2023 WSMO Team Round Problems

Problem 1

Bob has a number of pencils. When Bob splits them into groups of 10, he has 3 left over. When he splits them into groups of 12, he has 5 left over. Find the smallest number of pencils Bob can have.

Solution

Problem 2

Integers $W,S,M,$ and $O$ satisfy $W+S+M+O = 13.$ Find the maximum possible value of $WSMO.$

Solution

Problem 3

In the figure below, there are $1023$ total circles. The area between circles alternate between shaded and non-shaded. If the area of the shaded region is $k\pi,$ find the remainder when $k$ is divided by $1000.$ [asy] size(5cm); fill(circle((0,0),10),black); fill(circle((0,0),9),white); fill(circle((0,0),8),black); fill(circle((0,0),7),white); fill(circle((0,0),3),black); fill(circle((0,0),2),white); fill(circle((0,0),1),black); dot((0,6)); dot((0,5)); dot((0,4)); [/asy]

Solution

Problem 4

Honko the hamster is in his cage. He wants to find the smallest distance needed to travel to reach four tennis balls. His current position is $(0,0)$. The tennis balls are located at $(1, 1), (2, -2), (-3, -3),$ and $(-4, 4)$. The length of the shortest path can be expressed as $\sum_{1}^n \sqrt{a_i},$ where $n$ is minimal. Find $\sum_{1}^n a_i.$

Solution

Problem 5

A monkey is throwing darts at the dart board pictured below. The dart is equally likely to land anywhere on the board. Point values for the three regions are labeled and the radii the three circles are $1,2,3,$ respectively. If the expected value of points the monkey gets from 5 dart throws is $\frac{m\pi}{n},$ for relatively prime positive integers $m$ and $n,$ find $m+n.$ [asy] size(6cm); fill(circle((0,0), 6), red); fill(circle((0,0), 4), green); fill(circle((0,0), 2), yellow);  label("3",(0,5)); label("5",(0,3)); label("7",(0,0)); [/asy]

Solution

Problem 6

A quartic real polynomial $f(x)$ satisfying $f(3+2i) = 0$ has 3 distinct roots. If the sum of the three roots is $12,$ find their product.

Solution

Problem 7

In triangle $ABC$ with $AB = 13,AC = 14$, and $BC = 15,$ a rectangle $WXYZ$ is inscribed such that the area of $WXYZ$ is maximized. If the minimum possible value of $\frac{WX}{XY}$ is $\frac{m}{n},$ for relatively prime positive integers $m$ and $n,$ find $m+n.$

Solution

Problem 8

Let $f(x)=\sqrt{x-\sqrt{x-\sqrt{x-\ldots}}}.$ Find the modulo 1000 on the minimum integer $a$ such $f(f(f(f(f(a)))))$ is a positive integer.

Solution

Problem 9

A circle has three chords of equal length, $4 + 2 \sqrt{3}$ which intersect forming a triangle with side lengths 2, 2, and $2 \sqrt{3}$. If the radius of the circle is $r,$ then $r^2 = a-b\sqrt{c},$ for positive integers $a,b$ and squarefree $c.$ Find $a+b+c.$

Solution

Problem 10

Let square $ABCD$ be a square with side length $4$. Define ellipse $\omega$ as the ellipse that is able to be inscribed inside $ABCD$ such that 2 of its vertices on its minor axis and 1 of its vertices on its major axis form an equilateral triangle. The largest possible area of $\omega$ is $m\pi\sqrt{n},$ for squarefree $n.$ Find $m+n.$

Solution

Problem 11

When $\frac{1}{7}$ is expressed in base $k,$ the digits are $0.a_{k,1}a_{k,2}\dots,$ where $a_{k,1}a_{k,2},\dots$ are decimal digits. Let $f(p)$ denote the minimum positive integer $x\geq2$ such that $a_{p,1} = a_{p,x},$ for $k\not\equiv0\pmod{7},$ and $1$ for $k\equiv0\pmod{7}.$ Find $\sum_{i=2}^{100}f(i).$

Solution

Problem 12

Consider parabola $\mathcal P$ pointing upwards with vertex at the origin of the Cartesian plane. Denote the focus of it as $F$ and the directrix of it as $\mathcal L$. Point $P$ with $x$-coordinate $4$ is selected in $\mathcal P$. The perpendicular bisector of $FP$ meets $\mathcal L$ at $Q$. Given the $x$-coordinate of $Q$ is $3$, then $FP^2 = \frac{m}{n},$ for relatively prime positive integers $m$ and $n.$ Find $m+n.$

Solution

Problem 13

Gerry the Squirrel is at a corner of a $9\times9$ grid. 1 acorn $A_1$ is placed. There are four coins placed randomly on the $9\times 9$ grid. Gerry will go the shortest path to $A_1,$ and if there are multiple shortest paths, Gerry will pick one randomly. Given that Gerry does not start on top of a coin, the acorn is not on a coin, find the expected value of coins Gerry passes during his trip to $A_1$ is $\frac{m}{n},$ for relatively prime positive integers $m$ and $n.$ Find $m+n.$

Solution

Problem 14

Consider quadrilateral $ABCD$ with circumcircle centered at $O$. Given $AB=2<CD$, the circumcircle of $\triangle{ABO}$ is tangent to the circumcircle of $\triangle{COD}$. The circumcircle of $\triangle{AOD}$ passes through the center of the circumcircle of $\triangle{ABO}$. Given the radius of the circumcircle of $ABCD$ is $6$. Denote $M,N$ as the area and perimeter of $ABCD$ respectively, compute $\frac{M}{N}$. The answer can be expressed in the simplest form of $\frac{p\sqrt{q}}{r}$. Find $p+q+r$

Solution

Problem 15

On a number line labeled $0, 1, 2, 3, 4, 5,$ and old man starts at $0$ and tries to reach $5.$ Initially, he knows to walk right. However, he has dementia. On each move, there is a $\frac{1}{3}$ chance he forgets which direction he is supposed to go, resulting in him walking the opposite direction. If the probability the old man reaches $5$ without dying is $\frac{m}{n},$ for relatively prime positive integers $m$ and $n.$ Find $m+n.$ (Note: if the old man tries to walk left when he is at 0, he falls off a cliff and dies.)

Solution