Found problems: 146
2015 MMATHS, 3
Is there a number $s$ in the set $\{\pi,2\pi,3\pi,...,\} $ such that the first three digits after the decimal point of $s$ are $.001$? Fully justify your answer.
2019 MMATHS, 3
Let m and n be positive integers. Alice wishes to walk from the point $(0, 0)$ to the point $(m,n)$ in increments of $(1, 0)$ and $(0, 1)$, and Bob wishes to walk from the point $(0,1)$ to the point $(m, n + 1)$ in increments of$ (1, 0)$ and $(0,1)$. Find (with proof) the number of ways for Alice and Bob to get to their destinations if their paths never pass through the same point (even at different times).
2014 MMATHS, 4
Determine, with proof, the maximum and minimum among the numbers
$$\sqrt5 - \lfloor \sqrt5 \rfloor, 2\sqrt5 - \lfloor 2\sqrt5 \rfloor, 3\sqrt5 - \lfloor 3
\sqrt5\rfloor, ..., 2013\sqrt5 - \lfloor 2013\sqrt5\rfloor, 2014\sqrt5 - \lfloor 2014\sqrt5\rfloor $$
2022 MMATHS, 5
Holding a rectangular sheet of paper $ABCD$, Prair folds triangle $ABD$ over diagonal $BD$, so that the new location of point $A$ is $A'$. She notices that $A'C =\frac13 BD$. If the area of $ABCD$ is $27\sqrt2$, find $BD$.
MMATHS Mathathon Rounds, 2015
[u]Round 1[/u]
[b]p1.[/b] If this mathathon has $7$ rounds of $3$ problems each, how many problems does it have in total? (Not a trick!)
[b]p2.[/b] Five people, named $A, B, C, D,$ and $E$, are standing in line. If they randomly rearrange themselves, what’s the probability that nobody is more than one spot away from where they started?
[b]p3.[/b] At Barrios’s absurdly priced fish and chip shop, one fish is worth $\$13$, one chip is worth $\$5$. What is the largest integer dollar amount of money a customer can enter with, and not be able to spend it all on fish and chips?
[u]Round 2[/u]
[b]p4.[/b] If there are $15$ points in $4$-dimensional space, what is the maximum number of hyperplanes that these points determine?
[b]p5.[/b] Consider all possible values of $\frac{z_1 - z_2}{z_2 - z_3} \cdot \frac{z_1 - z_4}{z_2 - z_4}$ for any distinct complex numbers $z_1$, $z_2$, $z_3$, and $z_4$. How many complex numbers cannot be achieved?
[b]p6.[/b] For each positive integer $n$, let $S(n)$ denote the number of positive integers $k \le n$ such that $gcd(k, n) = gcd(k + 1, n) = 1$. Find $S(2015)$.
[u]Round 3 [/u]
[b]p7.[/b] Let $P_1$, $P_2$,$...$, $P_{2015}$ be $2015$ distinct points in the plane. For any $i, j \in \{1, 2, ...., 2015\}$, connect $P_i$ and $P_j$ with a line segment if and only if $gcd(i - j, 2015) = 1$. Define a clique to be a set of points such that any two points in the clique are connected with a line segment. Let $\omega$ be the unique positive integer such that there exists a clique with $\omega$ elements and such that there does not exist a clique with $\omega + 1$ elements. Find $\omega$.
[b]p8.[/b] A Chinese restaurant has many boxes of food. The manager notices that
$\bullet$ He can divide the boxes into groups of $M$ where $M$ is $19$, $20$, or $21$.
$\bullet$ There are exactly $3$ integers $x$ less than $16$ such that grouping the boxes into groups of $x$ leaves $3$ boxes left over.
Find the smallest possible number of boxes of food.
[b]p9.[/b] If $f(x) = x|x| + 2$, then compute $\sum^{1000}_{k=-1000} f^{-1}(f(k) + f(-k) + f^{-1}(k))$.
[u]Round 4 [/u]
[b]p10.[/b] Let $ABC$ be a triangle with $AB = 13$, $BC = 20$, $CA = 21$. Let $ABDE$, $BCFG$, and $CAHI$ be squares built on sides $AB$, $BC$, and $CA$, respectively such that these squares are outside of $ABC$. Find the area of $DEHIFG$.
[b]p11.[/b] What is the sum of all of the distinct prime factors of $7783 = 6^5 + 6 + 1$?
[b]p12.[/b] Consider polyhedron $ABCDE$, where $ABCD$ is a regular tetrahedron and $BCDE$ is a regular tetrahedron. An ant starts at point $A$. Every time the ant moves, it walks from its current point to an adjacent point. The ant has an equal probability of moving to each adjacent point. After $6$ moves, what is the probability the ant is back at point $A$?
PS. You should use hide for answers. Rounds 5-7 have been posted [url=https://artofproblemsolving.com/community/c4h2782011p24434676]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2022 MMATHS, 7
$\vartriangle ABC$ satisfies $AB = 16$, $BC = 30$, and $\angle ABC = 90^o$. On the circumcircle of $\vartriangle ABC$, let $P$ be the midpoint of arc $AC$ not containing $B$, and let $X$ and $Y$ lie on lines $AB$ and $BC$, respectively, with $PX \perp AB$ and $PY \perp BC$. Find $XY^2$.
2024 MMATHS, 1
On a planet, far, far away, the Yaliens have defined: $x$ "equals" $y$ if and only if $|x-y| \le 3.$ Let $S$ be a set of positive integers. What is the smallest possible number of elements in $S$ such that, for any positive integer $r,$ where $1 \le r \le 2024,$ $r$ "equals" some element in $S$?
2021 MMATHS, 3
Find the sum of all $x$ from $2$ to $1000$ inclusive such that $$\prod_{n=2}^x \log_{n^n}(n+1)^{n+2}$$ is an integer.
