If a point $A_1$ is in the interior of an equilateral triangle $ABC$ and point $A_2$ is in the interior of $\triangle{A_1BC}$, prove that \[\operatorname{I.Q.} (A_1BC) > \operatorname{I.Q.} (A_2BC),\] where the [i]isoperrimetric quotient[/i] of a figure $F$ is defined by \[\operatorname{I.Q.}(F) = \frac{\operatorname{Area}(F)}{[\operatorname{Perimeter}(F)]^2}.\]

In the below figure, there is a regular hexagon and three squares whose sides are equal to $4$ cm. Let $M,N$, and $P$ be the centers of the squares. The perimeter of the triangle $MNP$ can be written in the form $a + b\sqrt3$ (cm), where $a, b$ are integers. Compute the value of $a + b$. [img]https://cdn.artofproblemsolving.com/attachments/e/8/5996e994d4bbed8d3b3269d3e38fc2ec5d2f0b.png[/img]

As shown in the figure, in $\vartriangle ABC$, $AB>AC$, the inscribed circle $I$ is tangent to the sides $BC$, $CA$, $AB$ at points $D$, $E$, $F$ respectively, and the straight lines $BC$ and $EF$ intersect at point $K$, $DG \perp EF$ at point $G$, ray $IG$ intersects the circumscribed circle of $\vartriangle ABC$ at point $H$. Prove that points $H$, $G$, $D$, $K$ lie on a circle. [img]https://cdn.artofproblemsolving.com/attachments/5/e/804fb919e9c2f9cf612099e44bad9c75699b2e.png[/img]

Let $ABCDEFGHIJ$ be a regular $10$-gon. Let $T$ be a point inside the $10$-gon, such that the $DTE$ is isosceles: $DT = ET$ , and its angle at the apex is $72^\circ$. Prove that there exists a point $S$ such that $FTS$ and $HIS$ are both isosceles, and for both of them the angle at the apex is $72^\circ$.

$A(-1,1)$, $B,C$ are points on hyperbola $x^2-y^2=1$. If $\triangle ABC$ is a regular triangle, then the area of $\triangle ABC$ is $\text{(A)}\frac{\sqrt3}{3}\qquad\text{(B)}\frac{3\sqrt3}{2}\qquad\text{(C)}3\sqrt3\qquad\text{(D)}6\sqrt3\qquad$

A rectangle with integer side lengths has the property that its area minus $5$ times its perimeter equals $2023$. Find the minimum possible perimeter of this rectangle.

Consider an arbitrary (optional convex) polygon. It's [i]chord [/i] is a segment whose ends lie on the boundary of the polygon, and itself belongs entirely to the polygon. Will there always be a chord of a polygon that divides it into two equal parts? Is it true that any polygon can be divided by some chord into parts, the area of each of which is not less than $\frac13$ the area of the polygon?

A circle of perimeter $1$ has been dissected into four equal arcs $B_1, B_2, B_3, B_4$. A closed smooth non-selfintersecting curve $C$ has been composed of translates of these arcs (each $B_j$ possibly occurring several times). Prove that the length of $C$ is an integer.

Let $H$ be the orthocenter of triangle $ABC$. Let $D$ and $E$ be points on $AB$ and $AC$ such that $DE$ is parallel to $CH$. If the circumcircle of triangle $BDH$ passes through $M$, the midpoint of $DE$, then prove that $\angle ABM=\angle ACM$

The sum of the angles $A$ and $C$ of a convex quadrilateral $ABCD$ is less than $180^{\circ} .$ Prove that \[AB \cdot CD + AD \cdot BC < AC(AB + AD).\]

Hexagon $ARTSCI$ has side lengths $AR=RT=TS=SC=4\sqrt2$ and $CI=IA=10\sqrt2$. Moreover, the vertices $A$, $R$, $T$, $S$, $C$, and $I$ lie on a circle $\mathcal{K}$. Find the area of $\mathcal{K}$.

[u]Round 1[/u] [b]p1.[/b] Evaluate the sum $1-2+3-...-208+209-210$. [b]p2.[/b] Tony has $14$ beige socks, $15$ blue socks, $6$ brown socks, $8$ blond socks and $7$ black socks. If Tony picks socks out randomly, how many socks does he have to pick in order to guarantee a pair of blue socks? [b]p3.[/b] The price of an item is increased by $25\%$, followed by an additional increase of $20\%$. What is the overall percentage increase? [u]Round 2[/u] [b]p4.[/b] A lamp post is $20$ feet high. How many feet away from the base of the post should a person who is $5$ feet tall stand in order to cast an 8-foot shadow? [b]p5.[/b] How many positive even two-digit integers are there that do not contain the digits $0$, $1$, $2$, $3$ or $4$? [b]p6.[/b] In four years, Jack will be twice as old as Jill. Three years ago, Jack was three times as old as Jill. How old is Jack? [u]Round 3[/u] [b]p7.[/b] Let $x \Delta y = x y^2 -2y$. Compute $20\Delta 18$. [u]p8.[/u] A spider crawls $14$ feet up a wall. If Cheenu is standing $6$ feet from the wall, and is $6$ feet tall, how far must the spider jump to land on his head? [b]p9.[/b] There are fourteen dogs with long nails and twenty dogs with long fur. If there are thirty dogs in total, and three do not have long fur or long nails, how many dogs have both long hair and long nails? [u]Round 4[/u] [b]p10.[/b] Exactly $420$ non-overlapping square tiles, each $1$ inch by $1$ inch, tesselate a rectangle. What is the least possible number of inches in the perimeter of the rectangle? [b]p11.[/b] John drives $100$ miles at fifty miles per hour to see a cat. After he discovers that there was no cat, he drives back at a speed of twenty miles per hour. What was John’s average speed in the round trip? [b]p12.[/b] What percent of the numbers $1,2,3,...,1000$ are divisible by exactly one of the numbers $4$ and $5$? PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h3165992p28809294]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3166045p28809814]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

On grid paper, mark three nodes so that in the triangle they formed, the sum of the two smallest medians equals to half-perimeter.

