In triangle $ABC$, angle $C$ is three times the angle $A$, and side $AB$ is twice the side $BC$. What can be the angle $ABC$?

We are given a circle $K$ with center $S$ and radius $1$ and a square $Q$ with center $M$ and side $2$. Let $XY$ be the hypotenuse of an isosceles right triangle $XY Z$. Describe the locus of points $Z$ as $X$ varies along $K$ and $Y$ varies along the boundary of $Q.$

Consider a latticelike packing of translates of a convex region $K$. Let $t$ be the area of the fundamental parallelogram of the lattice defining the packing, and let $t_{\min} (K)$ denote the minimal value of $t$ taken for all latticelike packings. Is there a natural number $N$ such that for any $n>N$ and for any $K$ different from a parallelogram, $nt_{\min} (K)$ is smaller that the area of any convex domain in which $n$ translates to $K$ can be placed without overlapping? (By a [i]latticelike packing[/i] of $K$ we mean a set of nonoverlapping translates of $K$ obtained from $K$ by translations with all vectors of a lattice.) [G. and L. Fejes-Toth]

In a triangle $ABC$, $\angle C = 60^o$, $D, E, F$ are points on the sides $BC, AB, AC$ respectively, and $M$ is the intersection point of $AD$ and $BF$. Suppose that $CDEF$ is a rhombus. Prove that $DF^2 = DM \cdot DA$

Given a pentagon with all equal sides. a) Prove that there exist such a point on the maximal diagonal, that every side is seen from it inside a right angle. (side $AB$ is seen from the point $C$ inside an arbitrary angle that is greater or equal than $\angle ACB$) b) Prove that the circles constructed on its sides as on the diameters cannot cover the pentagon entirely.

Triangle $A_1 A_2 A_3$ has a right angle at $A_3$. A sequence of points is now defined by the following iterative process, where $n$ is a positive integer. From $A_n$ ($n \geq 3$), a perpendicular line is drawn to meet $A_{n-2}A_{n-1}$ at $A_{n+1}$. (a) Prove that if this process is continued indefinitely, then one and only one point $P$ is interior to every triangle $A_{n-2} A_{n-1} A_{n}$, $n \geq 3$. (b) Let $A_1$ and $A_3$ be fixed points. By considering all possible locations of $A_2$ on the plane, find the locus of $P$.

In $\triangle ABC$, $D$ is on $AC$ and $F$ is on $BC$. Also, $AB \perp AC, AF \perp BC$, and $BD = DC = FC = 1$. Find $AC$. A. $\sqrt{2}$ B. $\sqrt{3}$ C. $\sqrt[3] {2}$ D. $\sqrt[3] {3}$ E. $\sqrt[4] {3}$

Let $ABC$ be a triangle with $AC\neq BC$, and let $A^{\prime }B^{\prime }C$ be a triangle obtained from $ABC$ after some rotation centered at $C$. Let $M,E,F$ be the midpoints of the segments $BA^{\prime },AC$ and $CB^{\prime }$ respectively. If $EM=FM$, find $\widehat{EMF}$.

Each of two angles of a triangle is $60^{\circ}$ and the included side is $4$ inches. The area of the triangle, in square inches, is: $ \textbf{(A) }8\sqrt{3}\qquad\textbf{(B) }8\qquad\textbf{(C) }4\sqrt{3}\qquad\textbf{(D) }4\qquad\textbf{(E) }2\sqrt{3} $

