Found problems: 146
2014 MMATHS, 3
Let $f : R^+ \to R^+$ be a function satisfying $$f(\sqrt{x_1x_2}) =\sqrt{f(x_1)f(x_2)}$$ for all positive real numbers $x_1, x_2$. Show that $$f( \sqrt[n]{x_1x_2... x_n}) = \sqrt[n]{f(x_1)f(x_2) ... f(x_n)}$$ for all positive integers $n$ and positive real numbers $x_1, x_2,..., x_n$.
2021 MMATHS, Mixer Round
[b]p1.[/b] Prair takes some set $S$ of positive integers, and for each pair of integers she computes the positive difference between them. Listing down all the numbers she computed, she notices that every integer from $1$ to $10$ is on her list! What is the smallest possible value of $|S|$, the number of elements in her set $S$?
[b]p2.[/b] Jake has $2021$ balls that he wants to separate into some number of bags, such that if he wants any number of balls, he can just pick up some bags and take all the balls out of them. What is the least number of bags Jake needs?
[b]p3.[/b] Claire has stolen Cat’s scooter once again! She is currently at (0; 0) in the coordinate plane, and wants to ride to $(2, 2)$, but she doesn’t know how to get there. So each second, she rides one unit in the positive $x$ or $y$-direction, each with probability $\frac12$ . If the probability that she makes it to $(2, 2)$ during her ride can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $gcd(a, b) = 1$, then find $a + b$.
[b]p4.[/b] Triangle $ABC$ with $AB = BC = 6$ and $\angle ABC = 120^o$ is rotated about $A$, and suppose that the images of points $B$ and $C$ under this rotation are $B'$ and $C'$, respectively. Suppose that $A$, $B'$ and $C$ are collinear in that order. If the area of triangle $B'CC'$ can be expressed as $a - b\sqrt{c}$ for positive integers $a, b, c$ with csquarefree, find $a + b + c$.
[b]p5.[/b] Find the sum of all possible values of $a + b + c + d$ if $a, b, c, $d are positive integers satisfying
$$ab + cd = 100,$$
$$ac + bd = 500.$$
[b]p6.[/b] Alex lives in Chutes and Ladders land, which is set in the coordinate plane. Each step they take brings them one unit to the right or one unit up. However, there’s a chute-ladder between points $(1, 2)$ and $(2, 0)$ and a chute-ladder between points $(1, 3)$ and $(4, 0)$, whenever Alex visits an endpoint on a chute-ladder, they immediately appear at the other endpoint of that chute-ladder! How many ways are there for Alex to go from $(0, 0)$ to $(4, 4)$?
[b]p7.[/b] There are $8$ identical cubes that each belong to $8$ different people. Each person randomly picks a cube. The probability that exactly $3$ people picked their own cube can be written as $\frac{a}{b}$ , where $a$ and $b$ are positive integers with $gcd(a, b) = 1$. Find $a + b$.
[b]p8.[/b] Suppose that $p(R) = Rx^2 + 4x$ for all $R$. There exist finitely many integer values of $R$ such that $p(R)$ intersects the graph of $x^3 + 2021x^2 + 2x + 1$ at some point $(j, k)$ for integers $j$ and $k$. Find the sum of all possible values of $R$.
[b]p9.[/b] Let $a, b, c$ be the roots of the polynomial $x^3 - 20x^2 + 22$. Find $\frac{bc}{a^2} +\frac{ac}{b^2} +\frac{ab}{c^2}$.
[b]p10.[/b] In any finite grid of squares, some shaded and some not, for each unshaded square, record the number of shaded squares horizontally or vertically adjacent to it, this grid’s score is the sum of all numbers recorded this way. Deyuan shades each square in a blank $n \times n$ grid with probability $k$; he notices that the expected value of the score of the resulting grid is equal to $k$, too! Given that $k > 0.9999$, find the minimum possible value of $n$.
