[hide=D stands for Dedekind, Z stands for Zermelo]they had two problem sets under those two names[/hide] [u]Set 1[/u] [b]D1 / Z1.[/b] What is $1 + 2 \cdot 3$? [b]D2.[/b] What is the average of the first $9$ positive integers? [b]D3 / Z2.[/b] A square of side length $2$ is cut into $4$ congruent squares. What is the perimeter of one of the $4$ squares? [b]D4.[/b] Find the ratio of a circle’s circumference squared to the area of the circle. [b]D5 / Z3.[/b] $6$ people split a bag of cookies such that they each get $21$ cookies. Kyle comes and demands his share of cookies. If the $7$ people then re-split the cookies equally, how many cookies does Kyle get? [u]Set 2[/u] [b]D6.[/b] How many prime numbers are perfect squares? [b]D7.[/b] Josh has an unfair $4$-sided die numbered $1$ through $4$. The probability it lands on an even number is twice the probability it lands on an odd number. What is the probability it lands on either $1$ or $3$? [b]D8.[/b] If Alice consumes $1000$ calories every day and burns $500$ every night, how many days will it take for her to first reach a net gain of $5000$ calories? [b]D9 / Z4.[/b] Blobby flips $4$ coins. What is the probability he sees at least one heads and one tails? [b]D10.[/b] Lillian has $n$ jars and $48$ marbles. If George steals one jar from Lillian, she can fill each jar with $8$ marbles. If George steals $3$ jars, Lillian can fill each jar to maximum capacity. How many marbles can each jar fill? [u]Set 3[/u] [b]D11 / Z6.[/b] How many perfect squares less than $100$ are odd? [b]D12.[/b] Jash and Nash wash cars for cash. Jash gets $\$6$ for each car, while Nash gets $\$11$ per car. If Nash has earned $\$1$ more than Jash, what is the least amount of money that Nash could have earned? [b]D13 / Z5.[/b] The product of $10$ consecutive positive integers ends in $3$ zeros. What is the minimum possible value of the smallest of the $10$ integers? [b]D14 / Z7.[/b] Guuce continually rolls a fair $6$-sided dice until he rolls a $1$ or a $6$. He wins if he rolls a $6$, and loses if he rolls a $1$. What is the probability that Guuce wins? [b]D15 / Z8.[/b] The perimeter and area of a square with integer side lengths are both three digit integers. How many possible values are there for the side length of the square? PS. You should use hide for answers. D.16-30/Z.9-14, 17, 26-30 problems have been collected [url=https://artofproblemsolving.com/community/c3h2916250p26045695]here [/url]and Z.15-25 [url=https://artofproblemsolving.com/community/c3h2916258p26045774]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given is triangle $ABC$ with an inscribed circle with center $M$ and radius $r$. The tangent to this circle parallel to $BC$ intersects $AC$ in $D$ and $AB$ in $E$. The tangent to this circle parallel to $AC$ intersects $AB$ in $F$ and $BC$ in $G$. The tangent to this circle parallel to $AB$ intersects $BC$ in $H$ and $AC$ in $K$. Name the centers of the inscribed circles of triangle $AED$, triangle $FBG$ and triangle $KHC$ successively $M_A, M_B, M_C$ and the rays successively $r_A, r_B$ and $r_C$. Prove that $r_A + r_B + r_C = r$.

Let $ABC$ be a triangle such that $AB\ne AC$ and $\omega$ be the circumcircle of this triangle. Let $I$ be the center of the inscribed circle of $ABC$ which touches $BC$ at $D$. Let the circle with diameter $AI$ meets $\omega$ again at $K$. If the line $AI$ intersects $\omega$ again at $M$, show that $K, D, M$ are collinear.

$H$ is the intersection point of the heights $AA'$ and $BB'$ of the acute-angled triangle $ABC$. A straight line, perpendicular to $AB$, intersects these heights at points $D$ and $E$, and side $AB$ at point $P$. Prove that the orthocenter of the triangle $DEH$ lies on segment $CP$.

In a dihedral angle of measure $c$ two non-intersecting spheres are inscribed, the centers of which are located on a straight line perpendicular to the edge of the dihedral angle. The points of contact of these spheres with the edges of the corner are at distances $a$ and $b$ from the edge. Let us consider an arbitrary plane tangent to these spheres and intersecting the segment connecting their centers. Let us denote by $\phi$ the measure of the angle formed at the intersection of this plane with the faces of a given dihedral angle. Find the greatest value $\phi$.

