In the tetrahedron $SABC$, points $E, F, K, L$ are the midpoints of the sides $SA , BC, AC, SB$ respectively, . The lengths of the segments $EF$ and $KL$ are equal to $11 cm$ and $13 cm$ respectively, and the length of the segment $AB$ equals to $18 cm$. Find the length of the side $SC$ of the tetrahedron.

Let $O, I$ be the circumcenter and the incenter of triangle $ABC$, $P$ be an arbitrary point on segment $OI$, $P_A$, $P_B$, and $P_C$ be the second common points of lines $PA$, $PB$, and $PC$ with the circumcircle of triangle $ABC$. Prove that the bisectors of angles $BP_AC$, $CP_BA$, and $AP_CB$ concur at a point lying on $OI$.

[u]Round 1[/u] [b]p1.[/b] The temperature inside is $28^o$ F. After the temperature is increased by $5^o$ C, what will the new temperature in Fahrenheit be? [b]p2.[/b] Find the least positive integer value of $n$ such that $\sqrt{2021+n}$ is a perfect square. [b]p3.[/b] A heart consists of a square with two semicircles attached by their diameters as shown in the diagram. Given that one of the semicircles has a diameter of length $10$, then the area of the heart can be written as $a +b\pi$ where $a$ and $b$ are positive integers. Find $a +b$. [img]https://cdn.artofproblemsolving.com/attachments/7/b/d277d9ebad76f288504f0d5273e19df568bc44.png[/img] [u]Round 2[/u] [b]p4.[/b] An $L$-shaped tromino is a group of $3$ blocks (where blocks are squares) arranged in a $L$ shape, as pictured below to the left. How many ways are there to fill a $12$ by $2$ rectangle of blocks (pictured below to the right) with $L$-shaped trominos if the trominos can be rotated or reflected? [img]https://cdn.artofproblemsolving.com/attachments/d/c/cf37cdf9703ae0cd31c38af23b6874fddb3c12.png[/img] [b]p5.[/b] How many permutations of the word $PIKACHU$ are there such that no two vowels are next to each other? [b]p6.[/b] Find the number of primes $n$ such that there exists another prime $p$ such that both $n +p$ and $n-p$ are also prime numbers. [u]Round 3[/u] [b]p7.[/b] Maisy the Bear is at the origin of the Cartesian Plane. WhenMaisy is on the point $(m,n)$ then it can jump to either $(m,n +1)$ or $(m+1,n)$. Let $L(x, y)$ be the number of jumps it takes forMaisy to reach point (x, y). The sum of $L(x, y)$ over all lattice points $(x, y)$ with both coordinates between $0$ and $2020$, inclusive, is denoted as $S$. Find $\frac{S}{2020}$ . [b]p8.[/b] A circle with center $O$ and radius $2$ and a circle with center $P$ and radius $3$ are externally tangent at $A$. Points $B$ and $C$ are on the circle with center $O$ such that $\vartriangle ABC$ is equilateral. Segment $AB$ extends past $B$ to point $D$ and $AC$ extends past $C$ to point $E$ such that $BD = CE = \sqrt3$. The area of $\vartriangle DEP$ can be written as $\frac{a\sqrt{b}}{c}$ where $a$, $b$, and $c$ are integers such that $b$ is squarefree and $gcd (a,c) = 1$. Find $a +b +c$. [b]p9.[/b] Find the number of trailing zeroes at the end of $$\prod^{2021}_{i=1}(2021+i -1) = (2021)(2022)...(4041).$$ [u]Round 4[/u] [b]p10.[/b] Let $a, b$, and $c$ be side lengths of a rectangular prism with space diagonal $10$. Find the value of $$(a +b)^2 +(b +c)^2 +(c +a)^2 -(a +b +c)^2.$$ [b]p11.[/b] In a regular heptagon $ABCDEFG$, $\ell$ is a line through $E$ perpendicular to $DE$. There is a point $P$ on $\ell$ outside the heptagon such that $PA = BC$. Find the measure of $\angle EPA$. [b]p12.[/b] Dunan is being "$SUS$". The word "$SUS$" is a palindrome. Find the number of palindromes that can be written using some subset of the letters $\{S, U, S, S, Y, B, A, K, A\}$. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h3166494p28814284]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3166500p28814367]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A circumscribed quadrilateral $ABCD$ is given. It is known that $\angle{ACB} = \angle{ACD}$. On the angle bisector of $\angle{C}$, a point $E$ is marked such that $AE \bot BD$. Point $F$ is the foot of the perpendicular line from point $E$ to the side $BC$. Prove that $AB = BF$.

