B_2 \), and \( \triangle CC_1C_2 \) intersect at exactly two common points. [i]Proposed by Vadym Solomka[/i]

Let $ABC$ be a triangle. The tangent in $A$ of the circumcircle of $ABC$ cuts the line $(BC)$ in $X$. Let $A'$ be the symetric of $A$ by $X$ and $C'$ the symetric of $C$ by the line $(AX)$ Prove that the points $A, C', A'$ and $B$ are concyclic.