Let $R$ be the set of points $(x, y)$ such that $x$ and $y$ are positive, $x + y$ is at most 2013, and \[ \lceil x \rceil \lfloor y \rfloor = \lfloor x \rfloor \lceil y \rceil. \] Compute the area of set $R$. Recall that $\lfloor a \rfloor$ is the greatest integer that is less than or equal to $a$, and $\lceil a \rceil$ is the least integer that is greater than or equal to $a$.

Given a triangular pyramid $SABC$, in which $\angle BSC = \alpha$, $\angle CSA =\beta$, $\angle ASB = \gamma$, and the dihedral angles at the edges $SA$ and $SB$ have the value of $\phi$ and $\delta$, respectively. Prove that $\gamma > \alpha \cdot \cos \delta +\beta \cdot \cos \phi.$$

Let $f(x)=e^{-x}\ \forall\ x\geq 0$ and let $g$ be a function defined as for every integer $k \ge 0$, a straight line joining $(k,f(k))$ and $(k+1,f(k+1))$ . Find the area between the graphs of $f$ and $g$.

In $\triangle ABC$ $\angle B$ is obtuse and $AB \ne BC$. Let $O$ is the circumcenter and $\omega$ is the circumcircle of this triangle. $N$ is the midpoint of arc $ABC$. The circumcircle of $\triangle BON$ intersects $AC$ on points $X$ and $Y$. Let $BX \cap \omega = P \ne B$ and $BY \cap \omega = Q \ne B$. Prove that $P, Q$ and reflection of $N$ with respect to line $AC$ are collinear.

Construct outside the acute-angled triangle $ABC$ the isosceles triangles $ABA_B, ABB_A , ACA_C,ACC_A ,BCB_C$ and $BCC_B$, so that $$AB = AB_A = BA_B, AC = AC_A=CA_C, BC = BC_B = CB_C$$ and $$\angle BAB_A = \angle ABA_B =\angle CAC_A=\angle ACA_C= \angle BCB_C =\angle CBC_B = a < 90^o$$. Prove that the perpendiculars from $A$ to $B_AC_A$, from $B$ to $A_BC_B$ and from $C$ to $A_CB_C$ are concurrent

The Fibonacci numbers $F_n$ are defined as $F_1 = F_2 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for all $n> 2$. Let $A$ be the minimum area of a (possibly degenerate) convex polygon with $2020$ sides, whose side lengths are the first $2020$ Fibonacci numbers $F_1$, $F_2$, $...$ , $F_{2020}$ (in any order). A degenerate convex polygon is a polygon where all angles are $\le 180^o$. If $A$ can be expressed in the form $$\frac{\sqrt{(F_a-b)^2-c}}{d}$$ , where $a, b, c$ and $d$ are positive integers, compute the minimal possible value of $a + b + c + d$.

Given an integer $n \geqslant 2$. Suppose there is a point $P$ inside a convex cyclic $2n$-gon $A_1 \ldots A_{2n}$ satisfying $$\angle PA_1A_2 = \angle PA_2A_3 = \ldots = \angle PA_{2n}A_1,$$prove that $$ \prod_{i=1}^{n} \left|A_{2i - 1}A_{2i} \right| = \prod_{i=1}^{n} \left|A_{2i}A_{2i+1} \right|,$$where $A_{2n + 1} = A_1$.

Let $ABC$ be a triangle, and let $P$ be a point on side $BC$ such that $\frac{BP}{PC}=\frac{1}{2}$. If $\measuredangle ABC$ $=$ $45^{\circ}$ and $\measuredangle APC$ $=$ $60^{\circ}$, determine $\measuredangle ACB$ without trigonometry.

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 $O$ be the circumcenter of $\triangle ABC$, where $\angle ABC> 90^{\circ}$ and $M$ is the midpoint of $BC$. Point $P$ lies inside $\triangle ABC$ such that $PB\perp PC$. $D,E$ distinct from $P$ lies on the perpendicular to $AP$ through $P$ such that $BD=BP, CE=CP$. If quadrilateral $ADOE$ is a parallelogram, prove that $$\angle OPE = \angle AMB.$$

Let $S$ be a given sphere with center $O$ and radius $r$. Let $P$ be any point outside then sphere $S$, and let $S'$ be the sphere with center $P$ and radius $PO$. Denote by $F$ the area of the surface of the part of $S'$ that lies inside $S$. Prove that $F$ is independent of the particular point $P$ chosen.

On a plane, given a square $D$ with side length 1 and a line which intersects with $D$. For the solid obtained by a rotation of $D$ about the line as the axis, answer the following questions: (1) Suppose that the line $l$ on a plane the same with $D$ isn't parallel to any edges. Prove that the line by which the volume of the solid is maximized has only intersection point with $D$. Note that the line as axis of rotation is parallel to $l$. (2) Find the possible maximum volume for which all solid formed by the rotation axis as line intersecting with $D$. [i]2011 Tokyo Institute of Technology entrance exam, Problem 4[/i]

The sides of a triangle $T$, with vertices $(0,0)$,$(3,0)$ and $(0,3)$ are mirrors. Show that one of the images of the triagle $T_1$ with vertices $(0,0)$,$(0,1)$ and $(2,0)$ is the triangle with vertices $(24,36)$,$(24,37)$ and $(26,36)$.

