This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 161

2025 Harvard-MIT Mathematics Tournament, 13
Tags: guts
A number is [i]upwards[/i] if its digits in base $10$ are nondecreasing when read from left to right. Compute the number of positive integers less than $10^6$ that are both upwards and multiples of $11.$

2023 Harvard-MIT Mathematics Tournament, 31
Tags: guts
Let $$P=\prod_{i=0}^{2016} (i^3-i-1)^2.$$ The remainder when $P$ is divided by the prime $2017$ is not zero. Compute this remainder.

2024 HMNT, 24
Tags: guts
Let $f(x) = x^2 +6x+6.$ Compute the greatest real number $x$ such that $f(f(f(f(f(f(x)))))) = 0.$

2024 Harvard-MIT Mathematics Tournament, 28
Tags: guts
Given that the $32$-digit integer $$64 \ 312 \ 311 \ 692 \ 944 \ 269 \ 609 \ 355 \ 712 \ 372 \ 657$$ is the product of $6$ consecutive primes, compute the sum of these $6$ primes.

2023 Harvard-MIT Mathematics Tournament, 16
Tags: guts
The graph of the equation $x+y=\lfloor x^2+y^2 \rfloor$ consists of several line segments. Compute the sum of their lengths.

2024 Harvard-MIT Mathematics Tournament, 8
Tags: guts
Three points, $A, B,$ and $C,$ are selected independently and uniformly at random from the interior of a unit square. Compute the expected value of $\angle{ABC}.$

2024 HMNT, 3
Tags: guts
The graphs of the lines $$y=x+2, \quad y=3x+4, \quad y=5x+6,\quad y=7x+8,\quad y=9x+10,\quad y=11x+12$$ are drawn. These six lines divide the plane into several regions. Compute the number of regions the plane is divided into.

2025 Harvard-MIT Mathematics Tournament, 21
Tags: guts
Compute the unique five-digit positive integer $\underline{abcde}$ such that $a \neq 0, c \neq 0,$ and $$\underline{abcde}=(\underline{ab}+\underline{cde})^2.$$

2025 Harvard-MIT Mathematics Tournament, 26
Tags: guts
Isabella has a bag with $20$ blue diamonds and $25$ purple diamonds. She repeats the following process $44$ times: she removes a diamond from the bag uniformly at random, then puts one blue diamond and one purple diamond into the bag. Compute the expected number of blue diamonds in the bag after all $44$ repetitions.

2025 Harvard-MIT Mathematics Tournament, 31
Tags: guts
There exists a unique circle that is both tangent to the parabola $y=x^2$ at two points and tangent to the curve $x=\sqrt{\tfrac{y^3}{1-y}}.$ Compute the radius of this circle.

2024 HMNT, 8
Tags: guts
Derek is bored in math class and is drawing a flower. He first draws $8$ points $A_1, A_2, \ldots, A_8$ equally spaced around an enormous circle. He then draws $8$ arcs outside the circle where the $i$th arc for $i = 1, 2, \ldots, 8$ has endpoints $A_i, A_{i+1}$ with $A_9 = A_1,$ such that all of the arcs have radius $1$ and any two consecutive arcs are tangent. Compute the perimeter of Derek’s $8$-petaled flower (not including the central circle). [center] [img] https://cdn.artofproblemsolving.com/attachments/8/4/e8b23c587762c089adb77b29cae155209f5db5.png [/img] [/center]

2023 Harvard-MIT Mathematics Tournament, 13
Tags: guts
Suppose $a, b, c,$ and $d$ are pairwise distinct positive perfect squares such that $a^b = c^d.$ Compute the smallest possible value of $a + b + c + d.$

2024 HMNT, 31
Tags: guts
Positive integers $a, b,$ and $c$ have the property that $\text{lcm}(a,b), \text{lcm}(b,c),$ and $\text{lcm}(c,a)$ end in $4, 6,$ and $7,$ respectively, when written in base $10.$ Compute the minimum possible value of $a + b + c.$

2024 HMNT, 4
Tags: guts
The number $17^6$ when written out in base $10$ contains $8$ distinct digits from $1,2,\ldots,9,$ with no repeated digits or zeroes. Compute the missing nonzero digit.