An ant crawls along the edges of a cube turning only at its vertices. It has visited one of the vertices $25$ times. Is it possible that it has visited each of the other $7$ vertices exactly $20$ times? (S Tokarev)

In a triangle $ \vartriangle ABC $ with sides integer numbers, it is known that the radius of the circumcircle circumscribed to $ \vartriangle ABC $ measures $ \dfrac {65} {8} $ centimeters and the area is $84$ cm². Determine the lengths of the sides of the triangle.

Circles ${{w} _ {1}}$ and ${{w} _ {2}}$ with centers at points ${{O} _ {1}}$ and ${{ O} _ {2}}$ intersect at points $A$ and $B$, respectively. Around the triangle ${{O} _ {1}} {{O} _ {2}} B$ circumscribe a circle $w$ centered at the point $O$, which intersects the circles ${{w } _ {1}}$ and ${{w} _ {2}}$ for the second time at points $K$ and $L$, respectively. The line $OA$ intersects the circles ${{w} _ {1}}$ and ${{w} _ {2}}$ at the points $M$ and $N$, respectively. The lines $MK$ and $NL$ intersect at the point $P$. Prove that the point $P$ lies on the circle $w$ and $PM = PN$. (Vadym Mitrofanov)

The real root of the equation $8x^3 - 3x^2 - 3x - 1 = 0$ can be written in the form $\frac{\sqrt[3]a + \sqrt[3]b + 1}{c}$, where $a$, $b$, and $c$ are positive integers. Find $a+b+c$.

Find the perimeter of the convex polygon whose coordinates of the vertices are the set of pairs of the integer solutions of the equation $x^2+xy = x + 2y + 9$.

An ant decides to walk on the perimeter of an $ABC$ triangle. The ant can start at any vertex. Whenever the ant is in a vertex, it chooses one of the adjacent vertices and walks directly (in a straight line) to the chosen vertex. a) In how many ways can the ant walk around each vertex exactly twice? b) In how many ways can the ant walk around each vertex exactly three times? Note: For each item, consider that the starting vertex is visited.

Two circles in the plane do not intersect and do not lie inside each other. We choose diameters $A_1B_1$ and $A_2B_2$ of these circles such that the segments $A_1A_2$ and $B_1B_2'$ intersect. Let $A$ and $B$ be the midpoints of the segments $A_1A_2$ and $B_1B_2$, and $C$ be the intersection point of these segments. Prove that the orthocenter of the triangle $ABC$ belongs to a fixed line that does not depend on the choice of diameters.

Let $ABCD$ be a quadrilateral and $M$ the midpoint of the segment $AB$. Outside of the quadrilateral are constructed the equilateral triangles $BCE$, $CDF$ and $DAG$. Let $P$ and $N$ be the midpoints of the segments $GF$ and $EF$. Prove that the triangle $MNP$ is equilateral.

In triangle $ABC$ with $AB = 10$, let$ D$ be a point on side BC such that $AD$ bisects $\angle BAC$. If $\frac{CD}{BD} = 2$ and the area of $ABC$ is $50$, compute the value of $\angle BAD$ in degrees.

In the complex plane, consider squares having the following property: the complex numbers its vertex correspond to are exactly the roots of integer coefficients equation $ x^4 \plus{} px^3 \plus{} qx^2 \plus{} rx \plus{} s \equal{} 0$. Find the minimum of square areas.

Show that in a non-equilateral triangle, the following statements are equivalent: $(a)$ The angles of the triangle are in arithmetic progression. $(b)$ The common tangent to the Nine-point circle and the Incircle is parallel to the Euler Line.

Given five points inside an equilateral triangle of side length $2$, show that there are two points whose distance from each other is at most $ 1$.

Let two circles $S_{1}$ and $S_{2}$ meet at the points $A$ and $B$. A line through $A$ meets $S_{1}$ again at $C$ and $S_{2}$ again at $D$. Let $M$, $N$, $K$ be three points on the line segments $CD$, $BC$, $BD$ respectively, with $MN$ parallel to $BD$ and $MK$ parallel to $BC$. Let $E$ and $F$ be points on those arcs $BC$ of $S_{1}$ and $BD$ of $S_{2}$ respectively that do not contain $A$. Given that $EN$ is perpendicular to $BC$ and $FK$ is perpendicular to $BD$ prove that $\angle EMF=90^{\circ}$.

