There are $ n$ points in $ \mathbb R^3$ such that every three form an acute angled triangle. Find maximum of $ n$.

Show that the cube roots of three distinct primes cannot be terms in an arithmetic progression.

We have four wooden triangles with sides $3, 4, 5$ centimeters. How many convex polygons can we make by all of these triangles? (Just draw the polygons without any proof) A convex polygon is a polygon which all of it's angles are less than $180^o$ and there isn't any hole in it. For example: [img]https://1.bp.blogspot.com/-JgvF_B-uRag/W1R4f4AXxTI/AAAAAAAAIzc/Fo3qu3pxXcoElk01RTYJYZNwj0plJaKPQCK4BGAYYCw/s640/igo%2B2015.el1.png[/img]

[b]p1.[/b] Each box represents $1$ square unit. Find the area of the shaded region. [img]https://cdn.artofproblemsolving.com/attachments/0/0/f8f8ad6d771f3bbbc59b374a309017cecdce5a.png[/img] [b]p2.[/b] Evaluate $(3^3)\sqrt{5^2-2^4} -5 \cdot 9$. [b]p3.[/b] Find the last two digits of $21^3$. [b]p4.[/b] Let $L$, $M$, and $T$ be distinct prime numbers. Find the least possible odd value of$ L+M +T$ . [b]p5.[/b]Two circles have areas that sum to $20\pi$ and diameters that sum to $12$. Find the radius of the smaller circle. [b]p6.[/b] Zach and Evin each independently choose a date in the year $2022$, uniformly and randomly. The probability that at least one of the chosen dates is December $17$, $2022$ can be expressed as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $A$. [b]p7.[/b] Let $L$ be a list of $2023$ real numbers with medianm. When any two numbers are removed from $L$, its median is still $m$. Find the greatest possible number of distinct values in $L$. [b]p8.[/b] Some children and adults are eating a delicious pile of sand. Children comprise $20\%$ of the group and combined, they consume $80\%$ of the sand. Given that on average, each child consumes $N$ pounds of sand and on average, each adult consumes $M$ pounds of sand, find $\frac{N}{M}$. [b]p9.[/b] An integer $N$ is chosen uniformly and randomly from the set of positive integers less than $100$. The expectedm number of digits in the base-$10$-representation of $N$ can be expressed as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A+B$. [b]p10.[/b] Dunan is taking a calculus course in which the final exam counts for $15\%$ of the total grade. Dunan wishes to have an $A$ in the course, which is defined as a grade of $93\%$ or above. When counting everything but the final exam, he currently has a $92\%$ in the course. What is the minimum integer grade Dunan must get on the final exam in order to get an $A$ in the course? [b]p11.[/b] Norbert, Eorbert, Sorbert, andWorbert start at the origin of the Cartesian Plane and walk in the positive $y$, positive $x$, negative $y$, and negative $x$ directions respectively at speeds of $1$, $2$, $3$, and $4$ units per second respectively. After how many seconds will the quadrilateral with a vertex at each person’s location have area $300$? [b]p12.[/b] Find the sum of the unique prime factors of $1020201$. [b]p13.[/b] HacoobaMatata rewrites the base-$10$ integers from $0$ to $30$ inclusive in base $3$. How many times does he write the digit $1$? [b]p14.[/b] The fractional part of $x$ is $\frac17$. The greatest possible fractional part of $x^2$ can be written as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A+B$. [b]p15.[/b] For howmany integers $x$ is $-2x^2 +8 \ge x^2 -3x +2$? [b]p16.