[b]p1.[/b] The serpent of fire and the serpent of ice play a game. Since the serpent of ice loves the lucky number $6$, he will roll a fair $6$-sided die with faces numbered $1$ through $6$. The serpent of fire will pay him $\log_{10} x$, where $x$ is the number he rolls. The serpent of ice rolls the die $6$ times. His expected total amount of winnings across the $6$ rounds is $k$. Find $10^k$. [b]p2.[/b] Let $a = \log_3 5$, $b = \log_3 4$, $c = - \log_3 20$, evaluate $\frac{a^2+b^2}{a^2+b^2+ab} +\frac{b^2+c^2}{b^2+c^2+bc} +\frac{c^2+a^2}{c^2+a^2+ca}$. [b]p3.[/b] Let $\vartriangle ABC$ be an isosceles obtuse triangle with $AB = AC$ and circumcenter $O$. The circle with diameter $AO$ meets $BC$ at points $X, Y$ , where X is closer to $B$. Suppose $XB = Y C = 4$, $XY = 6$, and the area of $\vartriangle ABC$ is $m\sqrt{n}$ for positive integers $m$ and $n$, where $n$ does not contain any square factors. Find $m + n$. [b]p4.[/b] Alice is not sure what to have for dinner, so she uses a fair $6$-sided die to decide. She keeps rolling, and if she gets all the even numbers (i.e. getting all of $2, 4, 6$) before getting any odd number, she will reward herself with McDonald’s. Find the probability that Alice could have McDonald’s for dinner. [b]p5.[/b] How many distinct ways are there to split $50$ apples, $50$ oranges, $50$ bananas into two boxes, such that the products of the number of apples, oranges, and bananas in each box are nonzero and equal? [b]p6.[/b] Sujay and Rishabh are taking turns marking lattice points within a square board in the Cartesian plane with opposite vertices $(1, 1)$,$(n, n)$ for some constant $n$. Sujay loses when the two-point pattern $P$ below shows up:[img]https://cdn.artofproblemsolving.com/attachments/1/9/d1fe285294d4146afc0c7a2180b15586b04643.png[/img] That is, Sujay loses when there exists a pair of points $(x, y)$ and $(x + 2, y + 1)$. He and Rishabh stop marking points when the pattern $P$ appears on the board. If Rishabh goes first, let $S$ be the set of all integers $3 \le n \le 100$ such that Rishabh has a strategy to always trick Sujay into being the one who creates $P$. Find the sum of all elements of $S$. [b]p7.[/b] Let $a$ be the shortest distance between the origin $(0, 0)$ and the graph of $y^3 = x(6y -x^2)-8$. Find $\lfloor a^2 \rfloor $. ($\lfloor x\rfloor $ is the largest integer not exceeding $x$) [b]p8.[/b] Find all real solutions to the following equation: $$2\sqrt2x^2 + x -\sqrt{1 - x^2 } -\sqrt2 = 0.$$ [b]p9.[/b] Given the expression $S = (x^4 - x)(x^2 - x^3)$ for $x = \cos \frac{2\pi}{5 }+ i\sin \frac{2\pi}{5 }$, find the value of $S^2$ . [b]p10.[/b] In a $32$ team single-elimination rock-paper-scissors tournament, the teams are numbered from $1$ to $32$. Each team is guaranteed (through incredible rock-paper-scissors skill) to win any match against a team with a higher number than it, and therefore will lose to any team with a lower number. Each round, teams who have not lost yet are randomly paired with other teams, and the losers of each match are eliminated. After the $5$ rounds of the tournament, the team that won all $5$ rounds is ranked $1$st, the team that lost the 5th round is ranked $2$nd, and the two teams that lost the $4$th round play each other for $3$rd and $4$th place. What is the probability that the teams numbered $1, 2, 3$, and $4$ are ranked $1$st, 2nd, 3rd, and 4th respectively? If the probability is $\frac{m}{n}$ for relatively prime integers $m$ and $n$, find $m$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the least positive integer which is a multiple of $13$ and all its digits are the same. [i](Adapted from Gazeta Matematică 1/1982, Florin Nicolăită)[/i]

Find all prime numbers $p$ such that $16p + 1$ is a perfect cube.

Find all prime numbers of the form $\tfrac{1}{11} \cdot \underbrace{11\ldots 1}_{2n \textrm{ ones}}$, where $n$ is a natural number.

