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 $ BC$ of right triangle $ ABC$ be the diameter of a circle intersecting hypotenuse $ AB$ in $ D$. At $ D$ a tangent is drawn cutting leg $ CA$ in $ F$. This information is [u]not[/u] sufficient to prove that $ \textbf{(A)}\ DF \text{ bisects }CA \qquad \textbf{(B)}\ DF \text{ bisects }\angle CDA$ $ \textbf{(C)}\ DF \equal{} FA \qquad \textbf{(D)}\ \angle A \equal{} \angle BCD \qquad \textbf{(E)}\ \angle CFD \equal{} 2\angle A$

Let $ABCD$ be a convex quadrilateral with side lengths satisfying the equality: $$ AB \cdot CD = AD \cdot BC = AC \cdot BD.$$ Determine the sum of the acute angles of quadrilateral $ABCD$. [i]Proposed by Zaza Meliqidze, Georgia[/i]

Let $I$ and $O$ be respectively the incentre and circumcentre of a triangle $ABC$. If $AB = 2$, $AC = 3$ and $\angle AIO = 90^{\circ}$, find the area of $\triangle ABC$.

Let $I,\ O$ and $\Gamma$ be the incenter, circumcenter and the circumcircle of triangle $ABC$, respectively. Line $AI$ meets $\Gamma$ at $M$ $(M\neq A)$. The circumference $\omega$ is tangent internally to $\Gamma$ at $T$, and is tangent to the lines $AB$ and $AC$. The tangents through $A$ and $T$ to $\Gamma$ intersect at $P$. Lines $PI$ and $TM$ meet at $Q$. Prove that the lines $QA$ and $MO$ meet at a point on $\Gamma$.

In a circle $C_1$ with centre $O$, let $AB$ be a chord that is not a diameter. Let $M$ be the midpoint of this chord $AB$. Take a point $T$ on the circle $C_2$ with $OM$ as diameter. Let the tangent to $C_2$ at $T$ meet $C_1$ at $P$. Show that $PA^2 + PB^2 = 4 \cdot PT^2$.

Prove that if $a,b,c,d$ are lengths of the successive sides of a quadrangle (not necessarily convex) with the area equal to $S$, then the following inequality holds \[S \leq \frac{1}{2}(ac+bd).\] For which quadrangles does the inequality become equality?

$ ABC$ is an acute triangle. The points $ B'$ and $ C'$are the reflections of $ B$ and $ C$ across the lines $ AC$ and $ AB$ respectively. Suppose that the circumcircles of triangles$ ABB$' and $ ACC'$ meet at $ A$ and $ P$. Prove that the line $ PA$ passes through the circumcenter of triangle$ ABC.$

A square sheet of paper $ABCD$ is folded straight in such a way that point $B$ hits to the midpoint of side $CD$. In what ratio does the fold line divide side $BC$?

Square $ABCD$ has side length $13$, and points $E$ and $F$ are exterior to the square such that $BE=DF=5$ and $AE=CF=12$. Find $EF^{2}$. [asy] size(200); defaultpen(fontsize(10)); real x=22.61986495; pair A=(0,26), B=(26,26), C=(26,0), D=origin, E=A+24*dir(x), F=C+24*dir(180+x); draw(B--C--F--D--C^^D--A--E--B--A, linewidth(0.7)); dot(A^^B^^C^^D^^E^^F); pair point=(13,13); 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));[/asy]

Find the area of quadrilateral $ABCD$ if: two opposite angles are right;two sides which form right angle are of equal length and sum of lengths of other two sides is $10$

A crystal of pyrite is a parallelepiped with dashed faces. The dashes on any two adjacent faces are perpendicular. Does there exist a convex polytope with the number of faces not equal to 6, such that its faces can be dashed in such a manner?