[/b] In the figure below, circle $\omega$ is inscribed in square $EFGH$, which is inscribed in unit square $ABCD$ such that $\overline{EB} = 2\overline{AE}$. If the minimum distance from a point on $\omega$ to $ABCD$ can be written as $\frac{P-\sqrt{Q}}{R}$ with $Q$ square-free, find $10000P +100Q +R$. [img]https://cdn.artofproblemsolving.com/attachments/a/1/c6e5400bc508ab14f34987c9f5f4039daaa4d6.png[/img] [b]p17.[/b] There are two base number systems in use in the LHS Math Team. One member writes “$13$ people usemy base, while $23$ people use the other, base $12$.” Another member writes “out of the $34$ people in the club, $10$ use both bases while $9$ use neither.” Find the sum of all possible numbers ofMath Team members, as a regular decimal number. [b]p18.[/b] Sam is taking a test with $100$ problems. On this test the questions gradually get harder in such a way that for question $i$ , Sam has a $\frac{(101-i)^2}{ 100} \%$ chance to get the question correct. Suppose the expected number of questions Sam gets correct can be written as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A+B$. [b]p19.[/b] In an ordered $25$-tuple, each component is an integer chosen uniformly and randomly from $\{1,2,3,4,5\}$. Ephram and Zach both copy this tuple into a $5\times 5$ grid, both starting from the top-left corner. Ephram writes five components from left to right to fill one row before continuing down to the next row. Zach writes five components from top to bottom to fill one column before continuing right to the next column. Find the expected number of spaces on their grids where Zach and Ephram have the same integer written. [b]p20.[/b] In $\vartriangle ABC$ with circumcenter $O$ and circumradius $8$, $BC = 10$. Let $r$ be the radius of the circle that passes through $O$ and is tangent to $BC$ at $C$. The value of $r^2$ can be written as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $1000m+n$. [b]p21.[/b] Find the number of integer values of $n$ between $1$ and $100$ inclusive such that the sum of the positive divisors of $2n$ is at least $220\%$ of the sum of the divisors of $n$. [b]p22.[/b] Twenty urns containing one ball each are arranged in a circle. Ernie then moves each ball either $1$, $2$ or $3$ urns clockwise, chosen independently, uniformly, and randomly. The expected number of empty urns after this process is complete can be expressed as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A+B$. [b]p23.[/b] Hannah the cat begins at $0$ on a number line. Every second, Hannah jumps $1$ unit in the positive or negative direction, chosen uniformly at random. After $7$ seconds,Hannah‘s expected distance from $0$, in units, can be expressed as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A+B$. [b]p24.[/b] Find the product of all primes $p < 30$ for which there exists an integer $n$ such that $p$ divides $n +(n +1)^{-1}\,\, (mod \,\,p)$. [b]p25.[/b] In quadrilateral $ABCD$, $\angle ABD = \angle CBD = \angle C AD$, $AB = 9$, $BC = 6$, and $AC = 10$. The area of $ABCD$ can be expressed as $\frac{P\sqrt{Q}}{R}$ with $Q$ squarefree and $P$ and $R$ relatively prime. Find $10000P +100Q +R$. [img]https://cdn.artofproblemsolving.com/attachments/4/8/28569605b262c8f26e685e27f5f261c70a396c.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A special kind of parallelogram tile is made up by attaching the legs of two right isosceles triangles of side length $1$. We want to put a number of these tiles on the floor of an $n\times n$ room such that the distance from each vertex of each tile to the sides of the room is an integer and also no two tiles overlap. Prove that at least an area $n$ of the room will not be covered by the tiles. [i]Proposed by Ali Khezeli[/i]

