Found problems: 146
2024 MMATHS, 6
How many $7$ digit numbers are there that satisfy the following?
[list]
[*] All digits are distinct from $1-7.$
[*] The first digit (from the left) is divisible by $1.$
[*] The two-digit number formed by the first two digits is divisible by $2.$
[*] The three-digit number formed by the first three digits is divisible by $3.$
[*] The four-digit number formed by the first four digits is divisible by $4.$
[*] The five-digit number formed by the first five digits is divisible by $5.$
[*] The six-digit number formed by the first six digits is divisible by $6.$
[/list]
2015 MMATHS, Mixer Round
[b]p1.[/b] Let $a_0, a_1,...,a_n$ be such that $a_n \ne 0$ and
$$(1 + x + x^3)^{341}(1 + 2x + x^2 + 2x^3 + 2x^4 + x^6)^{342} =\sum^n_{i=0}a_ix^i,$$
Find the number of odd numbers in the sequence a0; a1; : : : an.
[b]p2.[/b] Let $F_0 = 1$, $F_1 = 1$ and F$_k = F_{k-1} + F_{k-2}$. Let $P(x) =\sum^{99}_{k=0} x^{F_k}$ . The remainder when $P(x)$ is divided by $x^3 - 1$ can be expressed as $ax^2 + bx + c$. Find $2a + b$.
[b]p3.[/b] Let $a_n$ be the number of permutations of the numbers $S = \{1, 2,...,n\}$ such that for all $k$ with $1 \le k \le n$, the sum of $k$ and the number in the $k$th position of the permutation is a power of $2$. Compute $a_{2^0} + a_{2^1} +... + a_{2^{20}}$ .
[b]p4.[/b] Three identical balls are painted white and black, so that half of each sphere is a white hemisphere, and the other half is a black one. The three balls are placed on a plane surface, each with a random orientation, so that each ball has a point of contact with the other two. What is the probability that at at least one point of contact between two of the balls, both balls are the same color?
[b]p5.[/b] Compute the greatest positive integer $n$ such that there exists an odd integer $a$, for which $\frac{a^{2^n}-1}{4^{4^4}}$ is not an integer.
[b]p6.[/b] You are blind and cannot feel the difference between a coin that is heads up or tails up. There are $100$ coins in front of you and are told that exactly $10$ of them are heads up. On the back of this paper, explain how you can split the otherwise indistinguishable coins into two groups so that both groups have the same number of heads.
[b]p7.[/b] On the back of this page, write the best math pun you can think of. You’ll get a point if we chuckle.
[b]p8.[/b] Pick an integer between $1$ and $10$. If you pick $k$, and $n$ total teams pick $k$, then you’ll receive $\frac{k}{10n}$ points.
[b]p9.[/b] There are four prisoners in a dungeon. Tomorrow, they will be separated into a group of three in one room, and the other in a room by himself. Each will be given a hat to wear that is either black or white – two will be given white and two black. None of them will be able to communicate with each other and none will see his or her own hat color. The group of three is lined up, so that the one in the back can see the other two, the second can see the first, but the first cannot see the others. If anyone is certain of their hat color, then they immediately shout that they know it to the rest of the group. If they can secretly prove it to the guard, they are saved. They only say something if they’re sure. Which person is sure to survive?
[b]p10.[/b] Down the road, there are $10$ prisoners in a dungeon. Tomorrow they will be lined up in a single room and each given a black or white hat – this time they don’t know how many of each. The person in the back can see everyone’s hat besides his own, and similarly everyone else can only see the hats of the people in front of them. The person in the back will shout out a guess for his hat color and will be saved if and only if he is right. Then the person in front of him will have to guess, and this will continue until everyone has the opportunity to be saved. Each person can only say his or her guess of “white” or “black” when their turn comes, and no other signals may be made. If they have the night before receiving the hats to try to devise some sort of code, how many people at a minimum can be saved with the most optimal code? Describe the code on the back of this paper for full points.
