Let $ABCDEFGH$ be a cube. Consider a plane whose intersection with the tetrahedron $ABDE$ is a triangle with an obtuse angle $\varphi.$ Determine all $\varphi>\pi/2$ for which there is such a plane.

Dagan has a wooden cube. He paints each of the six faces a different color. He then cuts up the cube to get eight identically-sized smaller cubes, each of which now has three painted faces and three unpainted faces. He then puts the smaller cubes back together into one larger cube such that no unpainted face is visible. Compute the number of different cubes that Dagan can make this way. Two cubes are considered the same if one can be rotated to obtain the other. You may express your answer either as an integer or as a product of prime numbers.

Points $A, B, C$ are chosen on the boundary of a circle with center $O$ so that $\angle BAC$ encloses an arc of $120$ degrees. Let $D$ be chosen on $\overline{BA}$ so that $\angle AOD$ is a right angle. Extend $\overline{CD}$ so that it intersects with $O$ again at point $P$. What is the measure of the arc, in degrees, that is enclosed by $\angle ACP$? Please use the $tan^{-1}$ function to express your answer.

Construct a triangle, the base of which lies on the given line, and the feet of the altitudes, drawn on the sides, coincide with the given points.

Consider a semi-circle with diameter $AB$. Let points $C$ and $D$ be on diameter $AB$ such that $CD$ forms the base of a square inscribed in the semicircle. Given that $CD = 2$, compute the length of $AB$.

Let $m<n$ be two positive integers, let $I$ and $J$ be two index sets such that $|I|=|J|=n$ and $|I\cap J|=m$, and let $u_k$, $k\in I\cup J$ be a collection of vectors in the Euclidean plane such that \[|\sum_{i\in I}u_i|=1=|\sum_{j\in J}u_j|.\] Prove that \[\sum_{k\in I\cup J}|u_k|^2\geq \frac{2}{m+n}\] and find the cases of equality.

In the triangle $ABC$ let $B'$ and $C'$ be the midpoints of the sides $AC$ and $AB$ respectively and $H$ the foot of the altitude passing through the vertex $A$. Prove that the circumcircles of the triangles $AB'C'$,$BC'H$, and $B'CH$ have a common point $I$ and that the line $HI$ passes through the midpoint of the segment $B'C'.$

Given is an isosceles trapezoid $ABCD$ with $AB \parallel CD$ and $AB> CD$. The projection from $D$ on $ AB$ is $E$. The midpoint of the diagonal $BD$ is $M$. Prove that $EM$ is parallel to $AC$. (Karl Czakler)

Inside the quadrangle, a point is taken and connected with the midpoint of all sides. Areas of the three out of four formed quadrangles are $S_1, S_2, S_3$. Find the area of the fourth quadrangle.

Point $A$ is outside of a given circle $\omega$. Let the tangents from $A$ to $\omega$ meet $\omega$ at $S, T$ points $X, Y$ are midpoints of $AT, AS$ let the tangent from $X$ to $\omega$ meet $\omega$ at $R\neq T$. points $P, Q$ are midpoints of $XT, XR$ let $XY\cap PQ=K, SX\cap TK=L$ prove that quadrilateral $KRLQ$ is cyclic.

Let $A_0A_1A_2$ be a triangle and let $P$ be a point in the plane, not situated on the circle $A_0A_1A_2$. The line $PA_k$ meets again the circle $A_0A_1A_2$ at point $B_k, k = 0, 1, 2$. A line $\ell$ through the point $P$ meets the line $A_{k+1}A_{k+2}$ at point $C_k, k = 0, 1, 2$. Show that the lines $B_kC_k, k = 0, 1, 2$, are concurrent and determine the locus of their concurrency point as the line $\ell$ turns about the point $P$.

In a triangle $ABC$, a point $D$ is on the segment $BC$, Let $X$ and $Y$ be the incentres of triangles $ACD$ and $ABD$ respectively. The lines $BY$ and $CX$ intersect the circumcircle of triangle $AXY$ at $P\ne Y$ and $Q\ne X$, respectively. Let $K$ be the point of intersection of lines $PX$ and $QY$. Suppose $K$ is also the reflection of $I$ in $BC$ where $I$ is the incentre of triangle $ABC$. Prove that $\angle BAC=\angle ADC=90^{\circ}$.

