Given a rectangle $ABCD$ with a point $P$ inside it. It is known that $PA = 17, PB = 15,$ and $PC = 6.$ What is the length of $PD$?

Given a parallelogram $ABCD$ in which $\angle ABD=90^\circ$ and $\angle CBD=45^\circ$. Point $E$ lies on segment $AD$ such that $BC=CE$. Determine the measure of angle $BCE$.

Let $ABCDE$ be a convex pentagon with $CD= DE$ and $\angle EDC \ne 2 \cdot \angle ADB$. Suppose that a point $P$ is located in the interior of the pentagon such that $AP =AE$ and $BP= BC$. Prove that $P$ lies on the diagonal $CE$ if and only if area $(BCD)$ + area $(ADE)$ = area $(ABD)$ + area $(ABP)$. (Hungary)

There are several discs whose radii are no more that $1$, and whose centers all lie on a segment with length ${l}$. Prove that the union of all the discs has a perimeter not exceeding $4l+8$. [i]Proposed by Morteza Saghafian - Iran[/i]

Placing $n \in {\mathbb N}$ circles with radius $1$ $unit$ inside a square with side $100$ $unit$ such that, whichever line segment with lenght $10$ $unit$ intersect at least one circle. Prove that $$n \geq 416$$

Consider a prism, not necessarily right, whose base is a rhombus $ABCD$ with side $AB = 5$ and diagonal $AC = 8$. A sphere of radius $r$ is tangent to the plane $ABCD$ at $C$ and tangent to the edges $AA_1$ , $BB _1$ and $DD_ 1$ of the prism. Calculate $r$ .

In quadrilateral $ABCD$, $AB=2$, $AD=3$, $BC=CD=\sqrt7$, and $\angle DAB=60^\circ$. Semicircles $\gamma_1$ and $\gamma_2$ are erected on the exterior of the quadrilateral with diameters $\overline{AB}$ and $\overline{AD}$; points $E\neq B$ and $F\neq D$ are selected on $\gamma_1$ and $\gamma_2$ respectively such that $\triangle CEF$ is equilateral. What is the area of $\triangle CEF$?

For which $n > 0$ it is possible to mark several different points and several different circles on the plane in such a way that: — exactly $n$ marked circles pass through each marked point; — exactly $n$ marked points lie on each marked circle; — the center of each marked circle is marked?

Two circles intersect at the points $A$ and $B$. Tangent lines drawn to both of the circles at the point $A$ intersect the circles at the points $M$ and $N$. The lines $BM$ and $BN$ intersect the circles once more at the points $P$ and $Q$ respectively. Prove that the segments $MP$ and $NQ$ are equal. (I Nagel)

Let $\Delta ABC$ be an acute triangle and $H$ its orthocenter. $B_1$ and $C_1$ are the feet of heights from $B$ and $C$, $M$ is the midpoint of $AH$. Point $K$ is on the segment $B_1C_1$, but isn't on line $AH$. Line $AK$ intersects the lines $MB_1$ and $MC_1$ in $E$ and $F$, the lines $BE$ and $CF$ intersect at $N$. Prove that $K$ is the orthocenter of $\Delta NBC$.

For a convex polygon with $n$ edges $F$, if all its diagonals have the equal length, then $\text{(A)}F\in \{\text{quadrilaterals}\}$ $\text{(B)}F\in \{\text{pentagons}\}$ $\text{(C)}F\in \{\text{pentagons}\} \cup\{\text{quadrilaterals}\}$ $\text{(D)}F\in \{\text{convex polygons that have all edges' length equal}\} \cup\{\text{convex polygons that have all inner angles equal}\}$

Consider a circle centered at $O$. Parallel chords $AB$ of length $8$ and $CD$ of length $10$ are of distance $2$ apart such that $AC < AD$. We can write $\tan \angle BOD =\frac{a}{b}$ , where $a, b$ are positive integers such that gcd $(a, b) = 1$. Compute $a + b$.