[b]p11.[/b] A few of the problems on this mixer contest were taken from last year’s event. One of them had fewer than $5$ correct answers, and most of the answers given were the same incorrect answer. Half a point will be given if you can guess the number of the problem on this test that corresponds to last year’s question, and another $.5$ points will be given if you can guess the very common incorrect answer.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2022 MMATHS, 11
Every time Josh and Ron tap their screens, one of three emojis appears, each with equal probability: barbecue, bacon, or burger. Josh taps his screen until he gets a sequence of barbecue, bacon, and burger consecutively (in that specific order.) Ron taps his screen until he gets a sequence of three bacons in a row. Let $M$ and $N$ be the expected number of times Josh and Ron tap their screens, respectively. What is $|M-N|$?
2022 MMATHS, 5
Equilateral triangle $\vartriangle ABC$ has side length $6$. Points $D$ and $E$ lie on $\overline{BC}$ such that $BD = CE$ and $B$, $D$, $E$, $C$ are collinear in that order. Points $F$ and $G$ lie on $\overline{AB}$ such that $\overline{FD} \perp \overline{BC}$, and $GF = GA$. If the minimum possible value of the sum of the areas of $\vartriangle BFD$ and $\vartriangle DGE$ can be expressed as $\frac{a\sqrt{b}}{c}$ for positive integers $a, b, c$ with $gcd (a, c) = 1$ and $b$ squarefree, find $a + b + c$.
2024 MMATHS, 5
Two subsets are called [i]disjoint[/i] if they do not share any common elements. Compute the number of ordered tuples $(A,B,C),$ where $A,B,$ and $C$ are subsets (not necessarily distinct or non-empty) of $\{1, 2, 3,4,5\}$ such that $A$ and $B$ are disjoint and $B$ and $C$ are disjoint.
2020 MMATHS, 6
Consider the function $f(n) = n^2 + n + 1$. For each $n$, let $d_n$ be the smallest positive integer with $gcd(n, dn) = 1$ and $f(n) | f(d_n)$. Find $d_6 + d_7 + d_8 + d_9 + d_{10}$.
MMATHS Mathathon Rounds, 2021
[u]Round 6[/u]
[b]p16.[/b] Let $ABC$ be a triangle with $AB = 3$, $BC = 4$, and $CA = 5$. There exist two possible points $X$ on $CA$ such that if $Y$ and $Z$ are the feet of the perpendiculars from $X$ to $AB$ and $BC,$ respectively, then the area of triangle $XY Z$ is $1$. If the distance between those two possible points can be expressed as $\frac{a\sqrt{b}}{c}$ for positive integers $a$, $b$, and $c$ with $b$ squarefree and $gcd(a, c) = 1$, then find $a +b+ c$.
[b]p17.[/b] Let $f(n)$ be the number of orderings of $1,2, ... ,n$ such that each number is as most twice the number preceding it. Find the number of integers $k$ between $1$ and $50$, inclusive, such that $f (k)$ is a perfect square.
[b]p18.[/b] Suppose that $f$ is a function on the positive integers such that $f(p) = p$ for any prime p, and that $f (xy) = f(x) + f(y)$ for any positive integers $x$ and $y$. Define $g(n) = \sum_{k|n} f (k)$; that is, $g(n)$ is the sum of all $f(k)$ such that $k$ is a factor of $n$. For example, $g(6) = f(1) + 1(2) + f(3) + f(6)$. Find the sum of all composite $n$ between $50$ and $100$, inclusive, such that $g(n) = n$.
[u]Round 7[/u]
[b]p19.[/b] AJ is standing in the center of an equilateral triangle with vertices labelled $A$, $B$, and $C$. They begin by moving to one of the vertices and recording its label; afterwards, each minute, they move to a different vertex and record its label. Suppose that they record $21$ labels in total, including the initial one. Find the number of distinct possible ordered triples $(a, b, c)$, where a is the number of $A$'s they recorded, b is the number of $B$'s they recorded, and c is the number of $C$'s they recorded.
[b]p20.[/b] Let $S = \sum_{n=1}^{\infty} (1- \{(2 + \sqrt3)^n\})$, where $\{x\} = x - \lfloor x\rfloor$ , the fractional part of $x$. If $S =\frac{\sqrt{a} -b}{c}$ for positive integers $a, b, c$ with $a $ squarefree, find $a + b + c$.
