In a convex quadrilateral $ABCD$, $E$ is the intersection of $AB$ and $CD$, $F$ is the intersection of $AD$ and $BC$ and $G$ is the intersection of $AC$ and $EF$. Prove that the following two claims are equivalent: $(i)$ $BD$ and $EF$ are parallel. $(ii)$ $G$ is the midpoint of $EF$.

[b]U[/b]ndecillion years ago in a galaxy far, far away, there were four space stations in the three-dimensional space, each pair spaced 1 light year away from each other. Admiral Ackbar wanted to establish a base somewhere in space such that the sum of squares of the distances from the base to each of the stations does not exceed 15 square light years. (The sizes of the space stations and the base are negligible.) Determine the volume, in cubic light years, of the set of all possible locations for the Admiral’s base.

Let $\Delta ABC$ be a triangle with $AB=AC$ and $\angle A = \alpha$, and let $O,H$ be its circumcenter and orthocenter, respectively. If $P,Q$ are points on $AB$ and $AC$, respectively, such that $APHQ$ forms a rhombus, determine $\angle POQ$ in terms of $\alpha$.

[u]Set 6[/u] [b]B16.[/b] Let $\ell_r$ denote the line $x + ry + r^2 = 420$. Jeffrey draws the lines $\ell_a$ and $\ell_b$ and calculates their single intersection point. [b]B17.[/b] Let set $L$ consist of lines of the form $3x + 2ay = 60a + 48$ across all real constants a. For every line $\ell$ in $L$, the point on $\ell$ closest to the origin is in set $T$ . The area enclosed by the locus of all the points in $T$ can be expressed in the form nπ for some positive integer $n$. Compute $n$. [b]B18.[/b] What is remainder when the $2020$-digit number $202020 ... 20$ is divided by $275$? [u]Set 7[/u] [b]B19.[/b] Consider right triangle $\vartriangle ABC$ where $\angle ABC = 90^o$, $\angle ACB = 30^o$, and $AC = 10$. Suppose a beam of light is shot out from point $A$. It bounces off side $BC$ and then bounces off side $AC$, and then hits point $B$ and stops moving. If the beam of light travelled a distance of $d$, then compute $d^2$. [b]B20.[/b] Let $S$ be the set of all three digit numbers whose digits sum to $12$. What is the sum of all the elements in $S$? [b]B21.[/b] Consider all ordered pairs $(m, n)$ where $m$ is a positive integer and $n$ is an integer that satisfy $$m! = 3n^2 + 6n + 15,$$ where $m! = m \times (m - 1) \times ... \times 1$. Determine the product of all possible values of $n$. [u]Set 8[/u] [b]B22.[/b] Compute the number of ordered pairs of integers $(m, n)$ satisfying $1000 > m > n > 0$ and $6 \cdot lcm(m - n, m + n) = 5 \cdot lcm(m, n)$. [b]B23.[/b] Andrew is flipping a coin ten times. After every flip, he records the result (heads or tails). He notices that after every flip, the number of heads he had flipped was always at least the number of tails he had flipped. In how many ways could Andrew have flipped the coin? [b]B24.[/b] Consider a triangle $ABC$ with $AB = 7$, $BC = 8$, and $CA = 9$. Let $D$ lie on $\overline{AB}$ and $E$ lie on $\overline{AC}$ such that $BCED$ is a cyclic quadrilateral and $D, O, E$ are collinear, where $O$ is the circumcenter of $ABC$. The area of $\vartriangle ADE$ can be expressed as $\frac{m\sqrt{n}}{p}$, where $m$ and $p$ are relatively prime positive integers, and $n$ is a positive integer not divisible by the square of any prime. What is $m + n + p$? [u]Set 9[/u] [i]This set consists of three estimation problems, with scoring schemes described.[/i] [b]B25.[/b] Submit one of the following ten numbers: $$3 \,\,\,\, 6\,\,\,\, 9\,\,\,\, 12\,\,\,\, 15\,\,\,\, 18\,\,\,\, 21\,\,\,\, 24\,\,\,\, 27\,\,\,\, 30.$$ The number of points you will receive for this question is equal to the number you selected divided by the total number of teams that selected that number, then rounded up to the nearest integer. For example, if you and four other teams select the number $27$, you would receive $\left\lceil \frac{27}{5}\right\rceil = 6$ points. [b]B26.[/b] Submit any integer from $1$ to $1,000,000$, inclusive. The standard deviation $\sigma$ of all responses $x_i$ to this question is computed by first taking the arithmetic mean $\mu$ of all responses, then taking the square root of average of $(x_i -\mu)^2$ over all $i$. More, precisely, if there are $N$ responses, then $$\sigma =\sqrt{\frac{1}{N} \sum^N_{i=1} (x_i -\mu)^2}.$$ For this problem, your goal is to estimate the standard deviation of all responses. An estimate of $e$ gives $\max \{ \left\lfloor 130 ( min \{ \frac{\sigma }{e},\frac{e}{\sigma }\}^{3}\right\rfloor -100,0 \}$ points. [b]B27.[/b] For a positive integer $n$, let $f(n)$ denote the number of distinct nonzero exponents in the prime factorization of $n$. For example, $f(36) = f(2^2 \times 3^2) = 1$ and $f(72) = f(2^3 \times 3^2) = 2$. Estimate $N = f(2) + f(3) +.. + f(10000)$. An estimate of $e$ gives $\max \{30 - \lfloor 7 log_{10}(|N - e|)\rfloor , 0\}$ points. PS. You had better use hide for answers. First sets have been posted [url=https://artofproblemsolving.com/community/c4h2777391p24371239]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In trapezoid $ABCD$, $AD$ is parallel to $BC$. $\angle A = \angle D = 45^o$, while $\angle B = \angle C = 135^o$. If $AB = 6$ and the area of $ABCD$ is $30$, find $BC$. [img]https://cdn.artofproblemsolving.com/attachments/0/8/d667522259c773435bc53f5988831aceaef7b7.png[/img]