Let $ABC$ be a triangle with integral side lengths such that $\angle A=3\angle B$. Find the minimum value of its perimeter.

In acute triangle $\triangle ABC$, $H$ is the orthocenter, $BD$,$CE$ are altitudes. $M$ is the midpoint of $BC$. $P$,$Q$ are on segment $BM$,$DE$, respectively. $R$ is on segment $PQ$ such that $\frac{BP}{EQ}=\frac{CP}{DQ}=\frac{PR}{QR}$. Suppose $L$ is the orthocenter of $\triangle AHR$, then prove: $QM$ passes through the midpoint of $RL$.

Nine distinct positive integers are arranged in a circle such that the product of any two non-adjacent numbers in the circle is a multiple of $n$ and the product of any two adjacent numbers in the circle is not a multiple of $n$, where $n$ is a fixed positive integer. Find the smallest possible value for $n$.

Let $ ABC$ be a triangle and the points $ D\in (BC)$, $ E\in (CA)$, $ F\in (AB)$ such that \[ \frac {BD}{DC} \equal{} \frac {CE}{EA} \equal{} \frac {AF}{FB}.\] Prove that if the circumcenters of the triangles $ DEF$ and $ ABC$ coincide then $ ABC$ is equilateral.

Let $ABC$ be an isosceles triangle with $AB=4$, $BC=CA=6$. On the segment $AB$ consecutively lie points $X_{1},X_{2},X_{3},\ldots$ such that the lengths of the segments $AX_{1},X_{1}X_{2},X_{2}X_{3},\ldots$ form an infinite geometric progression with starting value $3$ and common ratio $\frac{1}{4}$. On the segment $CB$ consecutively lie points $Y_{1},Y_{2},Y_{3},\ldots$ such that the lengths of the segments $CY_{1},Y_{1}Y_{2},Y_{2}Y_{3},\ldots$ form an infinite geometric progression with starting value $3$ and common ratio $\frac{1}{2}$. On the segment $AC$ consecutively lie points $Z_{1},Z_{2},Z_{3},\ldots$ such that the lengths of the segments $AZ_{1},Z_{1}Z_{2},Z_{2}Z_{3},\ldots$ form an infinite geometric progression with starting value $3$ and common ratio $\frac{1}{2}$. Find all triplets of positive integers $(a,b,c)$ such that the segments $AY_{a}$, $BZ_{b}$ and $CX_{c}$ are concurrent.

On a board there is a regular polygon $A_1A_2\ldots A_{99}.$ Ana and Barbu alternatively occupy empty vertices of the polygon and write down triangles on a list: Ana only writes obtuse triangles, while Barbu only writes acute ones. At the first turn, Ana chooses three vertices $X,Y$ and $Z$ and writes down $\triangle XYZ.$ Then, Barbu chooses two of $X,Y$ and $Z,$ for example $X$ and $Y$, and an unchosen vertex $T$, and writes down $\triangle XYT.$ The game goes on and at each turn, the player must choose a new vertex $R$ and write down $\triangle PQR$, where $P$ is the last vertex chosen by the other player, and $Q$ is one of the other vertices of the last triangle written down by the other player. If one player cannot perform a move, then the other one wins. If both people play optimally, determine who has a winning strategy.

Let $D$ be a regular dodecahedron, which is a polyhedron with $20$ vertices, $30$ edges, and $12$ regular pentagon faces. A tetrahedron is a polyhedron with $4$ vertices, $6$ edges, and $4$ triangular faces. Find the number of tetrahedra with positive volume whose vertices are vertices of $D$. [img]https://cdn.artofproblemsolving.com/attachments/c/d/44d11fa3326780941d0b6756fb2e5989c2dc5a.png[/img]

For an integer $m\geq 1$, we consider partitions of a $2^m\times 2^m$ chessboard into rectangles consisting of cells of chessboard, in which each of the $2^m$ cells along one diagonal forms a separate rectangle of side length $1$. Determine the smallest possible sum of rectangle perimeters in such a partition. [i]Proposed by Gerhard Woeginger, Netherlands[/i]

