A square plate has side length $L$ and negligible thickness. It is laid down horizontally on a table and is then rotating about the axis $\overline{MN}$ where $M$ and $N$ are the midpoints of two adjacent sides of the square. The moment of inertia of the plate about this axis is $kmL^2$, where $m$ is the mass of the plate and $k$ is a real constant. Find $k$. [color=red]Diagram will be added to this post very soon. If you want to look at it temporarily, see the PDF.[/color] [i]Problem proposed by Ahaan Rungta[/i]

Let $ABC$ be the triangle with $\angle ABC = 76^o$ and $\angle ACB = 72^o$. Points $P$ and $Q$ lie on the sides $(AB)$ and $(AC)$, respectively, such that $\angle ABQ = 22^o$ and $\angle ACP = 44^o$. Find the measure of angle $\angle APQ$.

[b]p1.[/b] $\vartriangle DEF$ is constructed from equilateral $\vartriangle ABC$ by choosing $D$ on $AB$, $E$ on $BC$ and $F$ on $CA$ so that $\frac{DB}{AB}=\frac{EC}{BC}=\frac{FA}{CA}=a$, where $a$ is a number between $0$ and $1/2$. (a) Show that $\vartriangle DEF$ is also equilateral. (b) Determine the value of $a$ that makes the area of $\vartriangle DEF$ equal to one half the area of $\vartriangle ABC$. [b]p2.[/b] A bowl contains some red balls and some white balls. The following operation is repeated until only one ball remains in the bowl: Two balls are drawn at random from the bowl. If they have different colors, then the red one is discarded and the white one is returned to the bowl. If they have the same color, then both are discarded and a red ball (from an outside supply of red balls) is added to the bowl. (Note that this operation—in either case—reduces the number of balls in the bowl by one.) (a) Show that if the bowl originally contained exactly $1$ red ball and $ 2$ white balls, then the color of the ball remaining at the end (i.e., after two applications of the operation) does not depend on chance, and determine the color of this remaining ball. (b) Suppose the bowl originally contained exactly $1986$ red balls and $1986$ white balls. Show again that the color of the ball remaining at the end does not depend on chance and determine its color. [b]p3.[/b] Let $a, b$, and $c$ be three consecutive positive integers, with $a < b < c.$ (a) Show that $ab$ cannot be the square of an integer. (b) Show that $ac$ cannot be the square of an integer. (c) Show that $abc$ cannot be the square of an integer. [b]p4.[/b] Consider the system of equations $$\sqrt{x}+\sqrt{y}=2$$ $$ x^2+y^2=5$$ (a) Show (algebraically or graphically) that there are two or more solutions in real numbers $x$ and $y$. (b) The graphs of the two given equations intersect in exactly two points. Find the equation of the straight line passing through these two points of intersection. [b]p5.[/b] Let $n$ and $m$ be positive integers. An $n \times m $ rectangle is tiled with unit squares. Let $r(n, m)$ denote the number of rectangles formed by the edges of these unit squares. Thus, for example, $r(2, 1) = 3$. (a) Find $r(2, 3)$. (b) Find $r(n, 1)$. (c) Find, with justification, a formula for $r(n, m)$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$ABCDE$ is convex pentagon. $\angle BCA=\angle BEA = \frac{\angle BDA}{2}, \angle BDC =\angle EDA$. Prove, that $\angle DEB=\angle DAC$

Let $ABC$ be an acute triangle with circumcenter $O$ such that $AB=4$, $AC=5$, and $BC=6$. Let $D$ be the foot of the altitude from $A$ to $BC$, and $E$ be the intersection of $AO$ with $BC$. Suppose that $X$ is on $BC$ between $D$ and $E$ such that there is a point $Y$ on $AD$ satisfying $XY\parallel AO$ and $YO\perp AX$. Determine the length of $BX$.

Let $L,M$ and $N$ be points on sides $AC,AB$ and $BC$ of triangle $ABC$, respectively, such that $BL$ is the bisector of angle $ABC$ and segments $AN,BL$ and $CM$ have a common point. Prove that if $\angle ALB=\angle MNB$ then $\angle LNM=90^{\circ}$.