In convex quadrilateral $ABCD, \angle BAC=\angle CAD.$ $E$ lies on segment $CD$, and $BE$ and $AC$ intersect at $F,$ $DF$ and $BC$ intersect at $G.$ Prove that $\angle GAC=\angle EAC.$

Let $O$ be the center of the circle of the triangle $ABC$ with $AB \ne AC$. Furthermore, let $S$ and $T$ be points on the rays $AB$ and $AC$, such that $\angle ASO = \angle ACO$ and $\angle ATO = \angle ABO$. Show that $ST$ bisects the segment $BC$.

[i]20 problems for 25 minutes.[/i] [b]p1.[/b] Evaluate $20 \times 21 + 2021$. [b]p2.[/b] Let points $A$, $B$, $C$, and $D$ lie on a line in that order. Given that $AB = 5CD$ and $BD = 2BC$, compute $\frac{AC}{BD}$. [b]p3.[/b] There are $18$ students in Vincent the Bug's math class. Given that $11$ of the students take U.S. History, $15$ of the students take English, and $2$ of the students take neither, how many students take both U.S. History and English? [b]p4.[/b] Among all pairs of positive integers $(x, y)$ such that $xy = 12$, what is the least possible value of $x + y$? [b]p5.[/b] What is the smallest positive integer $n$ such that $n! + 1$ is composite? [b]p6.[/b] How many ordered triples of positive integers $(a, b,c)$ are there such that $a + b + c = 6$? [b]p7.[/b] Thomas orders some pizzas and splits each into $8$ slices. Hungry Yunseo eats one slice and then finds that she is able to distribute all the remaining slices equally among the $29$ other math club students. What is the fewest number of pizzas that Thomas could have ordered? [b]p8.[/b] Stephanie has two distinct prime numbers $a$ and $b$ such that $a^2-9b^2$ is also a prime. Compute $a + b$. [b]p9.[/b] Let $ABCD$ be a unit square and $E$ be a point on diagonal $AC$ such that $AE = 1$. Compute $\angle BED$, in degrees. [b]p10.[/b] Sheldon wants to trace each edge of a cube exactly once with a pen. What is the fewest number of continuous strokes that he needs to make? A continuous stroke is one that goes along the edges and does not leave the surface of the cube. [b]p11.[/b] In base $b$, $130_b$ is equal to $3n$ in base ten, and $1300_b$ is equal to $n^2$ in base ten. What is the value of $n$, expressed in base ten? [b]p12.[/b] Lin is writing a book with $n$ pages, numbered $1,2,..., n$. Given that $n > 20$, what is the least value of $n$ such that the average number of digits of the page numbers is an integer? [b]p13.[/b] Max is playing bingo on a $5\times 5$ board. He needs to fill in four of the twelve rows, columns, and main diagonals of his bingo board to win. What is the minimum number of boxes he needs to fill in to win? [b]p14.[/b] Given that $x$ and $y$ are distinct real numbers such that $x^2 + y = y^2 + x = 211$, compute the value of $|x - y|$. [b]p15.[/b] How many ways are there to place 8 indistinguishable pieces on a $4\times 4$ checkerboard such that there are two pieces in each row and two pieces in each column? [b]p16.[/b] The Manhattan distance between two points $(a, b)$ and $(c, d)$ in the plane is defined to be $|a - c| + |b - d|$. Suppose Neil, Neel, and Nail are at the points $(5, 3)$, $(-2,-2)$ and $(6, 0)$, respectively, and wish to meet at a point $(x, y)$ such that their Manhattan distances to$ (x, y)$ are equal. Find $10x + y$. [b]p17.[/b] How many positive integers that have a composite number of divisors are there between $1$ and $100$, inclusive? [b]p18.[/b] Find the number of distinct roots of the polynomial $$(x - 1)(x - 2) ... (x - 90)(x^2 - 1)(x^2 - 2) ... (x^2 - 90)(x^4 - 1)(x^4 - 2)...(x^4 - 90)$$. [b]p19.[/b] In triangle $ABC$, let $D$ be the foot of the altitude from $ A$ to $BC$. Let $P,Q$ be points on $AB$, $AC$, respectively, such that $PQ$ is parallel to $BC$ and $\angle PDQ = 90^o$. Given that $AD = 25$, $BD = 9$, and $CD = 16$, compute $111 \times PQ$. [b]p20.[/b] The simplified fraction with numerator less than $1000$ that is closest but not equal to $\frac{47}{18}$ is $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Compute $p$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given a real number $t$ with $0 < t < 1$, define the real-valued function $f(t, \theta) = \sum^{\infty}_{n=-\infty} t^{|n|}\omega^n$, where $\omega = e^{i \theta} = \cos \theta + i\sin \theta$. For $\theta \in [0, 2\pi)$, the polar curve $r(\theta) = f(t, \theta)$ traces out an ellipse $E_t$ with a horizontal major axis whose left focus is at the origin. Let $A(t)$ be the area of the ellipse $E_t$. Let $A\left( \frac12 \right) = \frac{a\pi}{b}$ , where $a, b$ are relatively prime positive integers. Find $100a +b$ .