[b]p1.[/b] Find the area of a right triangle with legs of lengths $20$ and $18$. [b]p2.[/b] How many $4$-digit numbers (without leading zeros) contain only $2,0,1,8$ as digits? Digits can be used more than once. [b]p3.[/b] A rectangle has perimeter $24$. Compute the largest possible area of the rectangle. [b]p4.[/b] Find the smallest positive integer with $12$ positive factors, including one and itself. [b]p5.[/b] Sammy can buy $3$ pencils and $6$ shoes for $9$ dollars, and Ben can buy $4$ pencils and $4$ shoes for $10$ dollars at the same store. How much more money does a pencil cost than a shoe? [b]p6.[/b] What is the radius of the circle inscribed in a right triangle with legs of length $3$ and $4$? [b]p7.[/b] Find the angle between the minute and hour hands of a clock at $12 : 30$. [b]p8.[/b] Three distinct numbers are selected at random fromthe set $\{1,2,3, ... ,101\}$. Find the probability that $20$ and $18$ are two of those numbers. [b]p9.[/b] If it takes $6$ builders $4$ days to build $6$ houses, find the number of houses $8$ builders can build in $9$ days. [b]p10.[/b] A six sided die is rolled three times. Find the probability that each consecutive roll is less than the roll before it. [b]p11.[/b] Find the positive integer $n$ so that $\frac{8-6\sqrt{n}}{n}$ is the reciprocal of $\frac{80+6\sqrt{n}}{n}$. [b]p12.[/b] Find the number of all positive integers less than $511$ whose binary representations differ from that of $511$ in exactly two places. [b]p13.[/b] Find the largest number of diagonals that can be drawn within a regular $2018$-gon so that no two intersect. [b]p14.[/b] Let $a$ and $b$ be positive real numbers with $a > b $ such that $ab = a +b = 2018$. Find $\lfloor 1000a \rfloor$. Here $\lfloor x \rfloor$ is equal to the greatest integer less than or equal to $x$. [b]p15.[/b] Let $r_1$ and $r_2$ be the roots of $x^2 +4x +5 = 0$. Find $r^2_1+r^2_2$ . [b]p16.[/b] Let $\vartriangle ABC$ with $AB = 5$, $BC = 4$, $C A = 3$ be inscribed in a circle $\Omega$. Let the tangent to $\Omega$ at $A$ intersect $BC$ at $D$ and let the tangent to $\Omega$ at $B$ intersect $AC$ at $E$. Let $AB$ intersect $DE$ at $F$. Find the length $BF$. [b]p17.[/b] A standard $6$-sided die and a $4$-sided die numbered $1, 2, 3$, and $4$ are rolled and summed. What is the probability that the sum is $5$? [b]p18.[/b] Let $A$ and $B$ be the points $(2,0)$ and $(4,1)$ respectively. The point $P$ is on the line $y = 2x +1$ such that $AP +BP$ is minimized. Find the coordinates of $P$. [b]p19.[/b] Rectangle $ABCD$ has points $E$ and $F$ on sides $AB$ and $BC$, respectively. Given that $\frac{AE}{BE}=\frac{BF}{FC}= \frac12$, $\angle ADE = 30^o$, and $[DEF] = 25$, find the area of rectangle $ABCD$. [b]p20.[/b] Find the sum of the coefficients in the expansion of $(x^2 -x +1)^{2018}$. [b]p21.[/b] If $p,q$ and $r$ are primes with $pqr = 19(p+q+r)$, find $p +q +r$ . [b]p22.[/b] Let $\vartriangle ABC$ be the triangle such that $\angle B$ is acute and $AB < AC$. Let $D$ be the foot of altitude from $A$ to $BC$ and $F$ be the foot of altitude from $E$, the midpoint of $BC$, to $AB$. If $AD = 16$, $BD = 12$, $AF = 5$, find the value of $AC^2$. [b]p23.[/b] Let $a,b,c$ be positive real numbers such that (i) $c > a$ (ii) $10c = 7a +4b +2024$ (iii) $2024 = \frac{(a+c)^2}{a}+ \frac{(c+a)^2}{b}$. Find $a +b +c$. [b]p24.[/b] Let $f^1(x) = x^2 -2x +2$, and for $n > 1$ define $f^n(x) = f ( f^{n-1}(x))$. Find the greatest prime factor of $f^{2018}(2019)-1$. [b]p25.[/b] Let $I$ be the incenter of $\vartriangle ABC$ and $D$ be the intersection of line that passes through $I$ that is perpendicular to $AI$ and $BC$. If $AB = 60$, $C A =120$, and $CD = 100$, find the length of $BC$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Pentagon $SPEAK$ is inscribed in triangle $NOW$ such that $S$ and $P$ lie on segment $NO$, $K$ and $A$ lie on segment $NW$, and $E$ lies on segment $OW$. Suppose that $NS = SP = PO$ and $NK = KA = AW$. Given that $EP = EK = 5$ and $EA = ES = 6$, compute $OW$.