Pentagon $ABCDE$ is inscribed in a circle such that $ACDE$ is a square with area $12$. What is the largest possible area of pentagon $ABCDE$? $\text{(A) }9+3\sqrt{2}\qquad\text{(B) }13\qquad\text{(C) }12+\sqrt{2}\qquad\text{(D) }14\qquad\text{(E) }12+\sqrt{6}-\sqrt{3}$

In a triangle $ABC$, $I$ is the incenter. $D$ is the reflection of $A$ to $I$. the incircle is tangent to $BC$ at point $E$. $DE$ cuts $IG$ at $P$ ($G$ is centroid). $M$ is the midpoint of $BC$. prove that a) $AP||DM$.(15 points) b) $AP=2DM$. (10 points)

Let $z$ be a complex number with $|z| = 2014$. Let $P$ be the polygon in the complex plane whose vertices are $z$ and every $w$ such that $\tfrac{1}{z+w} = \tfrac{1}{z} + \tfrac{1}{w}$. Then the area enclosed by $P$ can be written in the form $n\sqrt{3},$ where $n$ is an integer. Find the remainder when $n$ is divided by $1000$.

In obtuse triangle $ABC$, with the obtuse angle at $A$, let $D$, $E$, $F$ be the feet of the altitudes through $A$, $B$, $C$ respectively. $DE$ is parallel to $CF$, and $DF$ is parallel to the angle bisector of $\angle BAC$. Find the angles of the triangle.

Let $ABCDE$ be a convex pentagon such that quadrilateral $ABDE$ is a parallelogram and quadrilateral $BCDE$ is inscribed in a circle. The circle with center $C$ and radius $CD$ intersects the line $BD, DE$ at points $F, G(\neq D)$, and points $A, F, G$ is on line l. Let $H$ be the intersection point of line $l$ and segment $BC$. Consider the set of circle $\Omega$ satisfying the following condition. Circle $\Omega$ passes through $A, H$ and intersects the sides $AB, AE$ at point other than $A$. Let $P, Q(\neq A)$ be the intersection point of circle $\Omega$ and sides $AB, AE$. Prove that $AP+AQ$ is constant.

Let $ABC$ be a triangle and $AD$ be the angle bisector of $<BAC$, with $D$ on $BC$. Let $E$ be a point on segment $BC$ such that $BD = EC$. Through $E$ draw $l$ a parallel line to $AD$ and let $P$ be a point in $l$ inside the triangle. Let $G$ be the point where $BP$ intersects $AC$ and $F$ be the point where $CP$ intersects $AB$. Show $BF = CG$.

Let $BD$ and $CE$ be the bisectors of the interior angles $\angle B$ and $\angle C$, respectively ($D\in AC$, $E\in AB$). Consider the circumcircle of $ABC$ with center $O$ and the excircle corresponding to the side $BC$ with center $I_a$. These two circles intersect at points $P$ and $Q$. (a) Prove that $PQ$ is parallel to $DE$. (b) Prove that $I_aO$ is perpendicular to $DE$.