[b]p21.[/b] Misaka likes coloring. For each square of a $1\times 8$ grid, she flips a fair coin and colors in the square if it lands on heads. Afterwards, Misaka places as many $1 \times 2$ dominos on the grid as possible such that both parts of each domino lie on uncolored squares and no dominos overlap. Given that the expected number of dominos that she places can be written as $\frac{a}{b}$, for positive integers $a$ and $b$ 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 4-5 [url=https://artofproblemsolving.com/community/c4h3131422p28368457]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2023 MMATHS, 5
We call $\triangle{ABC}$ with centroid $G$ [i]balanced[/i] on side $AB$ if the foot of the altitude from $G$ onto line $\overline{AB}$ lies between $A$ and $B.$ $\triangle{XYZ},$ with $XY=2023$ and $\angle{ZXY}=120^\circ,$ is balanced on $XY.$ What is the maximum value of $XZ$?
2023 MMATHS, 12
Let $ABC$ be a triangle with incenter $I.$ The incircle $\omega$ of $ABC$ is tangent to sides $BC, CA,$ and $AB$ at points $D, E,$ and $F,$ respectively. Let $D'$ be the reflection of $D$ over $I.$ Let $P$ be a point on $\omega$ such that $\angle{ADP}=90^\circ.$ $\mathcal{H}$ is a hyperbola passing through $D', E, F, I,$ and $P.$ Given that $\angle{BAD}=45^\circ$ and $\angle{CAD}=30^\circ,$ the acute angle between the asymptotes of $\mathcal{H}$ can be expressed as $\left(\tfrac{m}{n}\right)^\circ,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
MMATHS Mathathon Rounds, 2016
[u]Round 1[/u]
[b]p1.[/b] This year, the Mathathon consists of $7$ rounds, each with $3$ problems. Another math test, Aspartaime, consists of $3$ rounds, each with $5$ problems. How many more problems are on the Mathathon than on Aspartaime?
[b]p2.[/b] Let the solutions to $x^3 + 7x^2 - 242x - 2016 = 0 $be $a, b$, and $c$. Find $a^2 + b^2 + c^2$. (You might find it helpful to know that the roots are all rational.)
[b]p3.[/b] For triangle $ABC$, you are given $AB = 8$ and $\angle A = 30^o$ . You are told that $BC$ will be chosen from amongst the integers from $1$ to $10$, inclusive, each with equal probability. What is the probability that once the side length $BC$ is chosen there is exactly one possible triangle $ABC$?
[u]Round 2 [/u]
[b]p4.[/b] It’s raining! You want to keep your cat warm and dry, so you want to put socks, rain boots, and plastic bags on your cat’s four paws. Note that for each paw, you must put the sock on before the boot, and the boot before the plastic bag. Also, the items on one paw do not affect the items you can put on another paw. How many different orders are there for you to put all twelve items of rain footwear on your cat?
[b]p5.[/b] Let $a$ be the square root of the least positive multiple of $2016$ that is a square. Let $b$ be the cube root of the least positive multiple of $2016$ that is a cube. What is $ a - b$?
[b]p6.[/b] Hypersomnia Cookies sells cookies in boxes of $6, 9$ or $10$. You can only buy cookies in whole boxes. What is the largest number of cookies you cannot exactly buy? (For example, you couldn’t buy $8$ cookies.)
[u]Round 3 [/u]
[b]p7.[/b] There is a store that sells each of the $26$ letters. All letters of the same type cost the same amount (i.e. any ‘a’ costs the same as any other ‘a’), but different letters may or may not cost different amounts. For example, the cost of spelling “trade” is the same as the cost of spelling “tread,” even though the cost of using a ‘t’ may be different from the cost of an ‘r.’ If the letters to spell out $1$ cost $\$1001$, the letters to spell out $2$ cost $\$1010$, and the letters to spell out $11$ cost $\$2015$, how much do the letters to spell out $12$ cost?
[b]p8.[/b] There is a square $ABCD$ with a point $P$ inside. Given that $PA = 6$, $PB = 9$, $PC = 8$. Calculate $PD$.
