Found problems: 161
Let $ABC$ be an isosceles triangle with orthocenter $H.$ Let $M$ and $N$ be the midpoints of sides $\overline{AB}$ and $\overline{AC},$ respectively. The circumcircle of triangle $MHN$ intersects line $BC$ at two points $X$ and $Y.$ Given $XY=AB=AC=2,$ compute $BC^2.$
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 $p.$
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 $N.$
Let $ABC$ be an acute triangle and $D$ be the foot of altitude from $A$ to $BC.$ Let $X$ and $Y$ be points on the segment $BC$ such that $\angle{BAX} = \angle{YAC}, BX = 2, XY = 6,$ and $YC = 3.$ Given that $AD = 12,$ compute $BD.$
Five pairs of twins are randomly arranged around a circle. Then they perform zero or more swaps, where each swap switches the positions of two adjacent people. They want to reach a state where no one is adjacent to their twin. Compute the expected value of the smallest number of swaps needed to reach such a state.
Let $f$ be a function from nonnegative integers to nonnegative integers such that $f(0)=0$ and $$f(m)=f\left(\left\lfloor \frac{m}{2}\right\rfloor\right)+\left\lceil\frac{m}{2}\right\rceil^2$$ for all positive integers $m.$ Compute $$\frac{f(1)}{1\cdot2}+\frac{f(2)}{2\cdot3}+\frac{f(3)}{3\cdot4}+\cdots+\frac{f(31)}{31\cdot32}.$$(Here, $\lfloor z \rfloor$ is the greatest integer less than or equal to $z,$ and $\lceil z \rceil$ is the least positive integer greater than or equal to $z.$)
Elisenda has a piece of paper in the shape of a triangle with vertices $A, B,$ and $C$ such that $AB = 42.$ She chooses a point $D$ on segment $AC,$ and she folds the paper along line $BD$ so that $A$ lands at a point $E$ on segment $BC.$ Then, she folds the paper along line $DE.$ When she does this, $B$ lands at the midpoint of segment $DC.$ Compute the perimeter of the original unfolded triangle.
Let $a,b,$ and $c$ be real numbers satisfying the system of equations
\begin{align*}
a\sqrt{1+b^2}+b\sqrt{1+a^2}&=\tfrac{3}{4},\\
b\sqrt{1+c^2}+c\sqrt{1+b^2}&=\tfrac{5}{12}, \ \text{and} \\
c\sqrt{1+a^2}+a\sqrt{1+c^2}&=\tfrac{21}{20}.
\end{align*}
Compute $a.$
Given $\sum_{n=1}^{\infty} \tfrac{1}{n^2}=\tfrac{\pi^2}{6}$, find$$\sum_{j=1}^{\infty} \sum_{i=1}^j \frac{1}{ij(i+1)(j+1)}.$$
Let $\ell$ and $m$ be two non-coplanar lines in space, and let $P_1$ be a point on $\ell.$ Let $P_2$ be the point on $,m$ closest to $P_1,$ $P_3$ be the point on $\ell$ closest to $P_3,$ $P_4$ be the point on $m$ closest to $P_3,$ and $P_5$ be the point on $\ell$ closest to $P_4.$ Given that $P_1P_2=5, P_2P_3=3,$ and $P_3P_4=2,$ compute $P_4P_5.$
Find the least prime factor of $2024^{2024}-1$.
Let $ABCD$ be a convex quadrilateral such that $\angle{ABD}=\angle{BCD}=90^\circ,$ and let $M$ be the midpoint of segment $BD.$ Suppose that $CM=2$ and $AM=3.$ Compute $AD.$
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.
Let $\Omega$ be a sphere of radius $4$ and $\Gamma$ be a sphere of radius $2.$ Suppose that the center of $\Gamma$ lies on the surface of $\Omega.$ The intersection of the surfaces of $\Omega$ and $\Gamma$ is a circle. Compute this circle's circumfrence.
Let $\mathcal{P}$ be a regular $10$-gon in the coordinate plane. Mark computes the number of distinct $x$-coordinates that vertices of $\mathcal{P}$ take. Across all possible placements of $\mathcal{P}$ in the plane, compute the sum of all possible answers Mark could get.
Compute the largest prime factor of $3^{12}+3^9+3^5+1.$
Compute the largest possible radius of a circle contained in the region defined by $|x+|y|| \le 1$ in the coordinate plane.
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 $AZ.$
A circle is tangent to both branches of the hyperbola $x^2 - 20y^2 = 24$ as well as the $x$-axis. Compute the area of this circle.
Let $ABC$ be a triangle with $\angle{BAC}>90^\circ.$ Let $D$ be the foot of the perpendicular from $A$ to side $BC.$ Let $M$ and $N$ be the midpoints of segments $BC$ and $BD,$ respectively. Suppose that $AC=2, \angle{BAN}=\angle{MAC},$ and $AB \cdot BC = AM.$ Compute the distance from $B$ to line $AM.$
A Pokemon fan walks into a store. An employee tells them that there are $2$ Pikachus, $3$ Eevees, $4$ Snorlaxes, and $5$ Bulbasaurs remaining inside the gacha machine. Given that this fan cannot see what is inside the Poké Balls before opening them, find the least number of Poké Balls they must buy in order to be sure to get one Pikachu and one Snorlax.
Compute the number of ways to select $99$ cells in a $19 \times 19$ square grid such that no two selected cells share an edge or a vertex.
Consider a quarter-circle with center $O,$ arc $\widehat{AB},$ and radius $2.$ Draw a semicircle with diameter $\overline{OA}$ lying inside the quarter-circle. Points $P$ and $Q$ lie on the semicircle and segment $\overline{OB},$ respectively, such that line $PQ$ is tangent to the semicircle. As $P$ and $Q$ vary, compute the maximum possible area of triangle $BQP.$
Point $P$ is inside a square $ABCD$ such that $\angle APB = 135^\circ, PC=12,$ and $PD=15.$ Compute the area of this square.
Let $a$ and $b$ be positive integers such that\[a^2+(a+1)^2=b^4.\]Find the least possible value of $a+b$.