Let $A$, $B$, $C$, $D$, $E$, $F$, $G$, $H$ be the eight vertices of a $30 \times30\times30$ cube as shown. The two figures $ACFH$ and $BDEG$ are congruent regular tetrahedra. Find the volume of the intersection of these two tetrahedra. [asy] import graph; size(12.57cm); real labelscalefactor = 0.5; pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); pen dotstyle = black; real xmin = -3.79, xmax = 8.79, ymin = 0.32, ymax = 4.18; /* image dimensions */ pen ffqqtt = rgb(1,0,0.2); pen ffzzzz = rgb(1,0.6,0.6); pen zzzzff = rgb(0.6,0.6,1); draw((6,3.5)--(8,1.5), zzzzff); draw((7,3)--(5,1), blue); draw((6,3.5)--(7,3), blue); draw((6,3.5)--(5,1), blue); draw((5,1)--(8,1.5), blue); draw((7,3)--(8,1.5), blue); draw((4,3.5)--(2,1.5), ffzzzz); draw((1,3)--(2,1.5), ffqqtt); draw((2,1.5)--(3,1), ffqqtt); draw((1,3)--(3,1), ffqqtt); draw((4,3.5)--(1,3), ffqqtt); draw((4,3.5)--(3,1), ffqqtt); draw((-3,3)--(-3,1), linewidth(1.6)); draw((-3,3)--(-1,3), linewidth(1.6)); draw((-1,3)--(-1,1), linewidth(1.6)); draw((-3,1)--(-1,1), linewidth(1.6)); draw((-3,3)--(-2,3.5), linewidth(1.6)); draw((-2,3.5)--(0,3.5), linewidth(1.6)); draw((0,3.5)--(-1,3), linewidth(1.6)); draw((0,3.5)--(0,1.5), linewidth(1.6)); draw((0,1.5)--(-1,1), linewidth(1.6)); draw((-3,1)--(-2,1.5)); draw((-2,1.5)--(0,1.5)); draw((-2,3.5)--(-2,1.5)); draw((1,3)--(1,1), linewidth(1.6)); draw((1,3)--(3,3), linewidth(1.6)); draw((3,3)--(3,1), linewidth(1.6)); draw((1,1)--(3,1), linewidth(1.6)); draw((1,3)--(2,3.5), linewidth(1.6)); draw((2,3.5)--(4,3.5), linewidth(1.6)); draw((4,3.5)--(3,3), linewidth(1.6)); draw((4,3.5)--(4,1.5), linewidth(1.6)); draw((4,1.5)--(3,1), linewidth(1.6)); draw((1,1)--(2,1.5)); draw((2,3.5)--(2,1.5)); draw((2,1.5)--(4,1.5)); draw((5,3)--(5,1), linewidth(1.6)); draw((5,3)--(6,3.5), linewidth(1.6)); draw((5,3)--(7,3), linewidth(1.6)); draw((7,3)--(7,1), linewidth(1.6)); draw((5,1)--(7,1), linewidth(1.6)); draw((6,3.5)--(8,3.5), linewidth(1.6)); draw((7,3)--(8,3.5), linewidth(1.6)); draw((7,1)--(8,1.5)); draw((5,1)--(6,1.5)); draw((6,3.5)--(6,1.5)); draw((6,1.5)--(8,1.5)); draw((8,3.5)--(8,1.5), linewidth(1.6)); label("$ A $",(-3.4,3.41),SE*labelscalefactor); label("$ D $",(-2.16,4.05),SE*labelscalefactor); label("$ H $",(-2.39,1.9),SE*labelscalefactor); label("$ E $",(-3.4,1.13),SE*labelscalefactor); label("$ F $",(-1.08,0.93),SE*labelscalefactor); label("$ G $",(0.12,1.76),SE*labelscalefactor); label("$ B $",(-0.88,3.05),SE*labelscalefactor); label("$ C $",(0.17,3.85),SE*labelscalefactor); label("$ A $",(0.73,3.5),SE*labelscalefactor); label("$ B $",(3.07,3.08),SE*labelscalefactor); label("$ C $",(4.12,3.93),SE*labelscalefactor); label("$ D $",(1.69,4.07),SE*labelscalefactor); label("$ E $",(0.60,1.15),SE*labelscalefactor); label("$ F $",(2.96,0.95),SE*labelscalefactor); label("$ G $",(4.12,1.67),SE*labelscalefactor); label("$ H $",(1.55,1.82),SE*labelscalefactor); label("$ A $",(4.71,3.47),SE*labelscalefactor); label("$ B $",(7.14,3.10),SE*labelscalefactor); label("$ C $",(8.14,3.82),SE*labelscalefactor); label("$ D $",(5.78,4.08),SE*labelscalefactor); label("$ E $",(4.6,1.13),SE*labelscalefactor); label("$ F $",(6.93,0.96),SE*labelscalefactor); label("$ G $",(8.07,1.64),SE*labelscalefactor); label("$ H $",(5.65,1.90),SE*labelscalefactor); dot((-3,3),dotstyle); dot((-3,1),dotstyle); dot((-1,3),dotstyle); dot((-1,1),dotstyle); dot((-2,3.5),dotstyle); dot((0,3.5),dotstyle); dot((0,1.5),dotstyle); dot((-2,1.5),dotstyle); dot((1,3),dotstyle); dot((1,1),dotstyle); dot((3,3),dotstyle); dot((3,1),dotstyle); dot((2,3.5),dotstyle); dot((4,3.5),dotstyle); dot((4,1.5),dotstyle); dot((2,1.5),dotstyle); dot((5,3),dotstyle); dot((5,1),dotstyle); dot((6,3.5),dotstyle); dot((7,3),dotstyle); dot((7,1),dotstyle); dot((8,3.5),dotstyle); dot((8,1.5),dotstyle); dot((6,1.5),dotstyle); [/asy]

The incircle of the triangle $ABC$ touches the sides $AC$ and $BC$ at points $K$ and $L$, respectively. the $B$-excircle touches the side $AC$ of this triangle at point $P$. The segment $AL$ intersects the inscribed circle for the second time at point $S$. Line $KL$ intersects the circumscribed circle of triangle $ASK$ for the second at point $M$. Prove that $PL = PM$.