Let $p$ be a prime number. Prove that from a $p^2\times p^2$ array of squares, we can select $p^3$ of the squares such that the centers of any four of the selected squares are not the vertices of a rectangle with sides parallel to the edges of the array.

Let $B$ be a point on a circle $S_1$, and let $A$ be a point distinct from $B$ on the tangent at $B$ to $S_1$. Let $C$ be a point not on $S_1$ such that the line segment $AC$ meets $S_1$ at two distinct points. Let $S_2$ be the circle touching $AC$ at $C$ and touching $S_1$ at a point $D$ on the opposite side of $AC$ from $B$. Prove that the circumcentre of triangle $BCD$ lies on the circumcircle of triangle $ABC$.

In an equilateral triangle ${ABC}$ the segments ${ED}$ and ${GF}$ are drawn to obtain two equilateral triangles ${ADE}$ and ${GFC}$ with sides 1 and 100 (points $E$ and $G$ are on the side ${AC}$ ). The segments ${EF}$ and ${DG}$ meet at point $O$ so that the angle ${EOG}$ is equal to ${120}^{ \circ }$ . What is the length of the side of the triangle ${ABC}$ ? Mikhail Evdokimov

The triangle $ABC$ is isosceles with $AB=AC$, and $\angle{BAC}<60^{\circ}$. The points $D$ and $E$ are chosen on the side $AC$ such that, $EB=ED$, and $\angle{ABD}\equiv\angle{CBE}$. Denote by $O$ the intersection point between the internal bisectors of the angles $\angle{BDC}$ and $\angle{ACB}$. Compute $\angle{COD}$.

What is the largest number of squares on $9 \times 9$ square board that can be cut along their both diagonals so that the board does not fall apart into several pieces?

Given a cyclic quadrilateral $ABCD$, let the diagonals $AC$ and $BD$ meet at $E$ and the lines $AD$ and $BC$ meet at $F$. The midpoints of $AB$ and $CD$ are $G$ and $H$, respectively. Show that $EF$ is tangent at $E$ to the circle through the points $E$, $G$ and $H$. [i]Proposed by David Monk, United Kingdom[/i]

Circles $S_1$ and $S_2$ with centers $O_1$ and $O_2$ respectively intersect at $A$ and $B$. Ray $O_1B$ meets $S_2$ again at $F$, and ray $O_2B$ meets $ S_1$ again at $E$. The line through $B$ parallel to $ EF$ intersects $S_1$ and $S_2$ again at $M$ and $N$, respectively. Prove that $MN = AE +AF$.

