Found problems: 161
2024 Harvard-MIT Mathematics Tournament, 30
Let $ABC$ be an equilateral triangle with side length $1.$ Points $D, E,$ and $F$ lie inside triangle $ABC$ such that $A, E, F$ are collinear, $B, F, D$ are collinear, $C, D, E$ are collinear, and triangle $DEF$ is equilateral. Suppose that there exists a unique equilateral triangle $XYZ$ with $X$ on side $\overline{BC},$ $Y$ is on side $\overline{AB},$ and $Z$ is on side $\overline{AC}$ such that $D$ lies on side $\overline{XZ},$ $E$ lies on side $\overline{YZ},$ and $F$ lies on side $\overline{XY}.$ Compute $AZ.$
2024 Harvard-MIT Mathematics Tournament, 7
Positive integers $a, b,$ and $c$ have the property that $a^b, b^c,$ and $c^a$ end in $4, 2,$ and $9,$ respectively. Compute the minimum possible value of $a+b+c.$
2024 HMNT, 25
Let $ABC$ be an equilateral triangle. A regular hexagon $PXQYRZ$ of side length $2$ is placed so that $P, Q,$ and $R$ lie on segments $\overline{BC}, \overline{CA},$ and $\overline{AB}$, respectively. If points $A, X,$ and $Y$ are collinear, compute $BC.$
2024 LMT Fall, 32
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 Harvard-MIT Mathematics Tournament, 32
Over all pairs of complex numbers $(x,y)$ satisfying the equations $$x+2y^2=x^4 \quad \text{and} \quad y+2x^2=y^2,$$ compute the minimum possible real part of $x.$
2023 Harvard-MIT Mathematics Tournament, 24
Let $AXBY$ be a cyclic quadrilateral, and let line $AB$ and line $XY$ intersect at $C.$ Suppose $AX \cdot AY = 6, BX \cdot BY=5,$ and $CX \cdot CY=4.$ Compute $AB^2.$
2024 Harvard-MIT Mathematics Tournament, 19
let $A_1A_2\ldots A_{19}$ be a regular nonadecagon. Lines $A_1A_5$ and $A_3A_4$ meet at $X.$ Compute $\angle A_7 X A_5.$
2025 Harvard-MIT Mathematics Tournament, 11
Let $f(n)=n^2+100.$ Compute the remainder when $\underbrace{f(f(\cdots f(f(}_{2025 \ f\text{'s}}1))\cdots ))$ is divided by $10^4.$
2024 HMNT, 16
Compute $$\frac{2+3+\cdots+100}{1}+\frac{3+4+\cdots+100}{1+2}+\cdots+\frac{100}{1+2+\cdots+99}.$$
2023 Harvard-MIT Mathematics Tournament, 17
An equilateral triangle lies in the Cartesian plane such that the $x$-coordinates of its vertices are pairwise distinct and all satisfy the equation $x^3-9x^2 + 10x + 5 = 0.$ Compute the side length of the triangle.
2024 Harvard-MIT Mathematics Tournament, 28
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.