[b]p11.[/b] Find the sum of all $x$ from $2$ to $1000$ inclusive such that $$\prod^x_{n=2} \log_{n^n}(n + 1)^{n+2}$$ is an integer.
[b]p12.[/b] Let triangle $ABC$ with incenter $I$ and circumcircle $\Gamma$ satisfy $AB = 6\sqrt3$, $BC = 14$, and $CA = 22$. Construct points $P$ and $Q$ on rays $BA$ and $CA$ such that $BP = CQ = 14$. Lines $PI$ and $QI$ meet the tangents from $B$ and $C$ to $\Gamma$, respectively, at points $X$ and $Y$ . If $XY$ can be expressed as $a\sqrt{b}-c$ for positive integers $a, b, c$ with $c$ squarefree, find $a + b + c$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2023 MMATHS, 2
In the Game of Life, each square in an infinite grid of squares is either shaded or blank. Every day, if a square shares an edge with exactly zero or four shaded squares, it becomes blank the next day. If a square shares an edge with exactly two or three shaded squares, it becomes shaded the next day. Otherwise, it does not change. On day $1$, each square is randomly shaded or blank with equal probability. If the probability that a given square is shaded on day 2 is $\tfrac{a}{b},$ where $a$ and $b$ are relatively prime positive integers, find $a + b.$
2022 MMATHS, 6
Siva has the following expression, which is missing operations:
$$\frac12 \,\, \_ \,\,\frac14 \,\, \_ \,\, \frac18 \,\, \_ \,\,\frac{1}{16} \,\, \_ \,\,\frac{1}{32}.$$ For each blank, he flips a fair coin: if it comes up heads, he fills it with a plus, and if it comes up tails, he fills it with a minus. Afterwards, he computes the value of the expression. He then repeats the entire process with a new set of coinflips and operations. If the probability that the positive difference between his computed values is greater than $\frac12$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$, $b$, then find $a + b$.
MMATHS Mathathon Rounds, 2016
[u]Round 5[/u]
[b]p13.[/b] Let $\{a\} _{n\ge 1}$ be an arithmetic sequence, with $a_ 1 = 0$, such that for some positive integers $k$ and $x$ we have $a_{k+1} = {k \choose x}$, $a_{2k+1} ={k \choose {x+1}}$ , and $a_{3k+1} ={k \choose {x+2}}$. Let $\{b\}_{n\ge 1}$ be an arithmetic sequence of integers with $b_1 = 0$. Given that there is some integer $m$ such that $b_m ={k \choose x}$, what is the number of possible values of $b_2$?
[b]p14.[/b] Let $A = arcsin \left( \frac14 \right)$ and $B = arcsin \left( \frac17 \right)$. Find $\sin(A + B) \sin(A - B)$.
[b]p15.[/b] Let $\{f_i\}^{9}_{i=1}$ be a sequence of continuous functions such that $f_i : R \to Z$ is continuous (i.e. each $f_i$ maps from the real numbers to the integers). Also, for all $i$, $f_i(i) = 3^i$. Compute $\sum^{9}_{k=1} f_k \circ f_{k-1} \circ ... \circ f_1(3^{-k})$.
[u]Round 6[/u]
[b]p16.[/b] If $x$ and $y$ are integers for which $\frac{10x^3 + 10x^2y + xy^3 + y^4}{203}= 1134341$ and $x - y = 1$, then compute $x + y$.
[b]p17.[/b] Let $T_n$ be the number of ways that n letters from the set $\{a, b, c, d\}$ can be arranged in a line (some letters may be repeated, and not every letter must be used) so that the letter a occurs an odd number of times. Compute the sum $T_5 + T_6$.
[b]p18.[/b] McDonald plays a game with a standard deck of $52$ cards and a collection of chips numbered $1$ to $52$. He picks $1$ card from a fully shuffled deck and $1$ chip from a bucket, and his score is the product of the numbers on card and on the chip. In order to win, McDonald must obtain a score that is a positive multiple of $6$. If he wins, the game ends; if he loses, he eats a burger, replaces the card and chip, shuffles the deck, mixes the chips, and replays his turn. The probability that he wins on his third turn can be written in the form $\frac{x^2 \cdot y}{z^3}$ such that $x, y$, and $z$ are relatively prime positive integers. What is $x + y + z$?