[hide=instructions]Time limit: 20 minutes. Fill in the crossword above with answers to the problems below. Notice that there are three directions instead of two. You are probably used to "down" and "across," but this crossword has "1," $e^{4\pi i/3}$, and $e^{5\pi i/3}$. You can think of these labels as complex numbers pointing in the direction to fill in the spaces. In other words "1" means "across", $e^{4\pi i/3}$ means "down and to the left," and $e^{5\pi i/3}$ means "down and to the right." To fill in the answer to, for example, $12$ across, start at the hexagon labeled $12$, and write the digits, proceeding to the right along the gray line. (Note: $12$ across has space for exactly $5$ digits.) Each hexagon is worth one point, and must be filled by something from the set $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$. Note that $\pi$ is not in the set, and neither is $i$, nor $\sqrt2$, nor $\heartsuit$,etc. None of the answers will begin with a $0$. "Concatenate $a$ and $b$" means to write the digits of $a$, followed by the digits of $b$. For example, concatenating $10$ and $3$ gives $103$. (It's not the same as concatenating $3$ and $10$.) Calculators are allowed! THIS SHEET IS PROVIDED FOR YOUR REFERENCE ONLY. DO NOT TURN IN THIS SHEET. TURN IN THE OFFICIAL ANSWER SHEET PROVIDED TO THE TEAM. OTHERWISE YOU WILL GET A SCORE OF ZERO! ZERO! ZERO! AND WHILE SOMETIMES "!" MEANS FACTORIAL, IN THIS CASE IT DOES NOT. Good luck, and have fun![/hide] [img]https://cdn.artofproblemsolving.com/attachments/b/f/f7445136e40bf4889a328da640f0935b2b8b82.png[/img] [u][b][i]Across[/i][/b][/u] (1) [b]A 3.[/b] (3 digits) Suppose you draw $5$ vertices of a convex pentagon (but not the sides!). Let $N$ be the number of ways you can draw at least $0$ straight line segments between the vertices so that no two line segments intersect in the interior of the pentagon. What is $N - 64$? (Note what the question is asking for! You have been warned!) [b]A 5.[/b] (3 digits) Among integers $\{1, 2,..., 10^{2012}\}$, let $n$ be the number of numbers for which the sum of the digits is divisible by $5$. What are the first three digits (from the left) of $n$? [b]A 6.[/b] (3 digits) Bob is punished by his math teacher and has to write all perfect squares, one after another. His teacher's blackboard has space for exactly $2012$ digits. He can stop when he cannot fit the next perfect square on the board. (At the end, there might be some space left on the board - he does not write only part of the next perfect square.) If $n^2$ is the largest perfect square he writes, what is $n$? [b]A 8. [/b](3 digits) How many positive integers $n$ are there such that $n \le 2012$, and the greatest common divisor of $n$ and $2012$ is a prime number? [b]A 9.[/b] (4 digits) I have a random number machine generator that is very good at generating integers between $1$ and $256$, inclusive, with equal probability. However, right now, I want to produce a random number between $1$ and $n$, inclusive, so I do the following: $\bullet$ I use my machine to generate a number between $1$ and $256$. Call this $a$. $\bullet$ I take a and divide it by $n$ to get remainder $r$. If $r \ne 0$, then I record $r$ as the randomly generated number. If $r = 0$, then I record $n$ instead. Note that this process does not necessarily produce all numbers with equal probability, but that is okay. I apply this process twice to generate two numbers randomly between $1$ and $10$. Let $p$ be the probability that the two numbers are equal. What is $p \cdot 2^{16}$? [b]A 12.[/b] (5 digits) You and your friend play the following dangerous game. You two start off at some point $(x, y)$ on the plane, where $x$ and $y$ are nonnegative integers. When it is player $A$'s turn, A tells his opponent $B$ to move to another point on the plane. Then $A$ waits for a while. If $B$ is not eaten by a tiger, then $A$ moves to that point as well. From a point $(x, y)$ there are three places $A$ can tell $B$ to walk to: leftwards to $(x - 1, y)$, downwards to $(x, y-1)$, and simultaneously downwards and leftwards to $(x-1, y-1)$. However, you cannot move to a point with a negative coordinate. Now, what was this about being eaten by a tiger? There is a tiger at the origin, which will eat the first person that goes there! Needless to say, you lose if you are eaten. Consider all possible starting points $(x, y)$ with $0 \le x \le 346$ and $0 \le y \le 346$, and $x$ and $y$ are not both zero. Also suppose that you two play strategically, and you go first (i.e., by telling your friend where to go). For how many of the starting points do you win? [b][u][i]Down and to the left [/i][/u][/b] $e^{4\pi i/3}$ [b]DL 2.