The square $ABCD$ has side length $1$. The point $E$ lies on the side $CD$. The line through $A$ and $E$ intersects the line through $B$ and $C$ at the point $F$. Prove that $$\frac{1}{|AE|^2}+\frac{1}{|AF|^2}= 1.$$ [img]https://cdn.artofproblemsolving.com/attachments/5/8/4e803eb7748f7a72783065717044cfc06f565f.png[/img]

Say that a (nondegenerate) triangle is [i]funny[/i] if it satisfies the following condition: the altitude, median, and angle bisector drawn from one of the vertices divide the triangle into 4 non-overlapping triangles whose areas form (in some order) a 4-term arithmetic sequence. (One of these 4 triangles is allowed to be degenerate.) Find with proof all funny triangles.

Let the triangle $ABC$ have area $1$. The interior bisectors of the angles $\angle BAC,\angle ABC, \angle BCA$ intersect the sides $(BC), (AC), (AB) $ and the circumscribed circle of the respective triangle $ABC$ at the points $L$ and $G, N$ and $F, Q$ and $E$. The lines $EF, FG,GE$ intersect the bisectors $(AL), (CQ) ,(BN)$ respectively at points $P, M, R$. Determine the area of the hexagon $LMNPR$.

In quadrilateral $ABCD$, sides $AB$ and $CD$ are equal (but not parallel), points $M$ and $N$ are the midpoints of $AD$ and $BC$. The perpendicular bisector of $MN$ intersects sides $AB$ and $CD$ at points $P$ and $Q$, respectively. Prove that $AP = CQ$. (M. Kungozhin)

We are given a figure bounded by arc $AC$ of a circle, and a broken line $ABC$, with the arc and broken line being on opposite sides of the chord $AC$. Construct a line passing through the mid-point of arc $AC$ and dividing the area of the figure into two regions of equal area.

The semicircle with centre $O$ and the diameter $AC$ is divided in two arcs $AB$ and $BC$ with ratio $1: 3$. $M$ is the midpoint of the radium $OC$. Let $T$ be the point of arc $BC$ such that the area of the cuadrylateral $OBTM$ is maximum. Find such area in fuction of the radium.

Let $O$ be a point of three-dimensional space and let $l_1, l_2, l_3$ be mutually perpendicular straight lines passing through $O$. Let $S$ denote the sphere with center $O$ and radius $R$, and for every point $M$ of $S$, let $S_M$ denote the sphere with center $M$ and radius $R$. We denote by $P_1, P_2, P_3$ the intersection of $S_M$ with the straight lines $l_1, l_2, l_3$, respectively, where we put $P_i \neq O$ if $l_i$ meets $S_M$ at two distinct points and $P_i = O$ otherwise ($i = 1, 2, 3$). What is the set of centers of gravity of the (possibly degenerate) triangles $P_1P_2P_3$ as $M$ runs through the points of $S$?

Let $ABCD$ be a parallelogram and $P$ be an arbitrary point of the circumcircle of $\Delta ABD$, different from the vertices. Line $PA$ intersects the line $CD$ at point $Q$. Let $O$ be the center of the circumcircle $\Delta PCQ$. Prove that $\angle ADO = 90^o$.

Given an isosceles triangle $ABC$ with base $BC$. Inside the side $BC$ is given a point $D$. Let $E, F$ be respectively points on the sides $AB, AC$ that $|\angle BED | = |\angle DF C| > 90^o$ . Prove that the circles circumscribed around the triangles $ABF$ and $AEC$ intersect on the line $AD$ at a point different from point $A$. (Patrik Bak, Michal Rolínek)

In triangle $ABC$, let $\omega$ be the excircle opposite to $A$. Let $D, E$ and $F$ be the points where $\omega$ is tangent to $BC, CA$, and $AB$, respectively. The circle $AEF$ intersects line $BC$ at $P$ and $Q$. Let $M$ be the midpoint of $AD$. Prove that the circle $MPQ$ is tangent to $\omega$.

It is given a triangle $ABC$. Let $R$ be the radius of the circumcircle of the triangle and $O_1,O_2,O_3$ be the centers of excircles of the triangle $ABC$ and $q$ is the perimeter of the triangle $O_1O_2O_3$. Prove that $q\le6R\sqrt3$. When does equality hold?

The circular base of a hemisphere of radius $2$ rests on the base of a square pyramid of height $6$. The hemisphere is tangent to the other four faces of the pyramid. What is the edge-length of the base of the pyramid? $ \textbf{(A)}\ 3\sqrt{2} \qquad \textbf{(B)}\ \frac{13}{3} \qquad \textbf{(C)}\ 4\sqrt{2} \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ \frac{13}{2} $