A rectangular piece of paper with side lengths 5 by 8 is folded along the dashed lines shown below, so that the folded flaps just touch at the corners as shown by the dotted lines. Find the area of the resulting trapezoid. [asy] size(150); defaultpen(linewidth(0.8)); draw(origin--(8,0)--(8,5)--(0,5)--cycle,linewidth(1)); draw(origin--(8/3,5)^^(16/3,5)--(8,0),linetype("4 4")); draw(origin--(4,3)--(8,0)^^(8/3,5)--(4,3)--(16/3,5),linetype("0 4")); label("$5$",(0,5/2),W); label("$8$",(4,0),S); [/asy]

In an acute-angled triangle $ABC$, the altitudes $AA_1,BB_1,CC_1$ intersect at point $V$. If the triangles $AC_1V, BA_1V, CB_1V$ have the same area, does it follow that the triangle $ABC$ is equilateral?

A cube with side length 10 is suspended above a plane. The vertex closest to the plane is labelled $A$. The three vertices adjacent to vertex $A$ are at heights 10, 11, and 12 above the plane. The distance from vertex $A$ to the plane can be expressed as $\tfrac{r-\sqrt{s}}{t}$, where $r$, $s$, and $t$ are positive integers, and $r+s+t<1000$. Find $r+s+t$.

A black witch's hat is in the classic shape of a cone on top of a circular brim. The cone has a slant height of $18$ inches and a base radius of $3$ inches. The brim has a radius of $5$ inches. What is the total surface area of the hat?

Given a nonequilateral triangle $ABC$, the vertices listed counterclockwise, find the locus of the centroids of the equilateral triangles $A'B'C'$ (the vertices listed counterclockwise) for which the triples of points $A,B', C'; A',B, C';$ and $A',B', C$ are collinear. [i]Proposed by Poland.[/i]

Let $ABC$ be a triangle and on the sides we draw, externally, the squares $BADE, CBFG$ and $ACHI$. Determine the greatest positive real constant $k$ such that, for any triangle $\triangle ABC$, the following inequality is true: $[DEFGHI]\geq k\cdot [ABC]$ Note: $[X]$ denotes the area of polygon $X$.

Let $ C$ be the parameterized curve for a given positive number $ r$ and $ 0\leq t\leq \pi$, $ C: \left\{\begin{array}{ll} x \equal{} 2r(t \minus{} \sin t\cos t) & \quad \\ y \equal{} 2r\sin ^ 2 t & \quad \end{array} \right.$ When the point $ P$ moves on the curve $ C$, (1) Find the magnitude of acceleralation of the point $ P$ at time $ t$. (2) Find the length of the locus by which the point $ P$ sweeps for $ 0\leq t\leq \pi$. (3) Find the volume of the solid by rotation of the region bounded by the curve $ C$ and the $ x$-axis about the $ x$-axis. Edited.

Two planes, $P$ and $Q$, intersect along the line $p$. The point $A$ is given in the plane $P$, and the point $C$ in the plane $Q$; neither of these points lies on the straight line $p$. Construct an isosceles trapezoid $ABCD$ (with $AB \parallel CD$) in which a circle can be inscribed, and with vertices $B$ and $D$ lying in planes $P$ and $Q$ respectively.

Let $OX$ and $OY$ be non-collinear rays. Through a point $A$ on $OX$, draw two lines $r_1$ and $r_2$ that are antiparallel with respect to $\angle XOY$. Let $r_1$ cut $OY$ at $M$ and $r_2$ cut $OY$ at $N$. (Thus, $\angle OAM = \angle ONA$). The bisectors of $ \angle AMY$ and $\angle ANY$ meet at $P$. Determine the location of $P$.

