Let $R$ be the set of points $(x, y)$ such that $\lfloor x^2 \rfloor = \lfloor y \rfloor$ and $\lfloor y^2 \rfloor = \lfloor x \rfloor$. Compute the area of region $R$. Recall that $\lfloor z \rfloor$ is the greatest integer that is less than or equal to $z$.

All edges of a polyhedron are painted with red or yellow. For an angle of a facet, if the edges determining it are of different colors, then the angle is called [i]excentric[/i]. The[i] excentricity [/i]of a vertex $A$, namely $S_A$, is defined as the number of excentric angles it has. Prove that there exist two vertices $B$ and $C$ such that $S_B + S_C \leq 4$.

Equilateral triangle $ABC$ has side length $2$. A semicircle is drawn with diameter $BC$ such that it lies outside the triangle, and minor arc $BC$ is drawn so that it is part of a circle centered at $A$. The area of the “lune” that is inside the semicircle but outside sector $ABC$ can be expressed in the form $\sqrt{p}-\frac{q\pi}{r}$, where $p, q$, and $ r$ are positive integers such that $q$ and $r$ are relatively prime. Compute $p + q + r$. [img]https://cdn.artofproblemsolving.com/attachments/7/7/f349a807583a83f93ba413bebf07e013265551.png[/img]

Let $ABC$ be a triangle with incenter $I$, and suppose that $AI$, $BI$, and $CI$ intersect $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. Let the circumcircles of $BDF$ and $CDE$ intersect at $D$ and $P$, and let $H$ be the orthocenter of $DEF$. Prove that $HI=HP$.

Point $P$ is inside $\triangle ABC$. Line segments $APD$, $BPE$, and $CPF$ are drawn with $D$ on $BC$, $E$ on $AC$, and $F$ on $AB$ (see the figure at right). Given that $AP=6$, $BP=9$, $PD=6$, $PE=3$, and $CF=20$, find the area of $\triangle ABC$. [asy] size(200); pair A=origin, B=(7,0), C=(3.2,15), D=midpoint(B--C), F=(3,0), P=intersectionpoint(C--F, A--D), ex=B+40*dir(B--P), E=intersectionpoint(B--ex, A--C); draw(A--B--C--A--D^^C--F^^B--E); pair point=P; label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$E$", E, dir(point--E)); label("$F$", F, dir(point--F)); label("$P$", P, dir(0));[/asy]

Let $\Gamma_1$ and $\Gamma_2$ be two circles that tangents each other at point $N$, with $\Gamma_2$ located inside $\Gamma_1$. Let $A, B, C$ be distinct points on $\Gamma_1$ such that $AB$ and $AC$ tangents $\Gamma_2$ at $D$ and $E$, respectively. Line $ND$ cuts $\Gamma_1$ again at $K$, and line $CK$ intersects line $DE$ at $I$. (i) Prove that $CK$ is the angle bisector of $\angle ACB$. (ii) Prove that $IECN$ and $IBDN$ are cyclic quadrilaterals.

Let $B = (-1, 0)$ and $C = (1, 0)$ be fixed points on the coordinate plane. A nonempty, bounded subset $S$ of the plane is said to be [i]nice[/i] if $\text{(i)}$ there is a point $T$ in $S$ such that for every point $Q$ in $S$, the segment $TQ$ lies entirely in $S$; and $\text{(ii)}$ for any triangle $P_1P_2P_3$, there exists a unique point $A$ in $S$ and a permutation $\sigma$ of the indices $\{1, 2, 3\}$ for which triangles $ABC$ and $P_{\sigma(1)}P_{\sigma(2)}P_{\sigma(3)}$ are similar. Prove that there exist two distinct nice subsets $S$ and $S'$ of the set $\{(x, y) : x \geq 0, y \geq 0\}$ such that if $A \in S$ and $A' \in S'$ are the unique choices of points in $\text{(ii)}$, then the product $BA \cdot BA'$ is a constant independent of the triangle $P_1P_2P_3$.