Given an isosceles triangle $\triangle ABC$, $AB=AC$. A line passes through $M$, the midpoint of $BC$, and intersects segment $AB$ and ray $CA$ at $D$ and $E$, respectively. Let $F$ be a point of $ME$ such that $EF=DM$, and $K$ be a point on $MD$. Let $\Gamma_1$ be the circle passes through $B,D,K$ and $\Gamma_2$ be the circle passes through $C,E,K$. $\Gamma_1$ and $\Gamma_2$ intersect again at $L \neq K$. Let $\omega_1$ and $\omega_2$ be the circumcircle of $\triangle LDE$ and $\triangle LKM$. Prove that, if $\omega_1$ and $\omega_2$ are symmetric wrt $L$, then $BF$ is perpendicular to $BC$.

The altitudes of a tetrahedron intersect in a point. Prove that this point, the foot of one of the altitudes, and the points dividing the other three altitudes in the ratio $2 : 1$ (measuring from the vertices) lie on a sphere. [i]D. Tereshin[/i]

On the extension of the heights $AH_1$ and $BH_2$ of an acute $\triangle ABC$, after points $H_1$ and $H_2$, are chosen points $M$ and $N$ in such way that $\angle MCB = \angle NCA = 30^\circ$. We denote with $C_1$ the intersection point of the lines $MB$ and $NA$. Analogously we define $A_1$ and $B_1$. Prove that the straight lines $AA_1$, $BB_1$, and $CC_1$ intersect in one point.

[u]Round 5[/u] [b]p13.[/b] Jason flips a coin repeatedly. The probability that he flips $15$ heads before flipping $4$ tails can be expressed as $\frac{a}{2^b}$ where $a$ and $b$ are positive integers and $a$ is odd. Find $a +b$. [b]p14.[/b] Triangle $ABC$ has side lengths $AB = 3$, $BC = 3$, and $AC = 4$. Let D be the intersection of the angle bisector of $\angle B AC$ and segment $BC$. Let the circumcircle of $\vartriangle B AD$ intersect segment $AC$ at a point $E$ distinct from $A$. The length of $AE$ can be expressed as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a +b$. [b]p15.[/b] The sum of the squares of all values of $x$ such that $\{(x -2)(x -3)\} = \{(x -1)(x -6)\}$ and $\lfloor x^2 -6x +6 \rfloor= 9$ can be written as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a +b$. Note: $\{a\}$ is the fractional part function, and returns $a -\lfloor a \rfloor$ . [u]Round 6[/u] [b]p16.[/b] Maisy the Polar Bear is at the origin of the Polar Plane ($r = 0, \theta = 0$). Maisy’s location can be expressed as $(r,\theta)$, meaning it is a distance of $r$ away from the origin and at a angle of $\theta$ degrees counterclockwise from the $x$-axis. When Maisy is on the point $(m,n)$ then it can jump to either $(m,n +1)$ or $(m+1,n)$. Maisy cannot jump to any point it has been to before. Let $L(r,\theta)$ be the number of paths Maisy can take to reach point $(r,\theta)$. The sum of $L(r,\theta)$ over all points where $r$ is an integer between $1$ and $2020$ and $\theta$ is an integer between $0$ and $359$ can be written as $\frac{n^k-1}{m}$ for some minimum value of $n$, such that $n$, $k$, and $m$ are all positive integers. Find $n +k +m$. [b]p17.[/b] A circle with center $O$ and radius $2$ and a circle with center $P$ and radius $3$ are externally tangent at $A$. Points $B$ and $C$ are on the circle with center $O$ such that $\vartriangle ABC$ is equilateral. Segment $AB$ extends past $B$ to point $D$ and $AC$ extends past $C$ to point $E$ such that $BD = CE = \sqrt3$. A line through $D$ is tangent to circle $P$ at $F$. Find $DF^2$. [img]https://cdn.artofproblemsolving.com/attachments/2/7/0ee8716cebd6701fcae6544d9e39e68fff35f5.png[/img] [b]p18.[/b] Find the number of trailing zeroes at the end of $$\prod^{2021}_{i=1} (2021i -1) = (2020)(4041)...(2021^2 -1).$$ [u]Round 7[/u] [b]p19.[/b] A function $f (n)$ is defined as follows: $$f (n) = \begin{cases} \frac{n}{3} \,\,\, if \,\,\, n \equiv 0 (mod \, 3) \\ n^2 +4n -5 \,\,\,if \,\,\,n \equiv 1 (mod \, 3) \\ n^2 +n -2 \,\,\, if \,\,\,n \equiv 2 (mod \, 3) \end{cases}$$ Find the number of integer values of $n$ between $2$ and $1000$ inclusive such that $f ( f (... f (n))) = 1$ for some number of applications of $f (n)$. [b]p20.[/b] In the diagram below, the larger circle with diameter $AW$ has radius $16$. $ABCD$ and $WXY Z$ are rhombi where $\angle B AD = \angle XWZ = 60^o$ and $AC = CY = YW$. $M$ is the midpoint of minor arc $AW$, as shown. Let $I$ be the center of the circle with diameter $OM$. Circles with center $P$ and $G$ are tangent to lines $AD$ and $WZ$, respectively, and also tangent to the circle with center $I$ . Given that $IP \perp AD$ and $IG \perp WZ$, the area of $\vartriangle PIG$ can be written as $a +b\sqrt{c}$ where $a$, $b$, and $c$ are positive integers and $c$ is not divisible by the square of a prime. Find $a +b +c$. [b]p21.[/b] In a list of increasing consecutive positive integers, the first item is divisible by $1$, the second item is divisible by $4$, the third item is divisible by $7$, and this pattern increases up to the seventh item being divisible by $19$. Find the remainder when the least possible value of the first item in the list is divided by $100$. [u]Round 8[/u] [b]p22.[/b] Let the answer to Problem $24$ be $C$. Jacob never drinks more than $C$ cups of coffee in a day. He always drinks a positive integer number of cups. The probability that he drinks $C +1-X$ cups is $X$ times the probability he drinks $C$ cups of coffee for any positive number $X$ from $1$ to $C$ inclusive. Find the expected number of cups of coffee he drinks. [b]p23.[/b] Let the answer to Problem $22$ be $A$. Three lines are drawn intersecting the interior of a triangle with side lengths $26$, $28$, and $30$ such that each line is parallel and a distance A away from a respective side. The perimeter of the triangle formed by the three new lines can be expressed as $\frac{a}{b}$ for relatively prime integers $a$ and $b$. Find $a +b$. [b]p24.[/b] Let the answer to Problem $23$ be $B$. Given that $ab-c = bc-a = ca-b$ and $a^2+b^2+c^2 = B +2$, find the sum of all possible values of $|a +b +c|$. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3166489p28814241]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3166500p28814367]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all values of $k$ by which it is possible to divide any triangular region in $k$ quadrilaterals of equal area.