2023 Harvard-MIT Mathematics Tournament, 21
Tags: guts
Let $x, y,$ and $N$ be real numbers, with $y$ nonzero, such that the sets $\{(x+y)^2, (x-y)^2, xy, x/y\}$ and $\{4, 12.8, 28.8, N\}$ are equal. Compute the sum of the possible values of $N.$

2023 Harvard-MIT Mathematics Tournament, 4
Tags: guts
A [i]standard $n$-sided die[/i] has $n$ sides labeled $1$ to $n.$ Luis, Luke, and Sean play a game in which they roll a fair standard $4$-sided die, a fair standard $6$-sided die, and a fair standard $8$-sided die, respectively. They lose the game if Luis's roll is less than Luke's roll, and Luke's roll is less than Sean's roll. Compute the probability that they lose the game.

2024 Harvard-MIT Mathematics Tournament, 31
Tags: guts
Ash and Gary independently come up with their own lineups of $15$ fire, grass, and water monsters. Then, the first monster of both lineups will fight, with fire beating grass, grass beating water, and water beating fire. The defeated monster is then substituted with the next one from their team’s lineup; if there is a draw, both monsters get defeated. Gary completes his lineup randomly, with each monster being equally likely to be any of the three types. Without seeing Gary’s lineup, Ash chooses a lineup that maximizes the probability p that his monsters are the last ones standing. Compute $p.$

2025 Harvard-MIT Mathematics Tournament, 25
Tags: guts
Let $ABCD$ be a trapezoid such that $AB \parallel CD, AD=13, BC=15, AB=20,$ and $CD=34.$ Point $X$ lies inside the trapezoid such that $\angle{XAB}=2\angle{XBA}$ and $\angle{XDC}=2\angle{XCD}.$ Compute $XD-XA.$

2024 LMT Fall, 18
Tags: guts
In the electoral college, each of $51$ places get some positive number of electoral votes for a nationwide total of $538$. Thus, $270$ electoral votes guarantees a win. Across all distributions of electoral votes to each place, let $M$ be the maximum number of sets of places that combine to have at least $270$ electoral votes. Find $M$.

2024 Harvard-MIT Mathematics Tournament, 6
Tags: guts
In triangle $ABC,$ points $M$ and $N$ are the midpoints of $AB$ and $AC,$ respectively, and points $P$ and $Q$ trisect $BC.$ Given that $A, M, P, N,$ and $Q$ lie on a circle and $BC=1,$ compute the area of triangle $ABC.$

2024 Harvard-MIT Mathematics Tournament, 14
Tags: guts
Compute the smallest positive integer such that, no matter how you rearrange its digits (in base ten), the resulting number is a multiple of $63.$

2024 HMNT, 28
Tags: guts
The graph of the equation $\tan(x+y) = \tan(x)+2\tan(y),$ with its pointwise holes filled in, partitions the coordinate plane into congruent regions. Compute the perimeter of one of these regions.

2023 Harvard-MIT Mathematics Tournament, 27
Tags: guts
Suppose $m>n>1$ are positive integers such that there exist $n$ complex numbers $x_1, x_2, \ldots, x_n$ for which [list] [*]$x_1^k+x_2^k+\ldots+x_n^k=1$ for $k=1, 2, \ldots, n-1;$ [*]$x_1^n+x_2^n+\ldots+x_n^n=2;$ and [*]$x_1^m+x_2^m+\ldots+x_n^m= 4.$ [/list] Compute the smallest possible value of $m+n.$

2024 LMT Fall, 32
Tags: guts
Let $a$ and $b$ be positive integers such that\[a^2+(a+1)^2=b^4.\]Find the least possible value of $a+b$.

2024 HMNT, 15
Tags: guts
Compute the sum of the three smallest positive integers $n$ for which $$\frac{1+2+3+\cdots+(2024n-1)+2024n}{1+2+3+\cdots+(4n-1)+4n}$$ is an integer.