Let $ A$ be the subset of the set of positive integers, having the following $ 2$ properties: 1) If $ a$ belong to $ A$,than all of the divisors of $ a$ also belong to $ A$; 2) If $ a$ and $ b$, $ 1 < a < b$, belong to $ A$, than $ 1 \plus{} ab$ is also in $ A$; Prove that if $ A$ contains at least $ 3$ positive integers, than $ A$ contains all positive integers.

Pairwise distinct prime numbers $p, q, r$ satisfy the equality $$rp^3 + p^2 + p = 2rq^2 +q^2 + q.$$ Determine all possible values of the product $pqr$.

[b]6.1. [/b] Three workers can do some work. Second and the third can together complete it twice as fast as the first, the first and the third can together complete it three times faster than the second. At what time since the first and second can do this job faster than the third? [b]6.2.[/b] Prove that the greatest common divisor of the sum of two numbers and their least common multiple is equal to their greatest common divisor the numbers themselves. [b]6.3.[/b] There were 20 schoolchildren at the consultation and 20 problems were dealt with. It turned out that each student solved two problems and each problem was solved by two schoolchildren. Prove that it is possible to organize the analysis in this way tasks so that everyone solves one problem and all tasks are solved. [hide=original wording] Наконсультациибыло20школьниковиразбиралось20задач. Оказалось, что каждый школьник решил две задачи и каждую задачу решило два школьника. Докажите, что можно так организовать разбор задач, чтобыкаждыйрассказалоднузадачуивсезадачибылирассказаны.[/hide] [b]6.4[/b].Two people Α and Β must get from point Μ to point Ν,located 15 km from M. On foot they can move at a speed of 6 km/h. In addition, they have a bicycle at their disposal, on which υou can drive at a speed of 15 km/h. A and B depart from Μ at the same time, A walks, and B rides a bicycle until meeting pedestrian C, going from N to M. Then B walks and C rides a bicycle to meeting with A, hands him a bicycle, on which he arrives at N. When must pedestrian C leave Nfor A and B to arrive at N simultaneously if he walks at the same speed as A and B? [b]6.5./ 7.1[/b] Prove that out of any six people there will always be three pairs of acquaintances or three pairs of strangers. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983442_1961_leningrad_math_olympiad]here[/url].

Let $P(x)$ and $Q(x)$ be polynomials of degree $p$ and $q$ respectively such that every coefficient is $1$ or $2023$. If $P(x)$ divides $Q(x)$, prove that $p+1$ divides $q+1$.

Prove that there are infinitely many triples $(a, b, p)$ of positive integers with $p$ prime, $a < p$, and $b < p$, such that $(a + b)^p - a^p - b^p$ is a multiple of $p^3$. [i]Noam Elkies[/i]

A number which when divided by $ 10$ leaves a remainder of $ 9$, when divided by $ 9$ leaves a remainder of $ 8$, by $ 8$ leaves a remainder of $ 7$, etc., down to where, when divided by $ 2$, it leaves a remainder of $ 1$, is: $ \textbf{(A)}\ 59 \qquad\textbf{(B)}\ 419 \qquad\textbf{(C)}\ 1259 \qquad\textbf{(D)}\ 2519 \qquad\textbf{(E)}\ \text{none of these answers}$

Prove that every positive rational number can be expressed uniquely as a finite sum of the form $$a_1+\frac{a_2}{2!}+\frac{a_3}{3!}+\dots+\frac{a_n}{n!},$$ where $a_n$ are integers such that $0 \leq a_n \leq n-1$ for all $n > 1$.

Two sequences $(a_n)$ and $(b_n)$ satisfy that $a_0=1,a_1=4,a_2=49$, and $\begin{cases} a_{n+1}=7a_n+6b_n-3\\ b_{n+1}=8a_n+7b_n-4\\ \end{cases}$ for $n=0,1,2,\cdots,$. Prove that $a_n$ is a perfect square for $n=0,1,2,\cdots,$.

Find all natural numbers such that \[ n\sigma(n)\equiv 2\pmod {\phi( n)}\]

A [i]slip[/i] on an integer $n\geq 2$ is an operation that consists in choosing a prime divisor $p$ of $n$ and replacing $n$ by $\frac{n+p^2}{p}.$ Starting with an arbitrary integer $n\geq 5$, we successively apply the slip operation on it. Show that one eventually reaches $5$, no matter the slips applied.