The figure below is cut along the lines into polygons (which need not be convex). No polygon contains a $2 \times 2$ square. What is the smallest possible number of polygons? [missing figure]

Let $I$ be the incenter of a triangle $ABC$ with $\angle BAC=60^\circ$. A line through $I$ parallel to $AC$ intersects $AB$ at $F$. Let $P$ be the point on the side $BC$ such that $3BP=BC$. Prove that $\angle BFP=\frac12\angle ABC$.

[hide=D stands for Dedekind, Z stands for Zermelo]they had two problem sets under those two names[/hide] [b]D1.[/b] The product of two positive integers is $5$. What is their sum? [b]D2.[/b] Gavin is $4$ feet tall. He walks $5$ feet before falling forward onto a cushion. How many feet is the top of Gavin’s head from his starting point? [b]D3.[/b] How many times must Nathan roll a fair $6$-sided die until he can guarantee that the sum of his rolls is greater than $6$? [b]D4 / Z1.[/b] What percent of the first $20$ positive integers are divisible by $3$? [b]D5.[/b] Let $a$ be a positive integer such that $a^2 + 2a + 1 = 36$. Find $a$. [b]D6 / Z2.[/b] It is said that a sheet of printer paper can only be folded in half $7$ times. A sheet of paper is $8.5$ inches by $11$ inches. What is the ratio of the paper’s area after it has been folded in half $7$ times to its original area? [b]D7 / Z3.[/b] Boba has an integer. They multiply the number by $8$, which results in a two digit integer. Bubbles multiplies the same original number by 9 and gets a three digit integer. What was the original number? [b]D8.[/b] The average number of letters in the first names of students in your class of $24$ is $7$. If your teacher, whose first name is Blair, is also included, what is the new class average? [b]D9 / Z4.[/b] For how many integers $x$ is $9x^2$ greater than $x^4$? [b]D10 / Z5.[/b] How many two digit numbers are the product of two distinct prime numbers ending in the same digit? [b]D11 / Z6.[/b] A triangle’s area is twice its perimeter. Each side length of the triangle is doubled,and the new triangle has area $60$. What is the perimeter of the new triangle? [b]D12 / Z7.[/b] Let $F$ be a point inside regular pentagon $ABCDE$ such that $\vartriangle FDC$ is equilateral. Find $\angle BEF$. [b]D13 / Z8.[/b] Carl, Max, Zach, and Amelia sit in a row with $5$ seats. If Amelia insists on sitting next to the empty seat, how many ways can they be seated? [b]D14 / Z9.[/b] The numbers $1, 2, ..., 29, 30$ are written on a whiteboard. Gumbo circles a bunch of numbers such that for any two numbers he circles, the greatest common divisor of the two numbers is the same as the greatest common divisor of all the numbers he circled. Gabi then does the same. After this, what is the least possible number of uncircled numbers? [b]D15 / Z10.[/b] Via has a bag of veggie straws, which come in three colors: yellow, orange, and green. The bag contains $8$ veggie straws of each color. If she eats $22$ veggie straws without considering their color, what is the probability she eats all of the yellow veggie straws? [b]Z11.[/b] We call a string of letters [i]purple[/i] if it is in the form $CVCCCV$ , where $C$s are placeholders for (not necessarily distinct) consonants and $V$s are placeholders for (not necessarily distinct) vowels. If $n$ is the number of purple strings, what is the remainder when $n$ is divided by $35$? The letter $y$ is counted as a vowel. [b]Z12.[/b] Let $a, b, c$, and d be integers such that $a+b+c+d = 0$ and $(a+b)(c+d)(ab+cd) = 28$. Find $abcd$. [b]Z13.[/b] Griffith is playing cards. A $13$-card hand with Aces of all $4$ suits is known as a godhand. If Griffith and $3$ other players are dealt $13$-card hands from a standard $52$-card deck, then the probability that Griffith is dealt a godhand can be expressed in simplest form as $\frac{a}{b}$. Find $a$. [b]Z14.[/b] For some positive integer $m$, the quadratic $x^2 + 202200x + 2022m$ has two (not necessarily distinct) integer roots. How many possible values of $m$ are there? [b]Z15.[/b] Triangle $ABC$ with altitudes of length $5$, $6$, and $7$ is similar to triangle $DEF$. If $\vartriangle DEF$ has integer side lengths, find the least possible value of its perimeter. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Two circles intersect at $A$ and $B$. A line through $A$ meets the first circle again at $C$ and the second circle again at $D$. Let $M$ and $N$ be the midpoints of the arcs $BC$ and $BD$ not containing $A$, and let $K$ be the midpoint of the segment $CD$. Show that $\angle MKN =\pi/2$. (You may assume that $C$ and $D$ lie on opposite sides of $A$.) [i]D. Tereshin[/i]

