Olympiad problems

2024 Princeton University Math Competition, 4

Tags: Team Round

Consider the $100 \times 100$ grid of points with integer coordinates $S=\{(x,y) \in \mathbb{Z}^2\mid$ $1 \le x \le 100,$ $1 \le y$ $\le$ $100\}.$ A set $C$ is formed by selecting each $p \in S$ with probability $\tfrac{1}{2}$ uniformly at random. The [I]expansion[/I] of $C$ is defined as the set of points $q \in S$ such that $\min_{p \in C} d(q,p) \le 1,$ where $d(q,p)$ denotes the Euclidean distance between $q,p.$ If the expected size of the expansion of $C$ can be written as $\tfrac{A}{B}$ for relatively prime positive integers $A,B,$ find $A+B.$

2018 PUMaC Team Round, 16

Tags: PuMAC , Team Round , abstract algebra

Let $N$ be the number of subsets $B$ of the set $\{1,2,\dots,2018\}$ such that the sum of the elements of $B$ is congruent to $2018$ modulo $2048$. Find the remainder when $N$ is divided by $1000$.

2018 PUMaC Team Round, 6

Tags: PuMAC , Team Round

Let $\tau(n)$ be the number of distinct positive divisors of $n$ (including $1$ and itself). Find the sum of all positive integers $n$ satisfying $n=\tau(n)^3.$

2018 PUMaC Team Round, 11

Tags: PuMAC , Team Round

Let $\tfrac{a}{b}$ be a fraction such that $a$ and $b$ are positive integers and the first three digits of its decimal expansion are $527$. What is the smallest possible value of $a+b?$

2018 PUMaC Team Round, 9

Tags: PuMAC , Team Round

There are numerous sets of $17$ distinct positive integers that sum to $2018$, such that each integer has the same sum of digits in base $10$. Let $M$ be the maximum possible integer that could exist in any such set. Find the sum of $M$ and the number of such sets that contain $M$.

2024 Princeton University Math Competition, 10

Tags: Team Round

Suppose that $A$ is a set of real numbers between $3$ and $2024$ inclusive such that for any $x, y \in A$ with $x \neq y,$ we have $|x-y|>\tfrac{xy}{2+2xy}.$ What is the largest possible size of $A$?

2024 Princeton University Math Competition, 7

Tags: Team Round

Consider a regular $24$-gon $\mathcal{P}.$ A quadrilateral is said to be inscribed in $\mathcal{P}$ if its vertices are among those of $\mathcal{P}.$ We consider two inscribed quadrilaterals equivalent if one can be obtained from the other via a rotation about the center of $\mathcal{P}.$ How many distinct (i.e. not equivalent) quadrilaterals can be inscribed in $\mathcal{P}$?

2018 PUMaC Team Round, 7

Tags: PuMAC , Team Round

Let triangle $\triangle{MNP}$ have side lengths $MN=13$, $NP=89$, and $PM=100$. Define points $S$, $R$, and $B$ as the midpoints of $\overline{MN}$, $\overline{NP}$, and $\overline{PM}$ respectively. A line $\ell$ cuts lines $\overline{MN}$, $\overline{NP}$, and $\overline{PM}$ at points $I$, $J$, and $A$ respectively. Find the minimum value of $(SI+RJ+BA)^2.$

2018 PUMaC Team Round, 2

Tags: PuMAC , Team Round

Let triangle $\triangle{ABC}$ have $AB=90$ and $AC=66$. Suppose that the line $IG$ is perpendicular to side $BC$, where $I$ and $G$ are the incenter and centroid, respectively. Find the length of $BC$.

2024 Princeton University Math Competition, 3

Tags: Team Round

Let $f(x)=x^2-3x+1,$ and let $\alpha_1, \alpha_2, \alpha_3, \alpha_4$ be the $4$ roots of $f(f(x))=x.$ Evaluate $\lfloor 10\alpha_1\rfloor+ $ $\lfloor 10\alpha_2\rfloor$ $+$ $\lfloor 10\alpha_3\rfloor+\lfloor 10\alpha_4\rfloor.$

2018 PUMaC Team Round, 12

Tags: PuMAC , Team Round

In right triangle $\triangle{ABC}$, a square $WXYZ$ is inscribed such that vertices $W$ and $X$ lie on hypotenuse $\overline{AB}$, vertex $Y$ lies on leg $\overline{BC}$, and vertex $Z$ lies on leg $\overline{CA}$. Let $\overline{AY}$ and $\overline{BZ}$ intersect at some point $P$. If the length of each side of square $WXYZ$ is $4$, the length of the hypotenuse $\overline{AB}$ is $60$, and the distance between point $P$ and point $G$, where $G$ denotes the centroid of $\triangle{ABC}$, is $\tfrac{a}{b}$, compute the value of $a+b$.

2018 PUMaC Team Round, 4

Tags: PuMAC , Team Round

For how many positive integers $n$ less than $2018$ does $n^2$ have the same remainder when divided by $7$, $11$, and $13?$

2024 Princeton University Math Competition, 14

Tags: Team Round

What is the largest value for $m$ for which I can find nonnegative integers $a_1, a_2, \ldots, a_m < 2024$ such that for all indices $i>j,$ $17$ divides $\tbinom{a_i}{a_j}$?