Let $ABC$ be an isosceles obtuse-angled triangle, and $D$ be a point on its base $AB$ such that $AD$ equals to the circumradius of triangle $BCD$. Find the value of $\angle ACD$.

The circles $\omega_1$ and $\omega_2$ centered at $O_1$ and $O_2$ are externally tangent at the point $T$. The circle $\omega_3$ centered at $O_3$ is tangent to the line $AB$ (the external common tangent of $\omega_1$ and $\omega_2$) at $D$ and externally tangent to $\omega_1$ and to $\omega_2$. The line $TD$ intersects again at $\omega_1$. Prove that $O_1 C \parallel AB$. [I]Proposed by V. Rastorguev[/I]

The circles $ S_1 $ and $ S_2 $ intersect at points $ A $ and $ B $. Circle $ S_3 $ externally touches $ S_1 $ and $ S_2 $ at points $ C $ and $ D $ respectively. Let $ K $ be the midpoint of the chord cut by the line $ AB $ on circles $ S_3 $. Prove that $ \angle CKA = \angle DKA $.

In the isosceles triangle $ABC$, with $AB=BC$, points $X$ and $Y$ are the midpoints of the sides $AB$ and $AC$, respectively. Point $Z$ is the foot of the perpendicular from $B$ to $CX$. Prove that the circumcenter of the triangle $XYZ$ is of the line $AC$.

Let \(ABCD\) be a convex quadrilateral in the plane and let \(O_{A}, O_{B}, O_{C}\) and \(O_{D}\) be the circumcenters of the triangles \(BCD, CDA, DAB\) and \(ABC\), respectively. Suppose these four circumcenters are distinct points. Prove that these points are not on a same circle.

The base of a pyramid is a regular triangle having side of size $1$. Two of three angles at the vertex of the pyramid are right. Find the maximum value of the volume of the pyramid.

In $\triangle ABC$, $\angle BAC = 60^{\circ}$, point $D$ lies on side $BC$, $O_1$ and $O_2$ are the centers of the circumcircles of $\triangle ABD$ and $\triangle ACD$, respectively. Lines $BO_1$ and $CO_2$ intersect at point $P$. If $I$ is the incenter of $\triangle ABC$ and $H$ is the orthocenter of $\triangle PBC$, then prove that the four points $B,C,I,H$ are on the same circle.

A circle is inscribed in the triangle $ ABC$ and it's center $I$ and the points of tangency $P, Q, R$ with the sides $BC$, $C A$ and $AB$ are marked, respectively. With a single ruler, build a point $K$ at which the circle passing through the vertices B and $C$ touches (internally) the inscribed circle.

A terrain ( $ABCD$ ) has a rectangular trapezoidal shape. The angle in $A$ measures $90^o$. $AB$ measures $30$ m, $AD$ measures $20$ m and $DC$ measures 45 m. This land must be divided into two areas of the same area, drawing a parallel to the $AD$ side . At what distance from $D$ do we have to draw the parallel? [img]https://1.bp.blogspot.com/-DnyNY3x4XKE/XNYvRUrLVTI/AAAAAAAAKLE/gohd7_S9OeIi-CVUVw-iM63uXE5u-WmGwCK4BGAYYCw/s400/image002.gif[/img]