(NOTE: Use Ace as $1$, Jack as $11$, Queen as $12$, and King as $13$)
[u]Round 7[/u]
[b]p19.[/b] Let $f_n(x) = ln(1 + x^{2^n}+ x^{2^{n+1}}+ x^{3\cdot 2^n})$. Compute $\sum^{\infty}_{k=0} f_{2k} \left( \frac12 \right)$.
[b]p20.[/b] $ABCD$ is a quadrilateral with $AB = 183$, $BC = 300$, $CD = 55$, $DA = 244$, and $BD = 305$. Find $AC$.
[b]p21.[/b] Define $\overline{xyz(t + 1)} = 1000x + 100y + 10z + t + 1$ as the decimal representation of a four digit integer. You are given that $3^x5^y7^z2^t = \overline{xyz(t + 1)}$ where $x, y, z$, and t are non-negative integers such that $t$ is odd and $0 \le x, y, z,(t + 1) \le 9$. Compute$3^x5^y7^z$
PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c4h2782822p24445934]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2021 MMATHS, 9
Suppose that $P(x)$ is a monic cubic polynomial with integer roots, and suppose that $\frac{P(a)}{a}$ is an integer for exactly $6$ integer values of $a$. Suppose furthermore that exactly one of the distinct numbers $\frac{P(1) + P(-1)}{2}$ and $\frac{P(1) - P(-1)}{2}$ is a perfect square. Given that $P(0) > 0$, find the second-smallest possible value of $P(0).$
[i]Proposed by Andrew Wu[/i]
2023 MMATHS, 1
Lucy has $8$ children, each of whom has a distinct favorite integer from $1$ to $10,$ inclusive. The smallest number that is a perfect multiple of all of these favorite numbers is $1260,$ and the average of these favorite numbers is at most $5.$ Find the sum of the four largest numbers.
2018 MMATHS, 4
A sequence of integers fsng is defined as follows: fix integers $a$, $b$, $c$, and $d$, then set $s_1 = a$, $s_2 = b$, and $$s_n = cs_{n-1} + ds_{n-2}$$ for all $n \ge 3$. Create a second sequence $\{t_n\}$ by defining each $t_n$ to be the remainder when $s_n$ is divided by $2018$ (so we always have $0 \le t_n \le 2017$). Let $N = (2018^2)!$. Prove that $t_N = t_{2N}$ regardless of the choices of $a$, $b$, $c$, and $d$.
2024 MMATHS, 7
The sum $\sum_{x=-5}^5\sum_{y=-5}^5\frac{2^x3^y}{(1+2^x)(1+3^y)}$ can be expressed as a fraction $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
2020 MMATHS, 2
Suppose that points $A$ and $B$ lie on circle $\Omega$, and suppose that points $C$ and $D$ are the trisection points of major arc $AB$, with $C$ closer to $B$ than $A$. Let $E$ be the intersection of line $AB$ with the line tangent to $\Omega$ at $C$. Suppose that $DC = 8$ and $DB = 11$. If $DE = a\sqrt{b}$ for integers $a$ and $b$ with $b$ squarefree, find $a + b$.
2021 MMATHS, 2
Define the [i]digital reduction[/i] of a two-digit positive integer $\underline{AB}$ to be the quantity $\underline{AB} - A - B$. Find the greatest common divisor of the digital reductions of all the two-digit positive integers. (For example, the digital reduction of $62$ is $62 - 6 - 2 = 54.$)
[i]Proposed by Andrew Wu[/i]
2023 MMATHS, 3
Simon expands factored polynomials with his favorite AI, ChatSFFT. However, he has not paid for a premium ChatSFFT account, so when he goes to expand $(m - a)(n - b),$ where $a, b, m, n$ are integers, ChatSFFT returns the sum of the two factors instead of the product. However, when Simon plugs in certain pairs of integer values for $m$ and $n,$ he realizes that the value of ChatSFFT’s result is the same as the real result in terms of $a$ and $b$. How many such pairs are there?