A disk with radius $10$ and a disk with radius $8$ are drawn so that the distance between their centers is $3$. Two congruent small circles lie in the intersection of the two disks so that they are tangent to each other and to each of the larger circles as shown. The radii of the smaller circles are both $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [asy] size(150); defaultpen(linewidth(1)); draw(circle(origin,10)^^circle((3,0),8)^^circle((5,15/4),15/4)^^circle((5,-15/4),15/4)); [/asy]

Given triangle $ ABC$. Points $ M$, $ N$ are the projections of $ B$ and $ C$ to the bisectors of angles $ C$ and $ B$ respectively. Prove that line $ MN$ intersects sides $ AC$ and $ AB$ in their points of contact with the incircle of $ ABC$.

In the square $ABCD$ denote by $M$ the midpoint of the side $[AB]$, with $P$ the projection of point $B$ on the line $CM$ and with $N$ the midpoint of the segment $[CP]$, Bisector of the angle $DAN$ intersects the line $DP$ at point $Q$. Show that the quadrilateral $BMQN$ is a parallelogram.

Different points $A$ and $D$ are on the same side of the line $BC$, with $|AB| = | BC|= |CD|$ and lines $AD$ and $BC$ are perpendicular. Let $E$ be the intersection point of lines $AD$ and $BC$. Prove that $||BE| - |CE|| < |AD| \sqrt3$

On the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ find the point $T=(x_0,y_0)$ such that the triangle bounded by the axes of the ellipse and the tangent at that point has the least area.

In unequal triangle $ABC$ bisector $AK$ was drawn. Diameter $XY$ of its circumcircle is perpendicular to $AK$ (order of points on circumcircle is $B-X-A-Y-C$). A circle, passing on points $X$ and $Y$, intersect segments $BK$ and $CK$ in points $T$ and $Z$ respectively. Prove that if $KZ=KT$, then $XT \perp YZ$.

Let $ABC$ be a triangle with area $92$ square centimeters. Calculate the area of another triangle whose sides have the same lengths as the medians of triangle $ABC$.

Let there be an acute scalene triangle $ABC$ with circumcenter $O$. Denote $D,E$ be the reflection of $O$ with respect to $AB,AC$, respectively. The circumcircle of $ADE$ meets $AB$, $AC$, the circumcircle of $ABC$ at points $K,L,M$, respectively, and they are all distinct from $A$. Prove that the lines $BC,KL,AM$ are concurrent.

Given an acute triangle $ABC$ let $M$ be the midpoint of $AB$. Point $K$ is given on the other side of line $AC$ from that of point $B$ such that $\angle KMC = 90 ^ \circ $ and $\angle KAC = 180^\circ - \angle ABC$. The tangent to circumcircle of triangle $ABC$ at $A$ intersects line $CK$ at $E$. Prove that the reflection of line $BC$ with respect to $CM$ passes through the midpoint of line segment $ME$.

Some identically sized spheres are piled in $n$ layers in the shape of a square pyramid with one sphere in the top layer, 4 spheres in the second layer, 9 spheres in the third layer, and so forth so that the bottom layer has a square array of $n^2$ spheres. In each layer the centers of the spheres form a square grid so that each sphere is tangent to any sphere adjacent to it on the grid. Each sphere in an upper level is tangent to the four spheres directly below it. The diagram shows how the first three layers of spheres are stacked. A square pyramid is built around the pile of spheres so that the sides of the pyramid are tangent to the spheres on the outside of the pile. There is a positive integer $m$ such that as $n$ gets large, the ratio of the volume of the pyramid to the total volume inside all of the spheres approaches $\frac{\sqrt{m}}{\pi}$. Find $m$. [center][img]https://snag.gy/bIwyl6.jpg[/img][/center]

In triangle $ABC$ point $O$ is circumcenter. Point $T$ is centroid of $ABC$, and points $D$, $E$ and $F$ are circumcenters of triangles $TBC$, $TCA$ and $TAB$. Prove that $O$ is centroid of $DEF$

[color=blue][size=150]PERU TST IMO - 2006[/size] Saturday, may 20.[/color] [b]Question 04[/b] In an actue-angled triangle $ABC$ draws up: its circumcircle $w$ with center $O$, the circumcircle $w_1$ of the triangle $AOC$ and the diameter $OQ$ of $w_1$. The points are chosen $M$ and $N$ on the straight lines $AQ$ and $AC$, respectively, in such a way that the quadrilateral $AMBN$ is a parallelogram. Prove that the intersection point of the straight lines $MN$ and $BQ$ belongs the circumference $w_1.$ --- [url=http://www.mathlinks.ro/Forum/viewtopic.php?t=88513]Spanish version[/url] $\text{\LaTeX}{}$ed by carlosbr