Let $n\geqslant 3$ be an integer. Prove that there exists a set $S$ of $2n$ positive integers satisfying the following property: For every $m=2,3,...,n$ the set $S$ can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality $m$.

Let $A_1A_2A_3...A_n$ be a bicentric polygon with $n$ sides. Denote by $I_i$ the incenter of triangle $A_{i-1}A_iA_{i+1}, A_{i(i+1)}$ the intersection of $A_iA_{i+2}$ and $A_{i-1}A_{i+1},I_{i(i+1)}$ is the incenter of triangle $A_iA_{i(i+1)}A_{i+1}$ ($i = 1, n$). Prove that there exist $2n$ points $I_1, I_2, ..., I_n, I_{12}, I_{23}, ...., I_{n1}$ on the same circle. Nguyễn Văn Linh

Let $\triangle ABC$ and $A'$,$B'$,$C'$ the symmetrics of vertex over opposite sides.The intersection of the circumcircles of $\triangle ABB'$ and $\triangle ACC'$ is $A_1$.$B_1$ and $C_1$ are defined similarly.Prove that lines $AA_1$,$BB_1$ and $CC_1$ are concurent.

Convex quadrilateral $ABCD$ has right angles $\angle A$ and $\angle C$ and is such that $AB=BC$ and $AD=CD$. The diagonals $AC$ and $BD$ intersect at point $M$. Points $P$ and $Q$ lie on the circumcircle of triangle $AMB$ and segment $CD$, respectively, such that points $P$, $M$, and $Q$ are collinear. Suppose that $m\angle ABC=160^\circ$ and $m\angle QMC=40^\circ$. Find $MP\cdot MQ$, given that $MC=6$.

Six circles of unit radius lie in the plane so that the distance between the centers of any two of them is greater than $d$. What is the least value of $d$ such that there always exists a straight line which does not intersect any of the circles and separates the circles into two groups of three?

Three distinct points $A, B$, and $C$ lie on a circle with centre at $O$. The triangles $AOB, BOC$ , and $COA$ have equal area. What are the possible measures of the angles of the triangle $ABC$ ?

In acute triangle $ABC$ ($AB<AC$) point $O$ is center of its circumcircle $\Omega$. Let the tangent to $\Omega$ drawn at point $A$ intersect the line $BC$ at point $D$. Let the line $DO$ intersects the segments $AB$ and $AC$ at points $E$ and $F$, respectively. Point $G$ is constructed such that $AEGF$ is a parallelogram. Let $K$ and $H$ be points of intersection of segment $BC$ with segments $EG$ and $FG$, respectively. Prove that the circle $(GKH)$ touches the circle $\Omega$. [i] Proposed by Dong Luu [/i]

The circle divides each side of an equilateral triangle into three equal parts. Prove that the sum of the squares of the distances from any point of this circle to the vertices of the triangle is constant.

Two parallelograms $ABCD$ and $A'B'C'D'$ are given in space. Points $A'',B'',C'',D''$ divide the segments $AA',BB',CC',DD'$ in the same ratio. What can be said about the quadrilateral $A''B''C''D''$?

Four rectangular paper strips of length $10$ and width $1$ are put flat on a table and overlap perpendicularly as shown. How much area of the table is covered? [asy] size(120); draw((0,0)--(1,0)--(1,4)--(0,4)--(0,0)--(0,1)--(-1,1)--(-1,2)--(0,2)--(0,4)--(-1,4)--(-1,5)--(1,5)--(1,6)--(0,6)--(0,5)--(3,5)--(3,6)--(4,6)--(4,2)--(5,2)--(5,1)--(1,1)--(3,1)--(3,0)--(4,0)--(4,1)); draw((1,4)--(3,4)--(3,2)--(1,2)--(4,2)--(3,2)--(3,6)--(4,6)--(4,5)--(5,5)--(5,4)--(4,4));[/asy] $ \textbf{(A)}\ 36 \qquad \textbf{(B)}\ 40 \qquad \textbf{(C)}\ 44 \qquad \textbf{(D)}\ 98 \qquad \textbf{(E)}\ 100 $