2023 MMATHS, 7
A $2023 \times 2023$ grid of lights begins with every light off. Each light is assigned a coordinate $(x,y).$ For every distinct pair of lights $(x_1, y_1), (x_2, y_2),$ with $x_1<x_2$ and $y_1>y_2,$ all lights strictly between them (i.e. $x_1<x<x_2$ and $y_2<y<y_1$) are toggled. After this procedure is done, how many lights are on?
2017 MMATHS, 3
Let $f : R \to R$, and let $P$ be a nonzero polynomial with degree no more than $2015$. For any nonnegative integer $n$, $f^{(n)}(x)$ denotes the function defined as $f$ composed with itself $n$ times. For example, $f^{(0)}(x) = x$, $f^{(1)}(x) = f(x)$, $f^{(2)}(x) = f(f(x))$, etc. Show that there always exists a real number $q$ such that $$f^{((2017^{2017})!)(q)} \ne (q + 2017)(qP(q) - 1).$$
2021 MMATHS, 1
Let $a,b,c$ be the roots of the polynomial $x^3 - 20x^2 + 22.$ Find \[\frac{bc}{a^2}+\frac{ac}{b^2}+\frac{ab}{c^2}.\]
[i]Proposed by Deyuan Li and Andrew Milas[/i]
2022 MMATHS, 8
In the number puzzle below, each cell contains a digit, each cell in the same bolded region has the same digit, and cells in different bolded regions have different digits. The answers to the clues are to be read as three-, four-, or five-digit numbers. Find the unique solution to the puzzle, given that no answer to any clue has a leading $0$.
[img]https://cdn.artofproblemsolving.com/attachments/b/a/23514673819aea46c30fd2947f8c82710a1fb3.png[/img]
2020 MMATHS, 1
A positive integer $n$ is called an untouchable number if there is no positive integer $m$ for which the sum of the factors of $m$ (including $m$ itself) is $n + m$. Find the sum of all of the untouchable numbers between $1$ and $10$ (inclusive)
MMATHS Mathathon Rounds, 2021
[u]Round 4[/u]
[b]p10.[/b] How many divisors of $10^{11}$ have at least half as many divisors that $10^{11}$ has?
[b]p11.[/b] Let $f(x, y) = \frac{x}{y}+\frac{y}{x}$ and $g(x, y) = \frac{x}{y}-\frac{y}{x} $. Then, if $\underbrace{f(f(... f(f(}_{2021 fs} f(f(1, 2), g(2,1)), 2), 2)... , 2), 2)$ can be expressed in the form $a + \frac{b}{c}$, where $a$, $b$,$c$ are nonnegative integers such that $b < c$ and $gcd(b,c) = 1$, find $a + b + \lceil (\log_2 (\log_2 c)\rceil $
[b]p12.[/b] Let $ABC$ be an equilateral triangle, and let$ DEF$ be an equilateral triangle such that $D$, $E$, and $F$ lie on $AB$, $BC$, and $CA$, respectively. Suppose that $AD$ and $BD$ are positive integers, and that $\frac{[DEF]}{[ABC]}=\frac{97}{196}$. The circumcircle of triangle $DEF$ meets $AB$, $BC$, and $CA$ again at $G$, $H$, and $I$, respectively. Find the side length of an equilateral triangle that has the same area as the hexagon with vertices $D, E, F, G, H$, and $I$.
[u]Round 5 [/u]
[b]p13.[/b] Point $X$ is on line segment $AB$ such that $AX = \frac25$ and $XB = \frac52$. Circle $\Omega$ has diameter $AB$ and circle $\omega$ has diameter $XB$. A ray perpendicular to $AB$ begins at $X$ and intersects $\Omega$ at a point $Y$. Let $Z$ be a point on $\omega$ such that $\angle YZX = 90^o$. If the area of triangle $XYZ$ can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $gcd(a, b) = 1$, find $a + b$.