We say that a triangle $ABC$ is great if the following holds: for any point $D$ on the side $BC$, if $P$ and $Q$ are the feet of the perpendiculars from $D$ to the lines $AB$ and $AC$, respectively, then the reflection of $D$ in the line $PQ$ lies on the circumcircle of the triangle $ABC$. Prove that triangle $ABC$ is great if and only if $\angle A = 90^{\circ}$ and $AB = AC$. [i]Senior Problems Committee of the Australian Mathematical Olympiad Committee[/i]

Let $ABC$ be a triangle, and its incenter touches the sides $BC,CA,AB$ at $D,E,F$ respectively. Let $AD$ intersects the incircle of $ABC$ at $M$ distinct from $D$. Let $DF$ intersects the circumcircle of $CDM$ at $N$ distinct from $D$. Let $CN$ intersects $AB$ at $G$. Prove that $EC = 3GF$.

Let $ABC$ be a triangle with $AB = 5$, $BC = 6$, and $CA = 7$. Denote $\Gamma$ the incircle of $ABC$, let $I$ be the center of $\Gamma$ . The circumcircle of $BIC$ intersects $\Gamma$ at $X_1$ and $X_2$. The circumcircle of $CIA$ intersects $\Gamma$ at $Y_1$ and $Y_2$. The circumcircle of $AIB$ intersects $\Gamma$ at $Z_1$ and $Z_2$. The area of the triangle determined by $\overline{X_1X_2}$, $\overline{Y_1Y_2}$, and $\overline{Z_1Z_2}$ equals $\frac{m \sqrt{p}}{n}$ for positive integers $m, n$, and $p$, where $m$ and$ n$ are relatively prime and $p$ is squarefree. Compute $m+n+ p$.

Renata the robot packs boxes in a warehouse. Each box is a cube of side length $1$ foot. The warehouse floor is a square, $12$ feet on each side, and is divided into a $12$-by-$12$ grid of square tiles $1$ foot on a side. Each tile can either support one box or be empty. The warehouse has exactly one door, which opens onto one of the corner tiles. Renata fits on a tile and can roll between tiles that share a side. To access a box, Renata must be able to roll along a path of empty tiles starting at the door and ending at a tile sharing a side with that box. [list=a] [*]Show how Renata can pack $91$ boxes into the warehouse and still be able to access any box. [*]Show that Renata [b]cannot[/b] pack $95$ boxes into the warehouse and still be able to access any box.[/list]

Triangle $ABC$ has circumcenter $O$. $M$ is the midpoint of arc $BC$ not containing $A$. $S$ is a point on $(ABC)$ such that $AS$ and $BC$ intersect on the line passing through $O$ and perpendicular to $AM$. $D$ is a point such that $ABDC$ is a parallelogram. Prove that $D$ lies on the line $SM$.

In the exterior of an equilateral triangle $ABC$ of side $\alpha$ we construct an isosceles right-angled triangle $ACD$ with $\angle CAD=90^0.$The lines $DA$ and $CB$ meet at point $E$. (a) Find the angle $\angle DBC.$ (b) Express the area of triangle $CDE$ in terms of $\alpha.$ (c) Find the length of $BD.$

Let $H$ be the orthocenter of a nonisosceles triangle $ABC$. The bisector of angle $BHC$ meets $AB$ and $AC$ at points $P$ and $Q$ respectively. The perpendiculars to $AB$ and $AC$ from $P$ and $Q$ meet at $K$. Prove that $KH$ bisects the segment $BC$.

A straight line passing through the center of the circumscribed circle and the intersection point of the heights of the non-equilateral triangle $ABC$ divides its perimeter and area in the same ratio.Find this ratio.

Let $[ABC]$ be a acute-angled triangle and its circumscribed circle $\Gamma$. Let $D$ be the point on the line $AB$ such that $A$ is the midpoint of the segment $[DB]$ and $P$ is the point of intersection of $CD$ with $\Gamma$. Points $W$ and $L$ lie on the smaller arcs $\overarc{BC}$ and $\overarc{AB}$, respectively, and are such that $\overarc{BW} = \overarc{LA }= \overarc{AP}$. The $LC$ and $AW$ lines intersect at $Q$. Shows that $LQ = BQ$.

Let the circles $\omega$ and $\omega'$ intersect in points $A$ and $B$. The tangent to circle $\omega$ at $A$ intersects $\omega'$ at $C$ and the tangent to circle $\omega'$ at $A$ intersects $\omega$ at $D$. Suppose that the internal bisector of $\angle CAD$ intersects $\omega$ and $\omega'$ at $E$ and $F$, respectively, and the external bisector of $\angle CAD$ intersects $\omega$ and $\omega'$ at $X$ and $Y$, respectively. Prove that the perpendicular bisector of $XY$ is tangent to the circumcircle of triangle $BEF$. [i]Proposed by Mahdi Etesami Fard[/i]