Let $P$ be the set of all primes. Given any positive integer $n$, define $$\displaystyle f(n) = \max_{p \in P}v_p(n)$$ Prove that for any positive integer $k\ge 2$, there exists infinitely many positive integers $m$ such that \[ f(m+1) = f(m+2) = \cdots = f(m+k) \] [i]Proposed by Ivan Chan Guan Yu[/i]

Find all integer solutions of the equation $p (x + y) = xy$, where $p$ is a prime number.

Let $p > 5$ be a prime. Suppose that $$\frac{1}{2^2} + \frac{1}{4^2}+ \frac{1}{6^2}+ ...+ \frac{1}{(p -1)^2} =\frac{a}{b}$$ where $a/b$ is a fraction in lowest terms. Show that $p | a$.

Four dice are thrown. What is the probability that the product of the number equals $ 36?$

Let $ABCD$ be a square with side length $2$. Let $M$ and $N$ be the midpoints of $\overline{BC}$ and $\overline{CD}$ respectively, and let $X$ and $Y$ be the feet of the perpendiculars from $A$ to $\overline{MD}$ and $\overline{NB}$, also respectively. The square of the length of segment $\overline{XY}$ can be written in the form $\tfrac pq$ where $p$ and $q$ are positive relatively prime integers. What is $100p+q$? [i]Proposed by David Altizio[/i]

Let $a_1<a_2<a_3<\dots$ be positive integers such that $a_{k+1}$ divides $2(a_1+a_2+\dots+a_k)$ for every $k\geqslant 1$. Suppose that for infinitely many primes $p$, there exists $k$ such that $p$ divides $a_k$. Prove that for every positive integer $n$, there exists $k$ such that $n$ divides $a_k$.

$q$ is the smallest rational number having the following properties: (i) $q>\frac{31}{17}$ (ii) when $q$ is written in its reduced form $\frac{a}{b}$, then $b<17$ As in part (ii) above, find $a+b$.