Let $ABCD$ be a tetrahedron and suppose that $M$ is a point inside it such that $\angle MAD=\angle MBC$ and $\angle MDB=\angle MCA$. Prove that $$MA\cdot MB+MC\cdot MD<\max(AD\cdot BC,AC\cdot BD).$$

[b]p1.[/b] How many perfect squares are in the set: $\{1, 2, 4, 9, 10, 16, 17, 25, 36, 49\}$? [b]p2.[/b] If $a \spadesuit b = a^b - ab - 5$, what is the value of $2 \spadesuit 11$? [b]p3.[/b] Joe can catch $20$ fish in $5$ hours. Jill can catch $35$ fish in $7$ hours. If they work together, and the number of days it takes them to catch $900$ fish is represented by $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers, what is $m + n$? Assume that they work at a constant rate without taking breaks and that there are an infinite number of fish to catch. [b]p4.[/b] What is the units digit of $187^{10}$? [b]p5.[/b] What is the largest number of regions we can create by drawing $4$ lines in a plane? [b]p6.[/b] A regular hexagon is inscribed in a circle. If the area of the circle is $2025\pi$, given that the area of the hexagon can be expressed as $\frac{a\sqrt{b}}{c}$ for positive integers $a$, $b$, $c$ where $gcd(a, c) = 1$ and $b$ is not divisible by the square of any number other than $1$, find $a + b + c$. [b]p7.[/b] Find the number of trailing zeroes in the product $3! \cdot 5! \cdot 719!$. [b]p8.[/b] How many ordered triples $(x, y, z)$ of odd positive integers satisfy $x + y + z = 37$? [b]p9.[/b] Let $N$ be a number with $2021$ digits that has a remainder of $1$ when divided by $9$. $S(N)$ is the sum of the digits of $N$. What is the value of $S(S(S(S(N))))$? [b]p10.[/b] Ayana rolls a standard die $10$ times. If the probability that the sum of the $10$ die is divisible by $6$ is $\frac{m}{n}$ for relatively prime positive integers $m$, $n$, what is $m + n$? [b]p11.[/b] In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. The inscribed circle touches the side $BC$ at point $D$. The line $AI$ intersects side $BC$ at point $K$ given that $I$ is the incenter of triangle $ABC$. What is the area of the triangle $KID$? [b]p12.[/b] Given the cubic equation $2x^3+8x^2-42x-188$, with roots $a, b, c$, evaluate $|a^2b+a^2c+ab^2+b^2c+c^2a+bc^2|$. [b]p13.[/b] In tetrahedron $ABCD$, $AB=6$, $BC=8$, $CA=10$, and $DA$, $DB$, $DC=20$. If the volume of $ABCD$ is $a\sqrt{b}$ where $a$, $b$ are positive integers and in simplified radical form, what is $a + b$? [b]p14.[/b] A $2021$-digit number starts with the four digits $2021$ and the rest of the digits are randomly chosen from the set $0$,$1$,$2$,$3$,$4$,$5$,$6$. If the probability that the number is divisible by $14$ is $\frac{m}{n}$ for relatively prime positive integers $m$, $n$. what is $m + n$? [b]p15.[/b] Let $ABCD$ be a cyclic quadrilateral with circumcenter $O_1$ and circumradius $20$, Let the intersection of $AC$ and $BD$ be $E$. Let the circumcenter of $\vartriangle EDC$ be $O_2$. Given that the circumradius of 4EDC is $13$; $O_1O_2 = 11$, $BE = 11 \sqrt2$, find $O_1E^2$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given a rectangle $A_1A_2A_3A_4$. Four circles with $A_i$ as their centres have their radiuses $r_1, r_2, r_3, r_4$; and $r_1+r_3=r_2+r_4<d$, where d is a diagonal of the rectangle. Two pairs of the outer common tangents to {the first and the third} and {the second and the fourth} circumferences make a quadrangle. Prove that you can inscribe a circle into that quadrangle.

Given triangle $ ABC$ and $ AB\equal{}c$, $ AC\equal{}b$ and $ BC\equal{}a$ satisfying $ a \geq b \geq c$, $ BE$ and $ CF$ are two interior angle bisectors. $ P$ is a point inside triangle $ AEF$. $ R$ and $ Q$ are the projections of $ P$ on sides $ AB$ and $ AC$. Prove that $ PR \plus{} PQ \plus{} RQ < b$.