[b]p9.[/b] How many ordered pairs of positive integers $(x, y)$ are solutions to $x^2 - y^2 = 2016$?
[u]Round 4 [/u]
[b]p10.[/b] Given a triangle with side lengths $5, 6$ and $7$, calculate the sum of the three heights of the triangle.
[b]p11. [/b]There are $6$ people in a room. Each person simultaneously points at a random person in the room that is not him/herself. What is the probability that each person is pointing at someone who is pointing back to them?
[b]p12.[/b] Find all $x$ such that $\sum_{i=0}^{\infty} ix^i =\frac34$.
PS. You should use hide for answers. Rounds 5-7 have been posted [url=https://artofproblemsolving.com/community/c4h2782837p24446063]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2021 MMATHS, 5
Suppose that $a_1 = 1$, and that for all $n \ge 2$, $a_n = a_{n-1} + 2a_{n-2} + 3a_{n-3} + \ldots + (n-1)a_1.$ Suppose furthermore that $b_n = a_1 + a_2 + \ldots + a_n$ for all $n$. If $b_1 + b_2 + b_3 + \ldots + b_{2021} = a_k$ for some $k$, find $k$.
[i]Proposed by Andrew Wu[/i]
MMATHS Mathathon Rounds, Sample
[b]p1.[/b] What is the largest distance between any two points on a regular hexagon with a side length of one?
[b]p2.[/b] For how many integers $n \ge 1$ is $\frac{10^n - 1}{9}$ the square of an integer?
[b]p3.[/b] A vector in $3D$ space that in standard position in the first octant makes an angle of $\frac{\pi}{3}$ with the $x$ axis and $\frac{\pi}{4}$ with the $y$ axis. What angle does it make with the $z$ axis?
[b]p4.[/b] Compute $\sqrt{2012^2 + 2012^2 \cdot 2013^2 + 2013^2} - 2012^2$.
[b]p5.[/b] Round $\log_2 \left(\sum^{32}_{k=0} {{32} \choose k} \cdot 3^k \cdot 5^k\right)$ to the nearest integer.
[b]p6.[/b] Let $P$ be a point inside a ball. Consider three mutually perpendicular planes through $P$. These planes intersect the ball along three disks. If the radius of the ball is $2$ and $1/2$ is the distance between the center of the ball and $P$, compute the sum of the areas of the three disks of intersection.
[b]p7.[/b] Find the sum of the absolute values of the real roots of the equation $x^4 - 4x - 1 = 0$.
[b]p8.[/b] The numbers $1, 2, 3, ..., 2013$ are written on a board. A student erases three numbers $a, b, c$ and instead writes the number $$\frac12 (a + b + c)\left((a - b)^2 + (b - c)^2 + (c - a)^2\right).$$ She repeats this process until there is only one number left on the board. List all possible values of the remainder when the last number is divided by 3.
[b]p9.[/b] How many ordered triples of integers $(a, b, c)$, where $1 \le a, b, c \le 10$, are such that for every natural number $n$, the equation $(a + n)x^2 + (b + 2n)x + c + n = 0$ has at least one real root?
Problems' source (as mentioned on official site) is Gator Mathematics Competition.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2014 MMATHS, 1
Show that there does not exist a right triangle with all integer side lengths such that exactly one of the side lengths is odd.
2022 MMATHS, 1
Rectangle $ABCD$ has $AB = 8$ and $BC = 13$. Points $P_1$ and $P_2$ lie on $AB$ and $CD$ with $P_1P_2 \parallel BC$. Points $Q_1$ and $Q_2$ lie on $BC$ and $DA$ with $Q_1Q_2 \parallel AB$. Find the area of quadrilateral $P_1Q_1P_2Q_2$.
2024 MMATHS, 4
Consider a pattern of squares and triangles. The first move of the pattern is to place an isosceles right triangle with side lengths $1, 1, \sqrt{2}.$ For each subsequent move, you need to attach a square to every non-hypotenuse side of a triangle and attach the same isosceles right triangle to every side of a square. After $2024$ moves, what is smallest possible area of the resulting shape?