[i]Proposed by Deyuan Li and Andrew Milas[/i]
2017 MMATHS, 2
Suppose you are playing a game against Daniel. There are $2017$ chips on a table. During your turn, if you can write the number of chips on the table as a sum of two cubes of not necessarily distinct, nonnegative integers, then you win. Otherwise, you can take some number of chips between $1$ and $6$ inclusive off the table. (You may not leave fewer than $0$ chips on the table.) Daniel can also do the same on his turn. You make the first move, and you and Daniel always make the optimal move during turns. Who should win the game? Explain.
2023 MMATHS, 3
There are $360$ permutations of the letters in $MMATHS.$ When ordered alphabetically, starting from $AHMMST,$ $MMATHS$ is in the $n$th permutation. What is $n$?
2017 MMATHS, 1
For any integer $n > 4$, prove that $2^n > n^2$.
2022 MMATHS, 2
How many ways are there to fill in a three by three grid of cells with $0$’s and $2$’s, one number in each cell, such that each two by two contiguous subgrid contains exactly three $2$’s and one $0$?
2016 MMATHS, 3
Show that there are no integers $x, y, z$, and $t$ such that $$\sqrt[3]{x^5 + y^5 + z^5 + t^5} = 2016.$$
MMATHS Mathathon Rounds, 2014
[u]Round 5 [/u]
[b]p13.[/b] How many ways can we form a group with an odd number of members (plural) from $99$ people? Express your answer in the form $a^b + c$, where $a, b$, and $c$ are integers and $a$ is prime.
[b]p14.[/b] A cube is inscibed in a right circular cone such that the ratio of the height of the cone to the radius is $2:1$. Compute the fraction of the cone’s volume that the cube occupies.
[b]p15.[/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 $2a + b$.
[u]Round 6 [/u]
[b]p16.[/b] Ankit finds a quite peculiar deck of cards in that each card has n distinct symbols on it and any two cards chosen from the deck will have exactly one symbol in common. The cards are guaranteed to not have a certain symbol which is held in common with all the cards. Ankit decides to create a function f(n) which describes the maximum possible number of cards in a set given the previous constraints. What is the value of $f(10)$?
[b]p17.[/b] If $|x| <\frac14$ and $$X = \sum^{\infty}_{N=0} \sum^{N}_{n=0} {N \choose n}x^{2n}(2x)^{N-n}.$$ then write $X$ in terms of $x$ without any summation or product symbols (and without an infinite number of ‘$+$’s, etc.).
[b]p18.[/b] Dietrich is playing a game where he is given three numbers $a, b, c$ which range from $[0, 3]$ in a continuous uniform distribution. Dietrich wins the game if the maximum distance between any two numbers is no more than $1$. What is the probability Dietrich wins the game?
[u]Round 7 [/u]
[b]p19.[/b] Consider f defined by $$f(x) = x^6 + a_1x^5 + a_2x^4 + a_3x^3 + a_4x^2 + a_5x + a_6.$$ How many tuples of positive integers $(a_1, a_2, a_3, a_4, a_5, a_6)$ exist such that $f(-1) = 12$ and $f(1) = 30$?
[b]p20.[/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 $a_1 + a_2 + a_4 + a_8 + ... + a_{1048576}$.
[b]p21.[/b] A $4$-dimensional hypercube of edge length $1$ is constructed in $4$-space with its edges parallel to the coordinate axes and one vertex at the origin. Its coordinates are given by all possible permutations of $(0, 0, 0, 0)$,$(1, 0, 0, 0)$,$(1, 1, 0, 0)$,$(1, 1, 1, 0)$, and $(1, 1, 1, 1)$. The $3$-dimensional hyperplane given by $x+y+z+w = 2$ intersects the hypercube at $6$ of its vertices. Compute the 3-dimensional volume of the solid formed by the intersection.
PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c4h2781335p24424563]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2023 MMATHS, 2
The lengths of the altitudes of $\triangle{ABC}$ are the roots of the polynomial $x^3-34x^2+360x-1200.$ Find the area of $\triangle{ABC}.$
2018 MMATHS, 2
Prove that if a triangle has integer side lengths and the area (in square units) equals the perimeter (in units), then the perimeter is not a prime number.
2024 MMATHS, 11
Let $n$ be the least possible value of $$\sqrt{x^2+y^2-2x+6y+19}+\sqrt{x^2+y^2+8x-4y+21}.$$ Find $n^2.$
2024 MMATHS, 7
Bill has the expression $1+2+3+\cdots+8.$ He replaces two different addition symbols with multiplication symbols uniformly at random. The value that he obtains on average can be expressed as a common fraction $\tfrac{m}{n}.$ Find $m+n.$
2020 MMATHS, 5
Let $x, y$ be positive reals such that $x \ne y$. Find the minimum possible value of $(x + y)^2 + \frac{54}{xy(x-y)^2}$ .
2022 MMATHS, 12
Let triangle $ABC$ with incenter $I$ satisfy $AB = 3$, $AC = 4$, and $BC = 5$. Suppose that $D$ and $E$ lie on $AB$ and $AC$, respectively, such that $D$, $I$, and $E$ are collinear and $DE \perp AI$. Points $P$ and $Q$ lie on side $BC$ such that $IP = BP$ and $IQ = CQ$, and lines $DP$ and $EQ$ meet at $S$. Compute $SI^2$.
2021 MMATHS, 3
Let $ABCDEF$ be a regular hexagon with sidelength $6$, and construct squares $ABGH$, $BCIJ$, $CDKL$, $DEMN$, $EFOP$, and $FAQR$ outside the hexagon. Find the perimeter of dodecagon $HGJILKNMPORQ$.
[i]Proposed by Andrew Wu[/i]