Let the incircle of a nonisosceles triangle $ABC$ touch $AB$, $AC$ and $BC$ at points $D$, $E$ and $F$ respectively. The corresponding excircle touches the side $BC$ at point $N$. Let $T$ be the common point of $AN$ and the incircle, closest to $N$, and $K$ be the common point of $DE$ and $FT$. Prove that $AK||BC$.

Given an equilateral triangle we select an arbitrary point on its interior. We draw theperpendiculars from that point to the three sides of the triangle. Show that the sum of the lengths of these perpendiculars is equal to the height of the triangle.

A triangle is formed by joining three points whose coordinates are integers. If the $ x$-coordinate and the $ y$-coordinate each have a value of $ 1$, then the area of the triangle, in square units: $ \textbf{(A)}\ \text{must be an integer}\qquad \textbf{(B)}\ \text{may be irrational}\qquad \textbf{(C)}\ \text{must be irrational}\qquad \textbf{(D)}\ \text{must be rational}\qquad \\ \textbf{(E)}\ \text{will be an integer only if the triangle is equilateral.}$

Let $n$ be an integer with $n\geq 3$ and assume that $2n$ vertices of a regular $(4n + 1)-$gon are coloured. Show that there must exist three of the coloured vertices forming an isosceles triangle.

In a 100-dimensional hypercube, each edge has length $ 1$. The box contains $2^{100} + 1$ hyperspheres with the same radius $ r$. The center of one hypersphere is the center of the hypercube, and it touches all the other spheres. Each of the other hyperspheres is tangent to $100$ faces of the hypercube. Thus, the hyperspheres are tightly packed in the hypercube. Find $ r$.

The vertices of a regular hexagon $A_1,A_2,...,A_6$ lie respectively on the sides $B_1B_2$, $B_2B_3$, $B_3B_4$, $B_4B_5$, $B_5B_6$, $B_6B_1$ of a convex hexagon $B_1B_2B_3B_4B_5B_6$. Prove that $$S_{B_1B_2B_3B_4B_5B_6} \le \frac32 S_{A_1A_2A_3A_4A_5A_6}.$$

Let $ABC$ be a scalene triangle with circumcircle $\Omega$ and incenter $I$. Ray $AI$ meets $\overline{BC}$ at $D$ and meets $\Omega$ again at $M$; the circle with diameter $\overline{DM}$ cuts $\Omega$ again at $K$. Lines $MK$ and $BC$ meet at $S$, and $N$ is the midpoint of $\overline{IS}$. The circumcircles of $\triangle KID$ and $\triangle MAN$ intersect at points $L_1$ and $L_2$. Prove that $\Omega$ passes through the midpoint of either $\overline{IL_1}$ or $\overline{IL_2}$. [i]Proposed by Evan Chen[/i]

Given the areas of the three squares in the figure, what is the area of the interior triangle? [asy] real r=22.61986495; pair A=origin, B=(12,0), C=(12,5); draw(A--B--C--cycle); markscalefactor=0.1; draw(rightanglemark(C, B, A)); draw(scale(12)*shift(0,-1)*unitsquare); draw(scale(5)*shift(12/5,0)*unitsquare); draw(scale(13)*rotate(r)*unitsquare); pair P=shift(0,-1)*(13/sqrt(2) * dir(r+45)), Q=(14.5,1.2), R=(6, -7); label("169", P, N); label("25", Q, N); label("144", R, N); [/asy] $ \textbf{(A)}\ 13\qquad\textbf{(B)}\ 30\qquad\textbf{(C)}\ 60\qquad\textbf{(D)}\ 300\qquad\textbf{(E)}\ 1800$

Let $ABC$ be an acute triangle and $D$ a variable point on side $AC$ . Point $E$ is on $BD$ such that $BE =\frac{BC^2-CD\cdot CA}{BD}$ . As $D$ varies on side $AC$ prove that the circumcircle of $ADE$ passes through a fixed point other than $A$ .