[b]p1.[/b] A large square is cut into four smaller, congruent squares. If each of the smaller squares has perimeter $4$, what was the perimeter of the original square? [b]p2.[/b] Pie loves to bake apples so much that he spends $24$ hours a day baking them. If Pie bakes a dozen apples in one day, how many minutes does it take Pie to bake one apple, on average? [b]p3.[/b] Bames Jond is sent to spy on James Pond. One day, Bames sees James type in his $4$-digit phone password. Bames remembers that James used the digits $0$, $5$, and $9$, and no other digits, but he does not remember the order. How many possible phone passwords satisfy this condition? [b]p4.[/b] What do you get if you square the answer to this question, add $256$ to it, and then divide by $32$? [b]p5.[/b] Chloe the Horse and Flower the Chicken are best friends. When Chloe gets sad for any reason, she calls Flower, so Chloe must remember Flower's $3$ digit phone number, which can consist of any digits $0-5$. Given that the phone number's digits are unique and add to $5$, the number does not start with $0$, and the $3$ digit number is prime, what is the sum of all possible phone numbers? [b]p6.[/b] Anuj has a circular pizza with diameter $A$ inches, which is cut into $B$ congruent slices, where $A$,$B$ are positive integers. If one of Anuj's pizza slices has a perimeter of $3\pi + 30$ inches, find $A + B$. [b]p7.[/b] Bob really likes to study math. Unfortunately, he gets easily distracted by messages sent by friends. At the beginning of every minute, there is an $\frac{6}{10}$ chance that he will get a message from a friend. If Bob does get a message from a friend, there is a $\frac{9}{10}$ chance that he will look at the message, causing him to waste $30$ seconds before resuming his studying. If Bob doesn't get a message from a friend, there is a $\frac{3}{10}$ chance Bob will still check his messages hoping for a message from his friends, wasting $10$ seconds before he resumes his studying. What is the expected number of minutes in $100$ minutes for which Bob will be studying math? [b]p8.[/b] Suppose there is a positive integer $n$ with $225$ distinct positive integer divisors. What is the minimum possible number of divisors of n that are perfect squares? [b]p9.[/b] Let $a, b, c$ be positive integers. $a$ has $12$ divisors, $b$ has $8$ divisors, $c$ has $6$ divisors, and $lcm(a, b, c) = abc$. Let $d$ be the number of divisors of $a^2bc$. Find the sum of all possible values of $d$. [b]p10.[/b] Let $\vartriangle ABC$ be a triangle with side lengths $AB = 17$, $BC = 28$, $AC = 25$. Let the altitude from $A$ to $BC$ and the angle bisector of angle $B$ meet at $P$. Given the length of $BP$ can be expressed as $\frac{a\sqrt{b}}{c}$ for positive integers $a$, $b$, $c$ where $gcd(a, c) = 1$ and $b$ is not divisible by the square of any prime, find $a + b + c$. [b]p11.[/b] Let $a$, $b$, and $c$ be the roots of the cubic equation $x^3-5x+3 = 0$. Let $S = a^4b+ab^4+a^4c+ac^4+b^4c+bc^4$. Find $|S|$. [b]p12.[/b] Call a number palindromeish if changing a single digit of the number into a different digit results in a new six-digit palindrome. For example, the number $110012$ is a palindromeish number since you can change the last digit into a $1$, which results in the palindrome $110011$. Find the number of $6$ digit palindromeish numbers. [b]p13.[/b] Let $P(x)$ be a polynomial of degree $3$ with real coecients and leading coecient $1$. Let the roots of $P(x)$ be $a$, $b$, $c$. Given that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}= 4$ and $a^2 + b^2 + c^2 = 36$, the coefficient of $x^2$ is negative, and $P(1) = 2$, let the $S$ be the sum of possible values of $P(0)$. Then $|S|$ can be expressed as $\frac{a + b\sqrt{c}}{d}$ for positive integers $a$, $b$, $c$, $d$ such that $gcd(a, b, d) = 1$ and $c$ is not divisible by the square of any prime. Find $a + b + c + d$. [b]p14.[/b] Let $ABC$ be a triangle with side lengths $AB = 7$, $BC = 8$, $AC = 9$. Draw a circle tangent to $AB$ at $B$ and passing through $C$. Let the center of the circle be $O$. The length of $AO$ can be expressed as $\frac{a\sqrt{b}}{c\sqrt{d}}$ for positive integers $a$, $b$, $c$, $d$ where $gcd(a, c) = gcd(b, d) = 1$ and $b$,$ d$ are not divisible by the square of any prime. Find $a + b + c + d$. [b]p15.[/b] Many students in Mr. Noeth's BC Calculus class missed their first test, and to avoid taking a makeup, have decided to never leave their houses again. As a result, Mr. Noeth decides that he will have to visit their houses to deliver the makeup tests. Conveniently, the $17$ absent students in his class live in consecutive houses on the same street. Mr. Noeth chooses at least three of every four people in consecutive houses to take a makeup. How many ways can Mr. Noeth select students to take makeups? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A circle $\omega$ and a point $P$ not lying on it are given. Let $ABC$ be an arbitrary equilateral triangle inscribed into $\omega$ and $A', B', C'$ be the projections of $P$ to $BC, CA, AB$. Find the locus of centroids of triangles $A' B'C'$.