2020 MMATHS, 3
Let $a, b$ be two real numbers such that $$\sqrt[3]{a}- \sqrt[3]{b} = 10, ,\,\,\,\,\,\, ab = \left( \frac{8 - a - b}{6}\right)^3$$ Find $a - b$.
2014 MMATHS, 2
Let $(a_n)^{\infty}_{n =1}$ be a sequence of positive integers with $a_1 < a_2 < a_3 < ...$ , and for n = 1, 2, 3,..., $$a_{2n} = a_n + n.$$ Furthermore, whenever $n$ is prime, so is $a_n$. Prove that $a_n = n$.
2022 MMATHS, 9
Let $f$ be a monic cubic polynomial such that the sum of the coefficients of $f$ is $5$ and such that the sum of the roots of $f$ is $1$. Find the absolute value of the sum of the cubes of the roots of $f$.
2023 MMATHS, 4
How many distinct real numbers $x$ satisfy the equation $4\cos^3(x)+\sqrt{x}=3\sin(x)+\cos(3x)$?
2022 MMATHS, 9
Suppose sequence $\{a_i\} = a_1, a_2, a_3, ....$ satisfies $a_{n+1} = \frac{1}{a_n+1}$ for all positive integers $n$. Define $b_k$ for positive integers $k \ge 2$ to be the minimum real number such that the product $a_1 \cdot a_2 \cdot ...\cdot a_k$ does not exceed $b_k$ for any positive integer choice of $a_1$. Find $\frac{1}{b_2}+\frac{1}{b_3}+\frac{1}{b_4}+...+\frac{1}{b_{10}}.$
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2022 MMATHS, 11
Denote by $Re(z)$ and $Im(z)$ the real part and imaginary part, respectively, of a complex number $z$; that is, if $z = a + bi$, then $Re(z) = a$ and $Im(z) = b$. Suppose that there exists some real number $k$ such that $Im \left( \frac{1}{w} \right) = Im \left( \frac{k}{w^2} \right) = Im \left( \frac{k}{w^3} \right) $ for some complex number $w$ with $||w||=\frac{\sqrt3}{2}$ , $Re(w) > 0$, and $Im(w) \ne 0$. If $k$ can be expressed as $\frac{\sqrt{a}-b}{c}$ for integers $a$, $b$, $c$ with $a$ squarefree, find $a + b + c$.
2022 MMATHS, 8
Let $S = \{1, 2, 3, 5, 6, 10, 15, 30\}$. For each of the $64$ ordered pairs $(a, b)$ of elements of $S$, AJ computes $gcd(a, b)$. They then sum all of the numbers they computed. What is AJ’s sum?
2024 MMATHS, 6
Cat and Claire are having a discussion about their favorite positive two-digit numbers.
[b]Cat:[/b] My number has a $1$ in its tens digit. Is it possible that your number is a multiple of my number?
[b]Claire:[/b] No, however, my number is not prime. Additionally, if I told you the two digits of my number, you still wouldn't know my number.
[b]Cat:[/b] Aha, my number and your number aren't relatively prime!
[b]Claire:[/b] Then our numbers must share the same ones digit!
What is the product of Cat and Claire's numbers?
2016 MMATHS, 4
For real numbers $a, b, c$ with $a + b + c = 3$, prove that $$a^2(b - c)^2 + b^2(c - a)^2 + c^2(a - b)^2 \ge \frac9 2 abc(1 - abc)$$ and state when equality is reached.
2023 MMATHS, 12
Let $ABC$ be a triangle with incenter $I,$ circumcenter $O,$ and $A$-excenter $J_A.$ The incircle of $\triangle{ABC}$ touches side $BC$ at a point $D.$ Lines $OI$ and $J_AD$ meet at a point $K.$ Line $AK$ meets the circumcircle of $\triangle{ABC}$ again at a point $L \neq A.$ If $BD=11, CD=5,$ and $AO=10,$ the length of $DL$ can be expressed as $\tfrac{m\sqrt{p}}{n},$ where $m,n,p$ are positive integers, $m$ and $n$ are relatively prime, and $p$ is not divisible by the square of any prime. Find $m+n+p.$