Let $ABCD$ be a convex quadrilateral with $AB$ parallel to $CD$. Let $P$ and $Q$ be the midpoints of $AC$ and $BD$, respectively. Prove that if $\angle ABP=\angle CBD$, then $\angle BCQ=\angle ACD$.

A triangle with perimeter $7$ has integer sidelengths. What is the maximum possible area of such a triangle?

Let $ABC$ be an acute triangle. Let $M$ be the midpoint of side $BC$, and let $E$ and $F$ be the feet of the altitudes from $B$ and $C$, respectively. Suppose that the common external tangents to the circumcircles of triangles $BME$ and $CMF$ intersect at a point $K$, and that $K$ lies on the circumcircle of $ABC$. Prove that line $AK$ is perpendicular to line $BC$. [i]Kevin Cong[/i]

Let $K$ and $K'$ be two squares in the same plane, their sides of equal length. Is it possible to decompose $K$ into a finite number of triangles $T_1, T_2, \ldots, T_p$ with mutually disjoint interiors and find translations $t_1, t_2, \ldots, t_p$ such that \[K'=\bigcup_{i=1}^{p} t_i(T_i) \ ? \]

The vertices of a square sheet of paper are $ A$, $ B$, $ C$, $ D$. The sheet is folded in a way that the point $ D$ is mapped to the point $ D'$ on the side $ BC$. Let $ A'$ be the image of $ A$ after the folding, and let $ E$ be the intersection point of $ AB$ and $ A'D'$. Let $ r$ be the inradius of the triangle $ EBD'$. Prove that $ r\equal{}A'E$.

Higher Secondary P1 A polygon is called degenerate if one of its vertices falls on a line that joins its neighboring two vertices. In a pentagon $ABCDE$, $AB=AE$, $BC=DE$, $P$ and $Q$ are midpoints of $AE$ and $AB$ respectively. $PQ||CD$, $BD$ is perpendicular to both $AB$ and $DE$. Prove that $ABCDE$ is a degenerate pentagon.

Let $ABC$ be an acute scalene triangle and let $P$ be a point in the plane. For any point $Q\neq A,B,C$, define $T_A$ to be the unique point such that $\triangle T_ABP \sim \triangle T_AQC$ and $\triangle T_ABP, \triangle T_AQC$ are oriented in the same direction (clockwise or counterclockwise). Similarly define $T_B, T_C$. a) Find all $P$ such that there exists a point $Q$ with $T_A,T_B,T_C$ all lying on the circumcircle of $\triangle ABC$. Call such a pair $(P,Q)$ a [i]tasty pair[/i] with respect to $\triangle ABC$. b) Keeping the notations from a), determine if there exists a tasty pair which is also tasty with respect to $\triangle T_AT_BT_C$. [i]Proposed by Vincent Huang[/i]

The diagonals $ AC$ and $ BD$ of a convex quadrilateral $ ABCD$ intersect at point $ M$. The bisector of $ \angle ACD$ meets the ray $ BA$ at $ K$. Given that $ MA \cdot MC \plus{}MA \cdot CD \equal{} MB \cdot MD$, prove that $ \angle BKC \equal{} \angle CDB$.

Planes $\alpha,\beta,\gamma,\delta$ are tangent to the circumsphere of a tetrahedron $ABCD$ at points $A,B,C,D$, respectively. Line $p$ is the intersection of $\alpha$ and $\beta$, and line $q$ is the intersection of $\gamma$ and $\delta$. Prove that if lines $p$ and $CD$ meet, then lines $q$ and $AB$ lie on a plane.

A $6$-inch-wide rectangle is rotated $90$ degrees about one of its corners, sweeping out an area of $45\pi$ square inches, excluding the area enclosed by the rectangle in its starting position. Find the rectangle’s length in inches.

Let $ABCDEF$ be a convex hexagon such that $\angle A = \angle C = \angle E$ and $\angle B = \angle D = \angle F$ and the (interior) angle bisectors of $\angle A, ~\angle C,$ and $\angle E$ are concurrent. Prove that the (interior) angle bisectors of $\angle B, ~\angle D, $ and $\angle F$ must also be concurrent. [i]Note that $\angle A = \angle FAB$. The other interior angles of the hexagon are similarly described.[/i]