[hide=C stands for Cantor, G stands for Gauss]they had two problem sets under those two names[/hide] [u]Set 1[/u] [b]C.1 / G.1[/b] Daniel is exactly one year younger than his friend David. If David was born in the year $2008$, in what year was Daniel born? [b]C.2 / G.3[/b] Mr. Pham flips three coins. What is the probability that no two coins show the same side? [b]C.3 / G.2[/b] John has a sheet of white paper which is $3$ cm in height and $4$ cm in width. He wants to paint the sky blue and the ground green so the entire paper is painted. If the ground takes up a third of the page, how much space (in cm$^2$) does the sky take up? [b]C.4 / G.5[/b] Jihang and Eric are busy fidget spinning. While Jihang spins his fidget spinner at $15$ revolutions per second, Eric only manages $10$ revolutions per second. How many total revolutions will the two have made after $5$ continuous seconds of spinning? [b]C.5 / G.4[/b] Find the last digit of $1333337777 \cdot 209347802 \cdot 3940704 \cdot 2309476091$. [u]Set 2[/u] [b]C.6[/b] Evan, Chloe, Rachel, and Joe are splitting a cake. Evan takes $\frac13$ of the cake, Chloe takes $\frac14$, Rachel takes $\frac15$, and Joe takes $\frac16$. There is $\frac{1}{x}$ of the original cake left. What is $x$? [b]C.7[/b] Pacman is a $330^o$ sector of a circle of radius $4$. Pacman has an eye of radius $1$, located entirely inside Pacman. Find the area of Pacman, not including the eye. [b]C.8[/b] The sum of two prime numbers $a$ and $b$ is also a prime number. If $a < b$, find $a$. [b]C.9[/b] A bus has $54$ seats for passengers. On the first stop, $36$ people get onto an empty bus. Every subsequent stop, $1$ person gets off and $3$ people get on. After the last stop, the bus is full. How many stops are there? [b]C.10[/b] In a game, jumps are worth $1$ point, punches are worth $2$ points, and kicks are worth $3$ points. The player must perform a sequence of $1$ jump, $1$ punch, and $1$ kick. To compute the player’s score, we multiply the 1st action’s point value by $1$, the $2$nd action’s point value by $2$, the 3rd action’s point value by $3$, and then take the sum. For example, if we performed a punch, kick, jump, in that order, our score would be $1 \times 2 + 2 \times 3 + 3 \times 1 = 11$. What is the maximal score the player can get? [u]Set 3[/u] [b]C.11[/b] $6$ students are sitting around a circle, and each one randomly picks either the number $1$ or $2$. What is the probability that there will be two people sitting next to each other who pick the same number? [b]C.12 / G. 8[/b] You can buy a single piece of chocolate for $60$ cents. You can also buy a packet with two pieces of chocolate for $\$1.00$. Additionally, if you buy four single pieces of chocolate, the fifth one is free. What is the lowest amount of money you have to pay for $44$ pieces of chocolate? Express your answer in dollars and cents (ex. $\$3.70$). [b]C.13 / G.12[/b] For how many integers $k$ is there an integer solution $x$ to the linear equation $kx + 2 = 14$? [b]C.14 / G.9[/b] Ten teams face off in a swim meet. The boys teams and girls teams are ranked independently, each team receiving some number of positive integer points, and the final results are obtained by adding the points for the boys and the points for the girls. If Blair’s boys got $7$th place while the girls got $5$th place (no ties), what is the best possible total rank for Blair? [b]C.15 / G.11[/b] Arlene has a square of side length $1$, an equilateral triangle with side length $1$, and two circles with radius $1/6$. She wants to pack her four shapes in a rectangle without items piling on top of each other. What is the minimum possible area of the rectangle? PS. You should use hide for answers. C16-30/G10-15, G25-30 have been posted [url=https://artofproblemsolving.com/community/c3h2790676p24540145]here[/url] and G16-25 [url=https://artofproblemsolving.com/community/c3h2790679p24540159]here [/url] . Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $D$ be a point in the angle $ABC$. A circle $\gamma$ passing through $B$ and $D$ intersects the lines $AB$ and $BC$ at $M$ and $N$ respectively. Find the locus of the midpoint of $MN$ when circle $\gamma$ varies.

Distinct planes $p_1,p_2,....,p_k$ intersect the interior of a cube $Q$. Let $S$ be the union of the faces of $Q$ and let $ P =\bigcup_{j=1}^{k}p_{j} $. The intersection of $P$ and $S$ consists of the union of all segments joining the midpoints of every pair of edges belonging to the same face of $Q$. What is the difference between the maximum and minimum possible values of $k$? $ \textbf{(A)}\ 8\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 20\qquad\textbf{(D)}\ 23\qquad\textbf{(E)}\ 24 $

A parabola $p$ is drawn on the coordinate plane — the graph of the equation $y =-x^2$, and a point $A$ is marked that does not lie on the parabola $p$. All possible parabolas $q$ of the form $y = x^2+ax+b$ are drawn through point $A$, intersecting $p$ at two points $X$ and $Y$ . Prove that all possible $XY$ lines pass through a fixed point in the plane. [i]P.A.Kozhevnikov[/i]

Juri and Mari play the following game. Juri starts by drawing a random triangle on a piece of paper. Mari then draws a line on the same paper that goes through the midpoint of one of the midsegments of the triangle. Then Juri adds another line that also goes through the midpoint of the same midsegment. These two lines divide the triangle into four pieces. Juri gets the piece with maximum area (or one of those with maximum area) and the piece with minimum area (or one of those with minimum area), while Mari gets the other two pieces. The player whose total area is bigger wins. Does either of the players have a winning strategy, and if so, who has it?