Points $A,B,C,D$ have been marked on checkered paper (see fig.). Find the tangent of the angle $ABD$. [img]https://cdn.artofproblemsolving.com/attachments/6/1/eeb98ccdee801361f9f66b8f6b2da4714e659f.png[/img]

Let $O$ be the circumcenter of triangle $ABC$. A line through $O$ intersects the sides $CA$ and $CB$ at points $D$ and $E$ respectively, and meets the circumcircle of $ABO$ again at point $P \neq O$ inside the triangle. A point $Q$ on side $AB$ is such that $\frac{AQ}{QB}=\frac{DP}{PE}$. Prove that $\angle APQ = 2\angle CAP$. [i]Proposed by Dusan Djukic[/i]

Let $ABC$ be an acute triangle and let $D$ be the foot of the perpendicular from $A$ to side $BC$. Let $P$ be a point on segment $AD$. Lines $BP$ and $CP$ intersect sides $AC$ and $AB$ at $E$ and $F$, respectively. Let $J$ and $K$ be the feet of the peroendiculars from $E$ and $F$ to $AD$, respectively. Show that $\frac{FK}{KD}=\frac{EJ}{JD}$.

A circle of radius $10$ is tangent to two adjacent sides of a square and intersects its two remaining sides at the endpoints of a diameter of the circle. Find the side length of the square.

Let $ABC$ be a triangle. The angle bisector of $\angle B$ intersects $AC$ at point $P$, while the angle bisector of $\angle C$ intersects $AB$ at a point $Q$. Suppose the area of $\triangle ABP$ is 27, the area of $\triangle ACQ$ is 32, and the area of $\triangle ABC$ is $72$. The length of $\overline{BC}$ can be written in the form $m\sqrt n$ where $m$ and $n$ are positive integers with $n$ as small as possible. What is $m+n$?

5. In acute triangle $ABC$, the lines tangent to the circumcircle of $ABC$ at $A$ and $B$ intersect at point $D$. Let $E$ and $F$ be points on $CA$ and $CB$ such that $DECF$ forms a parallelogram. Given that $AB = 20$, $CA=25$ and $\tan C = 4\sqrt{21}/17$, the value of $EF$ may be expressed as $m/n$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by winnertakeover and Monkey_king1[/i]

Circles $k_1,k_2$ intersect at $B,C$ such that $BC$ is diameter of $k_1$.Tangent of $k_1$ at $C$ touches $k_2$ for the second time at $A$.Line $AB$ intersects $k_1$ at $E$ different from $B$, and line $CE$ intersects $k_2$ at F different from $C$. An arbitrary line through $E$ intersects segment $AF$ at $H$ and $k_1$ for the second time at $G$.If $BG$ and $AC$ intersect at $D$, prove $CH//DF$ .

An airplane wants to go from a point on the equator, and at each moment it will go to the northeast with speed $v$. Suppose the radius of earth is $R$. a) Will the airplane reach to the north pole? If yes how long it will take to reach the north pole? b) Will the airplne rotate finitely many times around the north pole? If yes how many times?

A triangle is given on a sphere of radius $1$, the sides of which are arcs of three different circles of radius $1$ centered in the center of a sphere having less than $\pi$ in length and an area equal to a quarter of the area of the sphere. Prove that four copies of such a triangle can cover the entire sphere. A. Zaslavsky