Olympiad problems

2023 UMD Math Competition Part I, #20

Tags: geometry , UMD

A strip is defined as the region between two parallel lines; the width of the strip is the distance between the two lines. Two strips of width $1$ intersect in a parallelogram whose area is $2.$ What is the angle between the strips? \[ \mathrm a. ~ 15^\circ\qquad \mathrm b.~30^\circ \qquad \mathrm c. ~45^\circ \qquad \mathrm d. ~60^\circ \qquad \mathrm e. ~90^\circ\]

2024 UMD Math Competition Part II, #2

Tags: number theory , UMD

Consider a set $S = \{a_1, \ldots, a_{2024}\}$ consisting of $2024$ distinct positive integers that satisfies the following property: [center] "For every positive integer $m < 2024,$ the sum of no $m$ distinct elements of $S$ is a multiple of $2024.$" [/center] Prove $a_1, \ldots, a_{2024}$ all leave the same remainder when divided by $2024.$ Justify your answer.

Maryland University HSMC part II, 2023.5

Tags: inequalities , AM-GM , UMD

Let $0 \le a_1 \le a_2 \le \dots \le a_n \le 1$ be $n$ real numbers with $n \ge 2$. Assume $a_1 + a_2 + \dots + a_n \ge n-1$. Prove that \[ a_2a_3\dots a_n \ge \left( 1 - \frac 1n \right)^{n-1} \]

2023 UMD Math Competition Part II, 5

Tags: inequalities , AM-GM , UMD

Let $0 \le a_1 \le a_2 \le \dots \le a_n \le 1$ be $n$ real numbers with $n \ge 2$. Assume $a_1 + a_2 + \dots + a_n \ge n-1$. Prove that \[ a_2a_3\dots a_n \ge \left( 1 - \frac 1n \right)^{n-1} \]

2024 UMD Math Competition Part I, #20

Tags: UMD , combinatorics

There are eight seats at a round table. Six adults $A_1, \ldots, A_6$ and two children sit around the table. The two children are not allowed to six next to each other. All the seating configurations where the children are not seated next to each other are equally likely. What is the probability that the adults $A_1$ and $A_2$ end up sitting next to each other?\[ \mathrm a. ~4/15\qquad \mathrm b. ~2/7 \qquad \mathrm c. ~2/9 \qquad\mathrm d. ~1/3\qquad\mathrm e. ~1/5\qquad\]

2024 UMD Math Competition Part I, #13

Tags: function , algebra , UMD

Consider the sets $A = \{0,1,2\},$ and $B = \{1,2,3,4,5\}.$ Find the number of functions $f: A \to B$ such that $x + f(x) + xf(x)$ is odd for all $x.$ (A function $f:A \to B$ is a rule that assigns to every number in $A$ a number in $B.$) \[\mathrm a. ~15\qquad \mathrm b. ~27 \qquad \mathrm c. ~30 \qquad\mathrm d. ~42\qquad\mathrm e. ~45\]

2023 UMD Math Competition Part I, #6

Tags: UMD , algebra

Let $$ A = \log (1) + \log 2 + \log(3) + \cdots + \log(2023) $$ and $$ B = \log(1/1) + \log(1/2) + \log(1/3) + \cdots + \log(1/2023). $$ What is the value of $A + B\ ?$ $($logs are logs base $10)$ $$ \mathrm a. ~ 0\qquad \mathrm b.~1\qquad \mathrm c. ~{-\log(2023!)} \qquad \mathrm d. ~\log(2023!) \qquad \mathrm e. ~{-2023} $$

2024 UMD Math Competition Part I, #21

Tags: UMD , Fein , geometry

The width of a lane in a circular running track is $1.22$ meters. One loop in the first lane (shortest lane) is $400$ meters. Thus $12.5$ loops makes it a $5{,}000$ meter distance. Which lane should an athlete run in if they want to make $12$ loops as close to the $5{,}000$ meter distance as possible? \[\rm a. ~second\qquad \mathrm b. ~third \qquad \mathrm c. ~fourth \qquad\mathrm d. ~fifth \qquad\mathrm e. ~sixth\]

2024 UMD Math Competition Part I, #19

Tags: UMD , rotation , combinatorics

A square-shaped quilt is divided into $16 = 4 \times 4$ equal squares. We say that the quilt is [i]UMD certified[/i] if each of these $16$ squares is colored red, yellow, or black, so that (i) all three colors are used at least once and (ii) the quilt looks the same when it is rotated $90, 180,$ or $270$ degrees about its center. How many distinct UMD certified quilts are there? \[\rm a. ~33\qquad \mathrm b. ~36 \qquad \mathrm c. ~45\qquad\mathrm d. ~54\qquad\mathrm e. ~81\]

2024 UMD Math Competition Part II, #5

Tags: combinatorics , UMD

Define two sequences $x_n, y_n$ for $n = 1, 2, \ldots$ by \[x_n = \left(\sum^n_{k=0} \binom{2n}{2k}49^k 48^{n-k} \right) -1, \quad \text{and} \quad y_n = \sum^{n-1}_{k=0} \binom{2n}{2k + 1} 49^k 48^{n-k}\] Prove there is a positive integer $m$ for which for every integer $n > m,$ the greatest common factor of $x_n$ and $y_n$ is more than $10^{2024}.$

2024 UMD Math Competition Part II, #1

Tags: number theory , UMD

Find the largest positive integer $n$ satisfying the following: [center] "There are precisely $53$ integers in the list of integers $1, 2, \ldots, n$ that are either perfect squares, perfect cubes or both."[/center]