Let $\mathcal{R}$ be the region in the plane bounded by the graphs of $y=x$ and $y=x^2$. Compute the volume of the region formed by revolving $\mathcal{R}$ around the line $y=x$.

Let $ABCD$ be a convex quadrilateral, and let $M$ and$ N$ be midpoints of sides $AB$ and $CD$ respectively. Point $P$ is chosen on $CD$ so that $MP \perp CD$, and point $Q$ is chosen on $AB$ so that $NQ \perp AB$. Show that $AD \parallel BC$ if and only if $\frac{AB}{CD} =\frac{MP}{NQ}$ .

Let $C$ and $D$ be points on the semicircle with center $O$ and diameter $AB$ such that $ABCD$ is a convex quadrilateral. Let $Q$ be the intersection of the diagonals $[AC]$ and $[BD]$, and $P$ be the intersection of the lines tangent to the semicircle at $C$ and $D$. If $m(\widehat{AQB})=2m(\widehat{COD})$ and $|AB|=2$, then what is $|PO|$? $ \textbf{(A)}\ \sqrt 2 \qquad\textbf{(B)}\ \sqrt 3 \qquad\textbf{(C)}\ \frac{1+\sqrt 3} 2 \qquad\textbf{(D)}\ \frac{1+\sqrt 3}{2\sqrt 2} \qquad\textbf{(E)}\ \frac{2\sqrt 3} 3 $

Let $ABCD$ be a parallelogram and a circle $k$ passes through $A, C$ and meets rays $AB, AD$ at $E, F$. If $BD, EF$ and the tangent at $C$ concur, show that $AC$ is diameter of $k$.

Let $ABCD$ be a tetrahedron with $AB=41$, $AC=7$, $AD=18$, $BC=36$, $BD=27$, and $CD=13$, as shown in the figure. Let $d$ be the distance between the midpoints of edges $AB$ and $CD$. Find $d^{2}$. [asy] pair C=origin, D=(4,11), A=(8,-5), B=(16,0); draw(A--B--C--D--B^^D--A--C); draw(midpoint(A--B)--midpoint(C--D), dashed); label("27", B--D, NE); label("41", A--B, SE); label("7", A--C, SW); label("$d$", midpoint(A--B)--midpoint(C--D), NE); label("18", (7,8), SW); label("13", (3,9), SW); pair point=(7,0); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D));[/asy]

Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be two circles, internally tangent at $P$ ($\mathcal{C}_2$ lies inside of $\mathcal{C}_1$). A chord $AB$ of $\mathcal{C}_1$ is tangent to $\mathcal{C}_2$ at $C.$ Let $D$ be the second point of intersection between the line $CP$ and $\mathcal{C}_1.$ A tangent from $D$ to $\mathcal{C}_2$ intersects $\mathcal{C}_1$ for the second time at $E$ and it intersects $\mathcal{C}_2$ at $F.$ Prove that $F$ is the incenter of triangle $ABE.$

Triangle $ ABC$ is isosceles with $ AC \equal{} BC$ and $ \angle{C} \equal{} 120^o$. Points $ D$ and $ E$ are chosen on segment $ AB$ so that $ |AD| \equal{} |DE| \equal{} |EB|$. Find the sizes of the angles of triangle $ CDE$.

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 $ABC$ be an acute scalene triangle with orthocenter $H$. Let $M$ be the midpoint of $BC$. $N$ is the point on line $AM$ such that $(BMN)$ is tangent to $AB$. Finally, let $H'$ be the reflection of $H$ in $B$. Prove that $\angle ANH'=90^{\circ}$. [i]Proposed by Malay Mahajan and Siddharth Choppara[/i]

Let $ABCDE$ be a convex pentagon such that angles $CAB$, $BCA$, $ECD$, $DEC$ and $AEC$ are equal. Prove that $CE$ bisects $BD$.

Consider two lines $\ell$ and $\ell ' $ and a fixed point $P$ equidistant from these lines. What is the locus of projections $M$ of $P$ on $AB$, where $A$ is on $\ell $, $B$ on $\ell ' $, and angle $\angle APB$ is right?