2024 Princeton University Math Competition, 9

Tags: Team Round

Define a sequence called the $2020$-nacci sequence. It is defined as follows: If $n \le 2020$ then $S_n=1,$ if $n>2020$ then $S_n=\sum_{i=n-2020}^{n-1} S_i.$ Find the last two digits of $S_{4040}.$

2024 Princeton University Math Competition, 12

Tags: Team Round

Find the number of positive integers $10 \le n \le 99$ with last digit at most $5$ such that the last two digits of $n^n$ are the same as $n.$

2024 Princeton University Math Competition, 13

Tags: Team Round

Consider the square with vertices $(0, 0),(1, 0),(1, 1),(0, 1).$ The line segments from $(t, 0)$ to $(0, 1 - t)$ are drawn for $0 \le t \le 1.$ The set of points inside the square but not on one of these line segments has area $\tfrac{m}{n}$ for coprime positive integers $m$ and $n.$ Find $m + n.$

2024 Princeton University Math Competition, 6

Tags: Team Round

Ben has a square of side length $2.$ He wants to put a circle and an equilateral triangle inside the square such that the circle and equilateral triangle do not overlap. The maximum possible sum of the areas of the circle and triangle is $\tfrac{a\pi+b\sqrt{c}+d\sqrt{e}}{f},$ where $a,c,e,f$ are positive integers, $b$ and $d$ are integers, $c$ and $e$ are square-free, and $\gcd(a,b,d,f)=1.$ Find $a+b+c+d+e+f.$

2024 Princeton University Math Competition, 5

Tags: Team Round

The [I]minkowski sausage[/I] is constructed as follows. $M_0$ is the line segment from $(0,0)$ to $(1,0).$ $M_{I+1}$ is constructed by replacing each segment in $M_i$ with eight segments, each of length $1/4_{I+1}$ (see figure below, where we have provided $M_0$ through $M_3$). Let $M_{\infty}$ denote the limiting shape of $M_0, M_1, \ldots.$ The area of the smallest convex polygon which encloses $M_{\infty}$ can be written as $\tfrac{a}{b}$ for relatively prime positive integers $a$ and $b.$ Find $a+b.$ [center] [img]https://cdn.artofproblemsolving.com/attachments/1/e/25c9980469584ce7ae4ab2ccb4ce80f3e5dfee.png[/img] [/center]

2018 PUMaC Team Round, 1

Tags: PuMAC , Team Round , probability , expected value

Let $T=\{a_1,a_2,\dots,a_{1000}\}$, where $a_1<a_2<\dots<a_{1000}$, be a uniformly randomly selected subset of $\{1,2,\dots,2018\}$ with cardinality $1000$. The expected value of $a_7$ can be written in reduced form as $\tfrac{m}{n}$. Find $m+n$.

2024 Princeton University Math Competition, 11

Tags: Team Round

Austen has a regular icosahedron ($20$-sided polyhedron with all triangular faces). He randomly chooses $3$ distinct points among the vertices and constructs the circle through these three points. The expected value of the total number of the icosahedron’s vertices that lie on this circle can be written as $m/n$ for relatively prime positive integers $m$ and $n.$ Find $m + n.$

2024 Princeton University Math Competition, 1

Tags: Team Round

Justin chooses a number $n$ uniformly at random from the set of integers between $90$ and $99,$ inclusive. He then chooses a positive divisor $d$ of $n$ uniformly at random. Justin notices that $d$ and $n/d$ are relatively prime. If the probability that $n = 90$ can be expressed as $a/b$ for relatively prime positive integers, find $a + b.$

2018 PUMaC Team Round, 15

Tags: PuMAC , Team Round

Aaron the Ant is somewhere on the exterior of a hollow cube of side length $2$ inches, and Fred the Flea is on the inside, at one of the vertices. At some instant, Fred flies in a straight line towards the opposite vertex, and simultaneously Aaron begins crawling on the exterior of the cube towards that same vertex. Fred moves at $\sqrt{3}$ inches per second and Aaron moves at $\sqrt{2}$ inches per second. If Aaron arrives before Fred, the area of the surface on the cube from which Aaron could have started can be written as $a\pi+\sqrt{b}+c$ where $a$, $b$, and $c$ are integers. Find $a+b+c.$

2024 Princeton University Math Competition, 2

Tags: Team Round

Let real number sequences $a_k$ and $x_k$ be defined for $1 \le k \le 7$ and suppose that $a_1=1$ and $a_{k+1}=a_k+x_k$ for $1 \le k \le 7.$ Let $x_k$ be chosen such that the quantity $S=\sum_{k=1}^7 (a_k^2+x_k^2)$ is minimized. Then $S=\tfrac{m}{n}$ for coprime positive integers $m$ and $n.$ Find $m+n.$

2018 PUMaC Team Round, 5

Tags: PuMAC , Team Round

There exist real numbers $a$, $b$, $c$, $d$, and $e$ such that for all positive integers $n$, we have $$\sqrt{n}=\sum_{i=0}^{n-1}\sqrt[5]{\sqrt{ai^5+bi^4+ci^3+di^2+ei+1}-\sqrt{ai^5+bi^4+ci^3+di^2+ei}}.$$ Find $a+b+c+d$.

2024 Princeton University Math Competition, 8

Tags: Team Round

Let $\triangle ABC$ be a triangle. Let points $D$ and $E$ be on segment $BC$ in the order $B, D, E, C$ such that $\angle BAD =$ $\angle DAE = \angle EAC.$ Suppose also that $BD = F_{2024}, DE = F_{2025}, EC = F_{2027},$ where $F_k$ is the $k$th Fibonacci number where $F_1 = F_2 = 1.$ To the nearest degree, $\angle BAC$ is $n^\circ.$ Find $n.$