A triangle is partitioned into three triangles and a quadrilateral by drawing two lines from vertices to their opposite sides. The areas of the three triangles are 3, 7, and 7, as shown. What is the area of the shaded quadrilateral? [asy] unitsize(1.5cm); defaultpen(.8); pair A = (0,0), B = (3,0), C = (1.4, 2), D = B + 0.4*(C-B), Ep = A + 0.3*(C-A); pair F = intersectionpoint( A--D, B--Ep ); draw( A -- B -- C -- cycle ); draw( A -- D ); draw( B -- Ep ); filldraw( D -- F -- Ep -- C -- cycle, mediumgray, black ); label("$7$",(1.25,0.2)); label("$7$",(2.2,0.45)); label("$3$",(0.45,0.35));[/asy] $ \textbf{(A) }15\qquad\textbf{(B) }17\qquad\textbf{(C) }\frac{35}{2}\qquad\textbf{(D) }18\qquad\textbf{(E) }\frac{55}{3} $

For a real number $ a$, let $ \lfloor a \rfloor$ denominate the greatest integer less than or equal to $ a$. Let $ \mathcal{R}$ denote the region in the coordinate plane consisting of points $ (x,y)$ such that \[\lfloor x \rfloor ^2 \plus{} \lfloor y \rfloor ^2 \equal{} 25.\] The region $ \mathcal{R}$ is completely contained in a disk of radius $ r$ (a disk is the union of a circle and its interior). The minimum value of $ r$ can be written as $ \tfrac {\sqrt {m}}{n}$, where $ m$ and $ n$ are integers and $ m$ is not divisible by the square of any prime. Find $ m \plus{} n$.

Let $ABC$ be a triangle with orthocenter $H,$ and let $M$ be the midpoint of $\overline{BC}.$ Suppose that $P$ and $Q$ are distinct points on the circle with diameter $\overline{AH},$ different from $A,$ such that $M$ lies on line $PQ.$ Prove that the orthocenter of $\triangle APQ$ lies on the circumcircle of $\triangle ABC.$ [i]Proposed by Michael Ren[/i]

As shown in the figure below, the in-circle of $ABC$ is tangent to sides $AB$ and $AC$ at $D$ and $E$ respectively, and $O$ is the circumcenter of $BCI$. Prove that $\angle ODB = \angle OEC$. [asy]import graph; size(5.55cm); pathpen=linewidth(0.7); pointpen=black; pen fp=fontsize(10); pointfontpen=fp; real xmin=-5.76,xmax=4.8,ymin=-3.69,ymax=3.71; pen zzttqq=rgb(0.6,0.2,0), wwwwqq=rgb(0.4,0.4,0), qqwuqq=rgb(0,0.39,0); pair A=(-2,2.5), B=(-3,-1.5), C=(2,-1.5), I=(-1.27,-0.15), D=(-2.58,0.18), O=(-0.5,-2.92); D(A--B--C--cycle,zzttqq); D(arc(D,0.25,-104.04,-56.12)--(-2.58,0.18)--cycle,qqwuqq); D(arc((-0.31,0.81),0.25,-92.92,-45)--(-0.31,0.81)--cycle,qqwuqq); D(A--B,zzttqq); D(B--C,zzttqq); D(C--A,zzttqq); D(CR(I,1.35),linewidth(1.2)+dotted+wwwwqq); D(CR(O,2.87),linetype("2 2")+blue); D(D--O); D((-0.31,0.81)--O); D(A); D(B); D(C); D(I); D(D); D((-0.31,0.81)); D(O); MP( "A", A, dir(110)); MP("B", B, dir(140)); D("C", C, dir(20)); D("D", D, dir(150)); D("E", (-0.31, 0.81), dir(60)); D("O", O, dir(290)); D("I", I, dir(100)); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]

Let $A$ and $B$ be two distinct points in the plane. Let $M$ be the midpoint of the segment $AB$, and let $\omega$ be a circle that goes through $A$ and $M$. Let $T$ be a point on $\omega$ such that the line $BT$ is tangent to $\omega$. Let $X$ be a point (other than $B$) on the line $AB$ such that $TB = TX$, and let $Y$ be the foot of the perpendicular from $A$ onto the line $BT$. Prove that the lines $AT$ and $XY$ are parallel. [i]Navneel Singhal, India[/i]