[b]p14.[/b] Andrew, Ben, and Clayton are discussing four different songs; for each song, each person either likes or dislikes that song, and each person likes at least one song and dislikes at least one song. As it turns out, Andrew and Ben don't like any of the same songs, but Clayton likes at least one song that Andrew likes and at least one song that Ben likes! How many possible ways could this have happened?
[b]p15.[/b] Let triangle $ABC$ with circumcircle $\Omega$ satisfy $AB = 39$, $BC = 40$, and $CA = 25$. Let $P$ be a point on arc $BC$ not containing $A$, and let $Q$ and $R$ be the reflections of $P$ in $AB$ and $AC$, respectively. Let $AQ$ and $AR$ meet $\Omega$ again at $S$ and $T$, respectively. Given that the reflection of $QR$ over $BC$ is tangent to $\Omega$ , $ST$ can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $gcd(a,b)= 1$. Find $a + b$.
PS. You should use hide for answers. Rounds 1-3 have been posted [url=https://artofproblemsolving.com/community/c4h3131401p28368159]here [/url] and 6-7 [url=https://artofproblemsolving.com/community/c4h3131434p28368604]here [/url],Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2017 MMATHS, 4
In a triangle $ABC$, let $A_0$ be the point where the interior angle bisector of angle $A$ meets with side $BC$. Similarly define $B_0$ and $C_0$. Prove that $\angle B_0A_0C_0 = 90^o$ if and only if $\angle BAC = 120^o$.
2024 MMATHS, 4
Let $ABC$ be an equilateral triangle with side length $1.$ Then, let $M$ be the midpoint of $\overline{BC}.$ The area of all points within $ABC$ that are closer to $M$ than either of $A, B,$ or $C$ can be expressed as the fraction $\tfrac{\sqrt{a}}{b}$ where $a$ is not divisible by the square of any prime and $b$ is a positive integer. Find $a+b.$
2024 MMATHS, 5
Amir and Bella play a game on a gameboard with $6$ spaces, labeled $0, 1, 2, 3, 4,$ and $5.$ Each turn, each player flips a coin. If it is heads, their character moves forward one space, and if it is tails, their character moves back one space, unless it was already at space $0,$ in which case it moves forward one space instead. If Amir and Bella each have a character that starts at space $0,$ the probability that they end turn $5$ on the same space can be expressed as a common fraction $\tfrac{a}{b}.$ Find $a+b.$
2016 MMATHS, 1
Let unit blocks be unit squares in the coordinate plane with vertices at lattice points (points $(a, b)$ such that $a$ and $b$ are both integers). Prove that a circle with area $\pi$ can cover parts of no more than $9$ unit blocks. The circle below covers part of $8$ unit blocks.
[img]https://cdn.artofproblemsolving.com/attachments/4/4/43da9abed06d0feba94012ba68c177e3c2835b.png[/img]
2023 MMATHS, 8
Find the number of ordered pairs of integers $(m,n)$ such that $0 \le m,n \le 2023$ and $$m^2 \equiv \sum_{d \mid 2023} n^d \pmod{2024}.$$
2021 MMATHS, 8
Consider a hexagon with vertices labeled $M$, $M$, $A$, $T$, $H$, $S$ in that order. Clayton starts at the $M$ adjacent to $M$ and $A$, and writes the letter down. Each second, Clayton moves to an adjacent vertex, each with probability $\frac{1}{2}$, and writes down the corresponding letter. Clayton stops moving when the string he's written down contains the letters $M, A, T$, and $H$ in that order, not necessarily consecutively (for example, one valid string might be $MAMMSHTH$.) What is the expected length of the string Clayton wrote?
[i]Proposed by Andrew Milas and Andrew Wu[/i]
2014 MMATHS, Mixer Round
[b]p1.[/b] How many real roots does the equation $2x^7 + x^5 + 4x^3 + x + 2 = 0$ have?