Given an acute and scalene triangle $ABC$ with $AB<AC$ and random line $(e)$ that passes throuh the center of the circumscribed circles $c(O,R)$. Line $(e)$, intersects sides $BC,AC,AB$ at points $A_1,B_1,C_1$ respectively (point $C_1$ lies on the extension of $AB$ towards $B$). Perpendicular from $A$ on line $(e)$ and $AA_1$ intersect circumscribed circle $c(O,R)$ at points $M$ and $A_2$ respectively. Prove that a) points $O,A_1,A_2, M$ are consyclic b) if $(c_2)$ is the circumcircle of triangle $(OBC_1)$ and $(c_3)$ is the circumcircle of triangle $(OCB_1)$, then circles $(c_1),(c_2)$ and $(c_3)$ have a common chord

In an $ ABC$ triangle such that $ m(\angle B)>m(\angle C)$, the internal and external bisectors of vertice $ A$ intersects $ BC$ respectively at points $ D$ and $ E$. $ P$ is a variable point on $ EA$ such that $ A$ is on $ [EP]$. $ DP$ intersects $ AC$ at $ M$ and $ ME$ intersects $ AD$ at $ Q$. Prove that all $ PQ$ lines have a common point as $ P$ varies.

Let $ a,\ b$ be positive real numbers. Prove that $ \int_{a \minus{} 2b}^{2a \minus{} b} \left|\sqrt {3b(2a \minus{} b) \plus{} 2(a \minus{} 2b)x \minus{} x^2} \minus{} \sqrt {3a(2b \minus{} a) \plus{} 2(2a \minus{} b)x \minus{} x^2}\right|dx$ $ \leq \frac {\pi}3 (a^2 \plus{} b^2).$ [color=green]Edited by moderator.[/color]

On sides $AB, BC$ and $CA$ of triangle $ABC$ are located points $P, M$ and $K$, respectively, so that $AM, BK$ and $CP$ intersect in one point and the sum of the vectors $\overrightarrow{AM}, \overrightarrow{BK}$ and $\overrightarrow{CP}$ equals $ \overrightarrow{0}$. Prove that $K, M$ and $P$ are midpoints of the sides of triangle $ABC$ on which they are located.

A trapezoid $ABCD$ is given with $BC // AD$. Assume that the bisectors of the angles $BAD$ and $CDA$ intersect on the perpendicular bisector of the line segment $BC$. Prove that $|AB|= |CD|$ or $|AB| +|CD| =|AD|$.

A triangle $ABC$ with circumcenter $U$ is given, so that $\angle CBA = 60^o$ and $\angle CBU = 45^o$ apply. The straight lines $BU$ and $AC$ intersect at point $D$. Prove that $AD = DU$. (Karl Czakler)

The diameter of a circle of radius $R$ is divided into $2n$ equal parts. The point $M$ is taken on the circle. Prove that the sum of the squares of the distances from the point $M$ to the points of division (together with the ends of the diameter) does not depend on the choice of the point $M$. Calculate this sum.

Trapezium $ABCD$, where $BC||AD$, is inscribed in a circle, $E$ is midpoint of the arc $AD$ of this circle not containing point $C$ . Let $F$ be the foot of the perpendicular drawn from $E$ on the line tangent to the circle at the point $C$ . Prove that $BC=2CF$.

Point \( A_1 \) inside the acute-angled triangle \( ABC \) is such that \[ \angle ACB = 2\angle A_1BC \quad \text{and} \quad \angle ABC = 2\angle A_1CB. \] Point \( A_2 \) is chosen so that points \( A \) and \( A_2 \) lie on opposite sides of line \( BC \), \( AA_2 \perp BC \), and the perpendicular bisector of \( AA_2 \) is tangent to the circumcircle of \( \triangle ABC \). Define points \( B_1, B_2, C_1, C_2 \) analogously. Prove that the circumcircles of \( \triangle AA_1A_2 \), \( \triangle BB_1B_2 \), and \( \triangle CC_1C_2 \) intersect at exactly two common points. [i]Proposed by Vadym Solomka[/i]