[/b] (2 digits) ABCDE is a pentagon with $AB = BC = CD = \sqrt2$, $\angle ABC = \angle BCD = 120$ degrees, and $\angle BAE = \angle CDE = 105$ degrees. Find the area of triangle $\vartriangle BDE$. Your answer in its simplest form can be written as $\frac{a+\sqrt{b}}{c}$ , where where $a, b, c$ are integers and $b$ is square-free. Find $abc$. [b]DL 3.[/b] (3 digits) Suppose $x$ and $y$ are integers which satisfy $$\frac{4x^2}{y^2} + \frac{25y^2}{x^2} = \frac{10055}{x^2} +\frac{4022}{y^2} +\frac{2012}{x^2y^2}- 20. $$ What is the maximum possible value of $xy -1$? [b]DL 5.[/b] (3 digits) Find the area of the set of all points in the plane such that there exists a square centered around the point and having the following properties: $\bullet$ The square has side length $7\sqrt2$. $\bullet$ The boundary of the square intersects the graph of $xy = 0$ at at least $3$ points. [b]DL 8.[/b] (3 digits) Princeton Tiger has a mom that likes yelling out math problems. One day, the following exchange between Princeton and his mom occurred: $\bullet$ Mom: Tell me the number of zeros at the end of $2012!$ $\bullet$ PT: Huh? $2012$ ends in $2$, so there aren't any zeros. $\bullet$ Mom: No, the exclamation point at the end was not to signify me yelling. I was not asking about $2012$, I was asking about $2012!$. What is the correct answer? [b]DL 9.[/b] (4 digits) Define the following: $\bullet$ $A = \sum^{\infty}_{n=1}\frac{1}{n^6}$ $\bullet$ $B = \sum^{\infty}_{n=1}\frac{1}{n^6+1}$ $\bullet$ $C = \sum^{\infty}_{n=1}\frac{1}{(n+1)^6}$ $\bullet$ $D = \sum^{\infty}_{n=1}\frac{1}{(2n-1)^6}$ $\bullet$ $E = \sum^{\infty}_{n=1}\frac{1}{(2n+1)^6}$ Consider the ratios $\frac{B}{A}, \frac{C}{A}, \frac{D}{A} , \frac{E}{A}$. Exactly one of the four is a rational number. Let that number be $r/s$, where $r$ and $s$ are nonnegative integers and $gcd \,(r, s) = 1$. Concatenate $r, s$. (It might be helpful to know that $A = \frac{\pi^6}{945}$ .) [b]DL 10.[/b] (3 digits) You have a sheet of paper, which you lay on the xy plane so that its vertices are at $(-1, 0)$, $(1, 0)$, $(1, 100)$, $(-1, 100)$. You remove a section of the bottom of the paper by cutting along the function $y = f(x)$, where $f$ satisfies $f(1) = f(-1) = 0$. (In other words, you keep the bottom two vertices.) You do this again with another sheet of paper. Then you roll both of them into identical cylinders, and you realize that you can attach them to form an $L$-shaped elbow tube. We can write $f\left( \frac13 \right)+f\left( \frac16 \right) = \frac{a+\sqrt{b}}{\pi c}$ , where $a, b, c$ are integers and $b$ is square-free. $Find a+b+c$. [b]DL 11.[/b] (3 digits) Let $$\Xi (x) = 2012(x - 2)^2 + 278(x - 2)\sqrt{2012 + e^{x^2-4x+4}} + 1392 + (x^2 - 4x + 4)e^{x^2-4x+4}$$ find the area of the region in the $xy$-plane satisfying: $$\{x \ge 0 \,\,\, and x \le 4 \,\,\, and \,\,\, y \ge 0 \,\,\, and \,\,\, y \le \sqrt{\Xi(x)}\}$$ [b]DL 13.[/b] (3 digits) Three cones have bases on the same plane, externally tangent to each other. The cones all face the same direction. Two of the cones have radii of $2$, and the other cone has a radius of $3$. The two cones with radii $2$ have height $4$, and the other cone has height $6$. Let $V$ be the volume of the tetrahedron with three of its vertices as the three vertices of the cones and the fourth vertex as the center of the base of the cone with height $6$. Find $V^2$. [b][u][i]Down and to the right[/i][/u][/b] $e^{5\pi i/3}$ [b]DR 1.[/b] (2 digits) For some reason, people in math problems like to paint houses. Alice can paint a house in one hour. Bob can paint a house in six hours. If they work together, it takes them seven hours to paint a house. You might be thinking "What? That's not right!" but I did not make a mistake. When Alice and Bob work together, they get distracted very easily and simultaneously send text messages to each other. When they are texting, they are not getting any work done. When they are not texting, they are painting at their normal speeds (as if they were working alone). Carl, the owner of the house decides to check up on their work. He randomly picks a time during the seven hours. The probability that they are texting during that time can be written as $r/s$, where r and s are integers and $gcd \,(r, s) = 1$. What is $r + s$? [b]DR 4.[/b] (3 digits) Let $a_1 = 2 +\sqrt2$ and $b_1 =\sqrt2$, and for $n \ge 1$, $a_{n+1} = |a_n - b_n|$ and $b_{n+1} = a_n + b_n$. The minimum value of $\frac{a^2_n+a_nb_n-6b^2_n}{6b^2_n-a^2_n}$ can be written in the form $a\sqrt{b} - c$, where $a, b, c$ are integers and $b$ is square-free. Concatenate $c, b, a$ (in that order!). [b]DR 7.[/b] (3 digits) How many solutions are there to $a^{503} + b^{1006} = c^{2012}$, where $a, b, c$ are integers and $|a|$,$|b|$, $|c|$ are all less than $2012$? PS. You should use hide for answers.