Consider points $D,E$ and $F$ on sides $BC,AC$ and $AB$, respectively, of a triangle $ABC$, such that $AD, BE$ and $CF$ concurr at a point $G$. The parallel through $G$ to $BC$ cuts $DF$ and $DE$ at $H$ and $I$, respectively. Show that triangles $AHG$ and $AIG$ have the same areas.

Let $ABC$ be a triangle with $AB < AC$ and let $D{}$ be the other intersection point of the angle bisector of $\angle A$ with the circumcircle of the triangle $ABC$. Let $E{}$ and $F{}$ be points on the sides $AB$ and $AC$ respectively, such that $AE = AF$ and let $P{}$ be the point of intersection of $AD$ and $EF$. Let $M{}$ be the midpoint of $BC{}$. Prove that $AM$ and the circumcircles of the triangles $AEF$ and $PMD$ pass through a common point.

Triangle $ABC$ has positive integer side lengths with $AB=AC$. Let $I$ be the intersection of the bisectors of $\angle B$ and $\angle C$. Suppose $BI=8$. Find the smallest possible perimeter of $\triangle ABC$.

Let $ABC$ be a triangle with $AB > AC$. Let $P$ be a point on the line $AB$ beyond $A$ such that $AP +P C = AB$. Let $M$ be the mid-point of $BC$ and let $Q$ be the point on the side $AB$ such that $CQ \perp AM$. Prove that $BQ = 2AP.$

$(YUG 2)$ A park has the shape of a convex pentagon of area $50000\sqrt{3} m^2$. A man standing at an interior point $O$ of the park notices that he stands at a distance of at most $200 m$ from each vertex of the pentagon. Prove that he stands at a distance of at least $100 m$ from each side of the pentagon.

A quadrilateral $ABCD$ with $AB \perp BC$ , $AD \perp DC$, $E$ is a point that is on the line $BD$ with $EC=CA$ , $F$, $G$ is on the line $AB$ $AD$ such that $EF\perp AC $ and $EG\perp AC$ ,let $X Y$ be the midpoint of segment $AF AG $ , let $Z W$ be the midpoint of segment $BE DE $ , try to proof that $(WBX)$ is tangent to $(ZDY)$

Point $X$ is chosen inside a convex $ABCD$ so that $\angle XBC = \angle XAD, \angle XCB = \angle XDA$. Rays $AB, DC$ intersect at point $O$, circumcircles of triangles $BCO, ADO$ intersect at point $T$. Prove that line $TX$ and the line through $O$ perpendicular to $BC$ intersect on the circumcircle of $\triangle AOD$. [i]Proposed by Anton Trygub[/i]

In the quadrilateral $ABCD$, the length of the sides $AB$ and $BC$ is equal to $1, \angle B= 100^o , \angle D= 130^o$ . Find the length of $BD$.

Let $D$ be the midpoint of the base $AB$ of the isosceles and acute angle triangle $ABC$, $E$ is a point on $AB$ and $O$ circumcenter of the triangle $ACE$. Prove that the line that passes through $D$ perpendicular to $DO$, the line that passes through $E$ perpendicular to $BC$ and the line that passes through$ B$ parallel to $AC$, they intersect at a point.