[b]p1.[/b] Lisa is playing the piano at a tempo of $80$ beats per minute. If four beats make one measure of her rhythm, how many seconds are in one measure? [b]p2.[/b] Compute the smallest integer $n > 1$ whose base-$2$ and base-$3$ representations both do not contain the digit $0$. [b]p3.[/b] In a room of $24$ people, $5/6$ of the people are old, and $5/8$ of the people are male. At least how many people are both old and male? [b]p4.[/b] Juan chooses a random even integer from $1$ to $15$ inclusive, and Gina chooses a random odd integer from $1$ to $15$ inclusive. What is the probability that Juan’s number is larger than Gina’s number? (They choose all possible integers with equal probability.) [b]p5.[/b] Set $S$ consists of all positive integers less than or equal to $ 2016$. Let $A$ be the subset of $S$ consisting of all multiples of $6$. Let $B$ be the subset of $S$ consisting of all multiples of $7$. Compute the ratio of the number of positive integers in $A$ but not $B$ to the number of integers in $B$ but not $A$. [b]p6.[/b] Three peas form a unit equilateral triangle on a flat table. Sebastian moves one of the peas a distance $d$ along the table to form a right triangle. Determine the minimum possible value of $d$. [b]p7.[/b] Oumar is four times as old as Marta. In $m$ years, Oumar will be three times as old as Marta will be. In another $n$ years after that, Oumar will be twice as old as Marta will be. Compute the ratio $m/n$. [b]p8.[/b] Compute the area of the smallest square in which one can inscribe two non-overlapping equilateral triangles with side length $ 1$. [b]p9.[/b] Teemu, Marcus, and Sander are signing documents. If they all work together, they would finish in $6$ hours. If only Teemu and Sander work together, the work would be finished in 8 hours. If only Marcus and Sander work together, the work would be finished in $10$ hours. How many hours would Sander take to finish signing if he worked alone? [b]p10.[/b]Triangle $ABC$ has a right angle at $B$. A circle centered at $B$ with radius $BA$ intersects side $AC$ at a point $D$ different from $A$. Given that $AD = 20$ and $DC = 16$, find the length of $BA$. [b]p11.[/b] A regular hexagon $H$ with side length $20$ is divided completely into equilateral triangles with side length $ 1$. How many regular hexagons with sides parallel to the sides of $H$ are formed by lines in the grid? [b]p12[/b]. In convex pentagon $PEARL$, quadrilateral $PERL$ is a trapezoid with side $PL$ parallel to side $ER$. The areas of triangle $ERA$, triangle $LAP$, and trapezoid $PERL$ are all equal. Compute the ratio $\frac{PL}{ER}$. [b]p13.[/b] Let $m$ and $n$ be positive integers with $m < n$. The first two digits after the decimal point in the decimal representation of the fraction $m/n$ are $74$. What is the smallest possible value of $n$? [b]p14.[/b] Define functions $f(x, y) = \frac{x + y}{2} - \sqrt{xy}$ and $g(x, y) = \frac{x + y}{2} + \sqrt{xy}$. Compute $g (g (f (1, 3), f (5, 7)), g (f (3, 5), f (7, 9)))$. [b]p15.[/b] Natalia plants two gardens in a $5 \times 5$ grid of points. Each garden is the interior of a rectangle with vertices on grid points and sides parallel to the sides of the grid. How many unordered pairs of two non-overlapping rectangles can Nataliia choose as gardens? (The two rectangles may share an edge or part of an edge but should not share an interior point.) PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ A,B,C,D,E$ be five distinct points, such that no three of them lie on the same line. Prove that \[ AB\plus{}BC\plus{}CA \plus{} DE < AD \plus{} AE \plus{} BD\plus{}BE \plus{} CD\plus{}CE .\]