a) Given a triangle $ABC$ with $\angle BAC=90^{\circ}$ and the altitude $AH(H$ is on the side $BC).$ Prove that: the [i]Lemoine[/i] point of the triangle $ABC$ is the midpoint of $AH.$ b) If a triangle has its [i]Lemoine[/i] point is the midpoint of $1$ in $3$ symmedian segments, does that triangle need to be a right triangle? Explain why.

Let $ABC$ be a triangle inscribed in a semicircle with center $O$ and diameter $BC.$ Two tangent lines to the semicircle at $A$ and $B$ intersect at $D.$ Prove that $DC$ goes through the midpoint of the altitude $AH$ of triangle $ABC.$

Let $ABC$ be an acute triangle such that $\overline{AB}=\overline{AC}$. Let $M, L, N$ be the midpoints of segment $BC, AM, AC$, respectively. The circumcircle of triangle $AMC$, denoted by $\Omega$, meets segment $AB$ at $P(\neq A)$, and the segment $BL$ at $Q$. Let $O$ be the circumcenter of triangle $BQC$. Suppose that the lines $AC$ and $PQ$ meet at $X$, $OB$ and $LN$ meet at $Y$, and $BQ$ and $CO$ meets at $Z$. Prove that the points $X, Y, Z$ are collinear.

Let $M$ be the midpoint of $BC$ of triangle $ABC$. The circle with diameter $BC$, $\omega$, meets $AB,AC$ at $D,E$ respectively. $P$ lies inside $\triangle ABC$ such that $\angle PBA=\angle PAC, \angle PCA=\angle PAB$, and $2PM\cdot DE=BC^2$. Point $X$ lies outside $\omega$ such that $XM\parallel AP$, and $\frac{XB}{XC}=\frac{AB}{AC}$. Prove that $\angle BXC +\angle BAC=90^{\circ}$.

Let $ABC$ be an acute-angled triangle with orthocenter $H$. Let $D$ be the foot of the altitude from $A$ to the line $BC$. Let $T$ be a point on the circle with diameter $AH$ such that this circle is internally tangent to the circumcircle of triangle $BDT$. Let $N$ be the midpoint of segment $AH$. Prove that $BT \perp CN$.

(a) Prove that the area of a given convex quadrilateral is at least twice the area of an arbitrary convex quadrilateral inscribed in it whose sides are parallel to the diagonals of the original one. (b) A tetrahedron with surface area $S$ is intersected by a plane perpendicular to two opposite edges. If the area of the cross-section is $Q$, prove that $S>4Q$.

Let $AC$ be a diameter of a circle and $AB$ be tangent to the circle. The segment $BC$ intersects the circle again at $D$. Show that if $AC = 1$, $AB = a$, and $CD = b$, then $$\frac{1}{a^2+ \frac12 }< \frac{b}{a}< \frac{1}{a^2}$$

Prove that the circumcircle of a triangle contains exactly 3 points whose Simson lines are tangent to the triangle's Euler circle and these points are the vertices of an equilateral triangle.

Let $P$ and $Q$ be interior points in $\Delta ABC$ such that $PQ$ doesn't contain any vertices of $\Delta ABC$. Let $A_1$, $B_1$, and $C_1$ be the points of intersection of $BC$, $CA$, and $AB$ with $AQ$, $BQ$, and $CQ$, respectively. Let $K$, $L$, and $M$ be the intersections of $AP$, $BP$, and $CP$ with $B_1C_1$, $C_1A_1$, and $A_1B_1$, respectively. Prove that $A_1K$, $B_1L$, and $C_1M$ are concurrent.

Let $Z$ and $Y$ be the tangency points of the incircle of the triangle $ABC$ with the sides $AB$ and $CA$, respectively. The parallel line to $Y Z$ through the midpoint $M$ of $BC$, meets $CA$ in $N$. Let $L$ be the point in $CA$ such that $NL = AB$ (and $L$ on the same side of $N$ than $A$). The line $ML$ meets $AB$ in $K$. Prove that $KA = NC$.

On a circle with diameter $AC$, let $B$ be an arbitrary point distinct from $A$ and $C$. Points $M, N$ are the midpoints of chords $AB, BC$, and points $P, Q$ are the midpoints of smaller arcs restricted by these chords. Lines $AQ$ and $BC$ meet at point $K$, and lines $CP$ and $AB$ meet at point $L$. Prove that lines $MQ, NP$ and $KL$ concur.

Two circles of radii $7$ and $17$ have a distance of $25$ between their centers. What is the difference between the lengths of their common internal and external tangents (positive difference)?