Let $ABCD$ be a cyclic quadrilateral. Let $E$ and $F$ be variable points on the sides $AB$ and $CD$, respectively, such that $AE:EB=CF:FD$. Let $P$ be the point on the segment $EF$ such that $PE:PF=AB:CD$. Prove that the ratio between the areas of triangles $APD$ and $BPC$ does not depend on the choice of $E$ and $F$.

Squares $ABB_{1}A_{2}$ and $BCC_{1}B_{2}$ are externally drawn on the hypotenuse $AB$ and on the leg $BC$ of a right triangle $ABC$ . Show that the lines $CA_{2}$ and $AB_{2}$ meet on the perimeter of a square with the vertices on the perimeter of triangle $ABC .$

Given an acute scalene triangle $ABC$ inscribed in circle $(O)$. Let $H$ be its orthocenter and $M$ be the midpoint of $BC$. Let $D$ lie on the opposite rays of $HA$ so that $BC=2DM$. Let $D'$ be the reflection of $D$ through line $BC$ and $X$ be the intersection of $AO$ and $MD$. a) Show that $AM$ bisects $D'X$. b) Similarly, we define the points $E,F$ like $D$ and $Y,Z$ like $X$. Let $S$ be the intersection of tangent lines from $B,C$ with respect to $(O)$. Let $G$ be the projection of the midpoint of $AS$ to the line $AO$. Show that there exists a point with the same power to all the circles $(BEY),(CFZ),(SGO)$ and $(O)$.

A non-equilateral triangle $\triangle ABC$ of perimeter $12$ is inscribed in circle $\omega$ .Points $P$ and $Q$ are arc midpoints of arcs $ABC$ and $ACB$ , respectively. Tangent to $\omega$ at $A$ intersects line $PQ$ at $R$. It turns out that the midpoint of segment $AR$ lies on line $BC$ . Find the length of the segment $BC$. [i] (А. Кузнецов)[/i]

Circles $\omega$ and $\Omega$ are inscribed into the same angle. Line $\ell$ meets the sides of angles, $\omega$ and $\Omega$ in points $A$ and $F, B$ and $C, D$ and $E$ respectively (the order of points on the line is $A,B,C,D,E, F$). It is known that$ BC = DE$. Prove that $AB = EF$.

A triangle with integer sides is called Heronian if its area is an integer. Does there exist a Heronian triangle whose sides are the arithmetic, geometric and harmonic means of two positive integers?

Denote by $ M$ midpoint of side $ BC$ in an isosceles triangle $ \triangle ABC$ with $ AC = AB$. Take a point $ X$ on a smaller arc $ \overarc{MA}$ of circumcircle of triangle $ \triangle ABM$. Denote by $ T$ point inside of angle $ BMA$ such that $ \angle TMX = 90$ and $ TX = BX$. Prove that $ \angle MTB - \angle CTM$ does not depend on choice of $ X$. [i]Author: Farzan Barekat, Canada[/i]

The diagonals of a convex quadrilateral divide it into four triangles. The radii of the circles circumscribed around these triangles are equal. Can such a property have a quadrilateral other than: a) parallelogram, b) rhombus? (Sharygin Igor)

Let $ABCD$ be a cyclic quadrilateral, $E = AC \cap BD$, $F = AD \cap BC$. The bisectors of angles $AFB$ and $AEB$ meet $CD$ at points $X, Y$ . Prove that $A, B, X, Y$ are concyclic.

Let $ABC$ be an acute-angled triangle with $AC > AB$, let $O$ be its circumcentre, and let $D$ be a point on the segment $BC$. The line through $D$ perpendicular to $BC$ intersects the lines $AO, AC,$ and $AB$ at $W, X,$ and $Y,$ respectively. The circumcircles of triangles $AXY$ and $ABC$ intersect again at $Z \ne A$. Prove that if $W \ne D$ and $OW = OD,$ then $DZ$ is tangent to the circle $AXY.$