$ABC$ and $A'B'C'$ are equilateral triangles with parallel sides and the same center, as in the figure. The distance between side $BC$ and side $B'C'$ is $\frac{1}{6}$ the altitude of $\triangle ABC$. The ratio of the area of $\triangle A'B'C'$ to the area of $\triangle ABC$ is [asy] size(170); defaultpen(linewidth(0.7)+fontsize(10)); pair H=origin, B=(1,-(1/sqrt(3))), C=(-1,-(1/sqrt(3))), A=(0,(2/sqrt(3))), E=(2,-(2/sqrt(3))), F=(-2,-(2/sqrt(3))), D=(0,(4/sqrt(3))); draw(A--B--C--A^^D--E--F--D); label("$A'$", A, dir(90)); label("$B'$", B, SE); label("$C'$", C, SW); label("$A$", D, dir(90)); label("$B$", E, dir(0)); label("$C$", F, W); [/asy] $ \textbf{(A)}\ \frac{1}{36}\qquad\textbf{(B)}\ \frac{1}{6}\qquad\textbf{(C)}\ \frac{1}{4}\qquad\textbf{(D)}\ \frac{\sqrt{3}}{4}\qquad\textbf{(E)}\ \frac{9+8\sqrt{3}}{36} $

Let $ABCDE$ be a convex pentagon such that $\angle ABC = \angle AED = 90^\circ$. Suppose that the midpoint of $CD$ is the circumcenter of triangle $ABE$. Let $O$ be the circumcenter of triangle $ACD$. Prove that line $AO$ passes through the midpoint of segment $BE$.

A circle touches the lateral sides of a trapezoid $ABCD$ at points $B$ and $C$, and its center lies on $AD$. Prove that the diameter of the circle is less than the medial line of the trapezoid.