[hide=D stands for Descartes, L stands for Leibniz]they had two problem sets under those two names[/hide] [u]Set 4[/u] [b]D.16 / L.6[/b] Alex has $100$ Bluffy Funnies in some order, which he wants to sort in order of height. They’re already almost in order: each Bluffy Funny is at most $1$ spot off from where it should be. Alex can only swap pairs of adjacent Bluffy Funnies. What is the maximum possible number of swaps necessary for Alex to sort them? [b]D.17[/b] I start with the number $1$ in my pocket. On each round, I flip a coin. If the coin lands heads heads, I double the number in my pocket. If it lands tails, I divide it by two. After five rounds, what is the expected value of the number in my pocket? [b]D.18 / L.12[/b] Point $P$ inside square $ABCD$ is connected to each corner of the square, splitting the square into four triangles. If three of these triangles have area $25$, $25$, and $15$, what are all the possible values for the area of the fourth triangle? [b]D.19[/b] Mr. Stein and Mr. Schwartz are playing a yelling game. The teachers alternate yelling. Each yell is louder than the previous and is also relatively prime to the previous. If any teacher yells at $100$ or more decibels, then they lose the game. Mr. Stein yells first, at $88$ decibels. What volume, in decibels, should Mr. Schwartz yell at to guarantee that he will win? [b]D.20 / L.15[/b] A semicircle of radius $1$ has line $\ell$ along its base and is tangent to line $m$. Let $r$ be the radius of the largest circle tangent to $\ell$, $m$, and the semicircle. As the point of tangency on the semicircle varies, the range of possible values of $r$ is the interval $[a, b]$. Find $b - a$. [u]Set 5[/u] [b]D.21 / L.14[/b] Hungryman starts at the tile labeled “$S$”. On each move, he moves $1$ unit horizontally or vertically and eats the tile he arrives at. He cannot move to a tile he already ate, and he stops when the sum of the numbers on all eaten tiles is a multiple of nine. Find the minimum number of tiles that Hungryman eats. [img]https://cdn.artofproblemsolving.com/attachments/e/7/c2ecc2a872af6c4a07907613c412d3b86cd7bc.png [/img] [b]D.22 / L.11[/b] How many triples of nonnegative integers $(x, y, z)$ satisfy the equation $6x + 10y +15z = 300$? [b]D.23 / L.16[/b] Anson, Billiam, and Connor are looking at a $3D$ figure. The figure is made of unit cubes and is sitting on the ground. No cubes are floating; in other words, each unit cube must either have another unit cube or the ground directly under it. Anson looks from the left side and says, “I see a $5 \times 5$ square.” Billiam looks from the front and says the same thing. Connor looks from the top and says the same thing. Find the absolute difference between the minimum and maximum volume of the figure. [b]D.24 / L.13[/b] Tse and Cho are playing a game. Cho chooses a number $x \in [0, 1]$ uniformly at random, and Tse guesses the value of $x(1 - x)$. Tse wins if his guess is at most $\frac{1}{50}$ away from the correct value. Given that Tse plays optimally, what is the probability that Tse wins? [b]D.25 / L.20[/b] Find the largest solution to the equation $$2019(x^{2019x^{2019}-2019^2+2019})^{2019}) = 2019^{x^{2019}+1}.$$ [u]Set 6[/u] [i]This round is an estimation round. No one is expected to get an exact answer to any of these questions, but unlike other rounds, you will get points for being close. In the interest of transparency, the formulas for determining the number of points you will receive are located on the answer sheet, but they aren’t very important when solving these problems.[/i] [b]D.26 / L.26[/b] What is the sum over all MBMT volunteers of the number of times that volunteer has attended MBMT (as a contestant or as a volunteer, including this year)? Last year there were $47$ volunteers; this is the fifth MBMT. [b]D.27 / L.27[/b] William is sharing a chocolate bar with Naveen and Kevin. He first randomly picks a point along the bar and splits the bar at that point. He then takes the smaller piece, randomly picks a point along it, splits the piece at that point, and gives the smaller resulting piece to Kevin. Estimate the probability that Kevin gets less than $10\%$ of the entire chocolate bar. [b]D.28 / L.28[/b] Let $x$ be the positive solution to the equation $x^{x^{x^x}}= 1.1$. Estimate $\frac{1}{x-1}$. [b]D.29 / L.29[/b] Estimate the number of dots in the following box: [img]https://cdn.artofproblemsolving.com/attachments/8/6/416ba6379d7dfe0b6302b42eff7de61b3ec0f1.png[/img] It may be useful to know that this image was produced by plotting $(4\sqrt{x}, y)$ some number of times, where x, y are random numbers chosen uniformly randomly and independently from the interval $[0, 1]$. [b]D.30 / L.30[/b] For a positive integer $n$, let $f(n)$ be the smallest prime greater than or equal to $n$. Estimate $$(f(1) - 1) + (f(2) - 2) + (f(3) - 3) + ...+ (f(10000) - 10000).$$ For $26 \le i \le 30$, let $E_i$ be your team’s answer to problem $i$ and let $A_i$ be the actual answer to problem $i$. Your score $S_i$ for problem $i$ is given by $S_{26} = \max(0, 12 - |E_{26} - A_{26}|/5)$ $S_{27} = \max(0, 12 - 100|E_{27} - A_{27}|)$ $S_{28} = \max(0, 12 - 5|E_{28} - A_{28}|))$ $S_{29} = 12 \max \left(0, 1 - 3 \frac{|E_{29} - A_{29}|}{A_{29}} \right)$ $S_{30} = \max (0, 12 - |E_{30} - A_{30}|/2000)$ PS. You should use hide for answers. D.1-15 / L1-9 problems have been collected [url=https://artofproblemsolving.com/community/c3h2790795p24541357]here [/url] and L10,16-30 [url=https://artofproblemsolving.com/community/c3h2790825p24541816]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given an integer $a>1$. Let $p_1 < p_2 <...< p_k$ be all prime divisors of $a$. For each positive integer $n$ we define: $C_0(n) = a^{2n}, C_1(n) =\frac{a^{2n}}{p^2_1}, .... , C_k(n) =\frac{a^{2_n}}{p^2_k}$ $A = a^2 + 1$ $T(n) = A^{C_0(n)} - 1$ $M(n) = LCM(a^{2n+2}, A^{C_1(n)} - 1, ..., A^{C_k(n)} - 1)$ $A_n =\frac{T(n)}{M(n)}$ Prove that the sequence $A_1, A_2, ... $ satisfies the properties: (i) Every number in the sequence is an integer greater than $1$ and has only prime divisors of the form $am + 1$. (ii) Any two different numbers in the sequence are coprime.

Let $a>2$, $n>1$ integers such that $a^n-2^n$ is a perfect square. Prove that $a$ is a even number.

Michael writes down all the integers between $1$ and $N$ inclusive on a piece of paper and discovers that exactly $40\%$ of them have leftmost digit $1$. Given that $N > 2017$, find the smallest possible value of $N$.