[b]p2.[/b] Given that $f(n) = 1 +\sum^n_{j=1}(1 +\sum^j_{i=1}(2i + 1))$, find the value of $f(99)-\sum^{99}_{i=1} i^2$.
[b]p3.[/b] A rectangular prism with dimensions $1\times a \times b$, where $1 < a < b < 2$, is bisected by a plane bisecting the longest edges of the prism. One of the smaller prisms is bisected in the same way. If all three resulting prisms are similar to each other and to the original box, compute $ab$. Note: Two rectangular prisms of dimensions $p \times q\times r$ and$ x\times y\times z$ are similar if $\frac{p}{x} = \frac{q}{y} = \frac{r}{z}$ .
[b]p4.[/b] For fixed real values of $p$, $q$, $r$ and $s$, the polynomial $x^4 + px^3 + qx^2 + rx + s$ has four non real roots. The sum of two of these roots is $4 + 7i$, and the product of the other two roots is $3 - 4i$. Compute $q$.
[b]p5.[/b] There are $10$ seats in a row in a theater. Say we have an infinite supply of indistinguishable good kids and bad kids. How many ways can we seat $10$ kids such that no two bad kids are allowed to sit next to each other?
[b]p6.[/b] There are an infinite number of people playing a game. They each pick a different positive integer $k$, and they each win the amount they chose with probability $\frac{1}{k^3}$ . What is the expected amount that all of the people win in total?
[b]p7.[/b] There are $100$ donuts to be split among $4$ teams. Your team gets to propose a solution about how the donuts are divided amongst the teams. (Donuts may not be split.) After seeing the proposition, every team either votes in favor or against the propisition. The proposition is adopted with a majority vote or a tie. If the proposition is rejected, your team is eliminated and will never receive any donuts. Another remaining team is chosen at random to make a proposition, and the process is repeated until a proposition is adopted, or only one team is left. No promises or deals need to be kept among teams besides official propositions and votes. Given that all teams play optimally to maximize the expected value of the number of donuts they receive, are completely indifferent as to the success of the other teams, but they would rather not eliminate a team than eliminate one (if the number of donuts they receive is the same either way), then how much should your team propose to keep?
[b]p8.[/b] Dominic, Mitchell, and Sitharthan are having an argument. Each of them is either credible or not credible – if they are credible then they are telling the truth. Otherwise, it is not known whether they are telling the truth. At least one of Dominic, Mitchell, and Sitharthan is credible. Tim knows whether Dominic is credible, and Ethan knows whether Sitharthan is credible. The following conversation occurs, and Tim and Ethan overhear:
Dominic: “Sitharthan is not credible.”
Mitchell: “Dominic is not credible.”
Sitharthan: “At least one of Dominic or Mitchell is credible.”
Then, at the same time, Tim and Ethan both simultaneously exclaim: “I can’t tell exactly who is credible!”
They each then think for a moment, and they realize that they can. If Tim and Ethan always tell the truth, then write on your answer sheet exactly which of the other three are credible.
[b]p9.[/b] Pick an integer $n$ between $1$ and $10$. If no other team picks the same number, we’ll give you $\frac{n}{10}$ points.
[b]p10.[/b] Many quantities in high-school mathematics are left undefined. Propose a definition or value for the following expressions and justify your response for each. We’ll give you $\frac15$ points for each reasonable argument.
$$(i) \,\,\,(.5)! \,\,\, \,\,\,(ii) \,\,\,\infty \cdot 0 \,\,\, \,\,\,(iii) \,\,\,0^0 \,\,\, \,\,\,(iv)\,\,\, \prod_{x\in \emptyset}x \,\,\, \,\,\,(v)\,\,\, 1 - 1 + 1 - 1 + ...$$
[b]p11.[/b] On the back of your answer sheet, write the “coolest” math question you know, and include the solution. If the graders like your question the most, then you’ll get a point. (With your permission, we might include your question on the Mixer next year!)
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].