Let $\omega$ be a circle with center $O$ and let $A$ be a point outside $\omega$. The tangents from $A$ touch $\omega$ at points $B$, and $C$. Let $D$ be the point at which the line $AO$ intersects the circle such that $O$ is between $A$ and $D$. Denote by $X$ the orthogonal projection of $B$ onto $CD$, by $Y$ the midpoint of the segment $BX$ and by $Z$ the second point of intersection of the line $DY$ with $\omega$. Prove that $ZA$ and $ZC$ are perpendicular to each other.

A convex quadrilateral $ABCD$ has an inscribed circle with center $I$. Let $I_a, I_b, I_c$ and $I_d$ be the incenters of the triangles $DAB, ABC, BCD$ and $CDA$, respectively. Suppose that the common external tangents of the circles $AI_bI_d$ and $CI_bI_d$ meet at $X$, and the common external tangents of the circles $BI_aI_c$ and $DI_aI_c$ meet at $Y$. Prove that $\angle{XIY}=90^{\circ}$.

Circles ω[size=35]1[/size] and ω[size=35]2[/size] have no common point. Where is outerior tangents a and b, interior tangent c. Lines a, b and c touches circle ω[size=35]1[/size] respectively on points A[size=35]1[/size], B[size=35]1[/size] and C[size=35]1[/size], and circle ω[size=35]2[/size] – respectively on points A[size=35]2[/size], B[size=35]2[/size] and C[size=35]2[/size]. Prove that triangles A[size=35]1[/size]B[size=35]1[/size]C[size=35]1[/size] and A[size=35]2[/size]B[size=35]2[/size]C[size=35]2[/size] area ratio is the same as ratio of ω[size=35]1[/size] and ω[size=35]2[/size] radii.

We are given an isosceles triangle $ABC$ such that $BC=a$ and $AB=BC=b$. The variable points $M\in (AC)$ and $N\in (AB)$ satisfy $a^2\cdot AM \cdot AN = b^2 \cdot BN \cdot CM$. The straight lines $BM$ and $CN$ intersect in $P$. Find the locus of the variable point $P$. [i]Dan Branzei[/i]

Four circles of radius $3$ are arranged as shown. Their centers are the vertices of a square. The area of the shaded region is closest to [asy] fill((3,3)--(3,-3)--(-3,-3)--(-3,3)--cycle,lightgray); fill(arc((3,3),(0,3),(3,0),CCW)--(3,3)--cycle,white); fill(arc((3,-3),(3,0),(0,-3),CCW)--(3,-3)--cycle,white); fill(arc((-3,-3),(0,-3),(-3,0),CCW)--(-3,-3)--cycle,white); fill(arc((-3,3),(-3,0),(0,3),CCW)--(-3,3)--cycle,white); draw(circle((3,3),3)); draw(circle((3,-3),3)); draw(circle((-3,-3),3)); draw(circle((-3,3),3)); draw((3,3)--(3,-3)--(-3,-3)--(-3,3)--cycle); [/asy] $\text{(A)}\ 7.7 \qquad \text{(B)}\ 12.1 \qquad \text{(C)}\ 17.2 \qquad \text{(D)}\ 18 \qquad \text{(E)}\ 27$

Rectangular sheet of paper $ABCD$ is folded as shown in the figure. Find the rato $DK: AB$, given that $C_1$ is the midpoint of $AD$. [img]https://3.bp.blogspot.com/-9EkSdxpGnPU/W6dWD82CxwI/AAAAAAAAJHw/iTkEOejlm9U6Dbu427vUJwKMfEOOVn0WwCK4BGAYYCw/s400/Yasinsky%2B2017%2BVIII-IX%2Bp1.png[/img]