2023 UMD Math Competition Part I, #22

Tags: algebra , UMD

A sequence $a_1, a_2, \ldots$ satisfies $a_1 = \dfrac 52$ and $a_{n + 1} = {a_n}^2 - 2$ for all $n \ge 1.$ Let $M$ be the integer which is closest to $a_{2023}.$ The last digit of $M$ equals $$ \mathrm a. ~ 0\qquad \mathrm b.~2\qquad \mathrm c. ~4 \qquad \mathrm d. ~6 \qquad \mathrm e. ~8 $$

2023 UMD Math Competition Part I, #15

Tags: UMD , algebra

What is the least positive integer $m$ such that the following is true? [i]Given $\it m$ integers between $\it1$ and $\it{2023},$ inclusive, there must exist two of them $\it a, b$ such that $1 < \frac ab \le 2.$ [/i] \[\mathrm a. ~ 10\qquad \mathrm b.~11\qquad \mathrm c. ~12 \qquad \mathrm d. ~13 \qquad \mathrm e. ~1415\]

2023 UMD Math Competition Part I, #7

Tags: UMD , algebra

Suppose $S = \{1, 2, 3, x\}$ is a set with four distinct real numbers for which the difference between the largest and smallest values of $S$ is equal to the sum of elements of $S.$ What is the value of $x?$ $$ \mathrm a. ~ {-1}\qquad \mathrm b.~{-3/2}\qquad \mathrm c. ~{-2} \qquad \mathrm d. ~{-2/3} \qquad \mathrm e. ~{-3} $$

2024 UMD Math Competition Part II, #3

Tags: UMD , geometry , 3D geometry , sphere

A right triangle $A_1 A_2 A_3$ with side lengths $6,\,8,$ and $10$ on a plane $\mathcal P$ is given. Three spheres $S_1,S_2$ and $S_3$ with centers $O_1, O_2,$ and $O_3,$ respectively, are located on the same side of the plane $\mathcal P$ in such a way that $S_i$ is tangent to $\mathcal P$ at $A_i$ for $i = 1, 2, 3.$ Assume $S_1, S_2, S_3$ are pairwise externally tangent. Find the area of triangle $O_1O_2O_3.$

2023 UMD Math Competition Part I, #12

Tags: UMD , algebra

Suppose for real numbers $a, b, c$ we know $a + \dfrac 1b = 3,$ and $b + \dfrac 3c = \dfrac 13.$ What is the value of $c + \dfrac{27}a?$ $$ \mathrm a. ~ 1\qquad \mathrm b.~3\qquad \mathrm c. ~8 \qquad \mathrm d. ~9 \qquad \mathrm e. ~21 $$

2023 UMD Math Competition Part I, #16

Tags: UMD , algebra

How many integers between $123$ and $789$ have at least two identical digits, when written in base $10?$ $$ \mathrm a. ~ 180\qquad \mathrm b.~184\qquad \mathrm c. ~186 \qquad \mathrm d. ~189 \qquad \mathrm e. ~191 $$

2023 UMD Math Competition Part I, #4

Tags: Euler , UMD , algebra

Euler is selling Mathematician cards to Gauss. Three Fermat cards plus $5$ Newton cards costs $95$ Euros, while $5$ Fermat cards plus $2$ Newton cards also costs $95$ Euros. How many Euroes does one Fermat card cost? $$ \mathrm a. ~ 10\qquad \mathrm b.~15\qquad \mathrm c. ~20 \qquad \mathrm d. ~30 \qquad \mathrm e. ~35 $$

2023 UMD Math Competition Part I, #1

Tags: UMD , algebra

An ant walks a distance $A = 10^9$ millimeters. A bear walks $B = 10^6$ feet. A chicken walks $C = 10^8$ inches. What is the correct ordering of $A, B, C?$ (Note there are $25.4$ millimeters in an inch, and there are $12$ inches in a foot.) $$ \mathrm a. ~ A<B<C\qquad \mathrm b.~A<C<B\qquad \mathrm c. ~C<B<A \qquad \mathrm d. ~B<A<C \qquad \mathrm e. ~B<C<A $$

2023 UMD Math Competition Part II, 2

Tags: combinatorics , induction , UMD

Let $n \ge 2$ be an integer. There are $n$ houses in a town. All distances between pairs of houses are different. Every house sends a visitor to the house closest to it. Find all possible values of $n$ (with full justification) for which we can design a town with $n$ houses where every house is visited.

2023 UMD Math Competition Part I, #9

Tags: UMD , geometry

The Amazing Prime company ships its products in boxes whose length, width, and height (in inches) are prime numbers. If the volume of one of their boxes is $105$ cubic inches, what is its surface area (that is, the sum of the areas of the 6 sides of the box) in square inches? $$ \mathrm a. ~ 21\qquad \mathrm b.~71\qquad \mathrm c. ~77 \qquad \mathrm d. ~05 \qquad \mathrm e. ~142 $$

2023 UMD Math Competition Part I, #3

Tags: UMD , geometry

Adam is walking in the city. In order to get around a large building, he walks $12$ miles east and then $5$ miles north, then stop. His friend Neutrino, who can go through buildings, starts in the same place as Adam but walks in a straight line to where Adam stops. How much farther than Neutrino does Adam walk? $$ \mathrm a. ~ 1~\mathrm{mile}\qquad \mathrm b.~2 ~\mathrm{miles}\qquad \mathrm c. ~3~\mathrm{miles} \qquad \mathrm d. ~4~\mathrm{miles} \qquad \mathrm e. ~5~\mathrm{miles} $$

2023 UMD Math Competition Part I, #8

Tags: UMD , number theory

How many positive integers less than $1$ million have exactly $5$ positive divisors? $$ \mathrm a. ~ 1\qquad \mathrm b.~5\qquad \mathrm c. ~11 \qquad \mathrm d. ~23 \qquad \mathrm e. ~24 $$

2023 UMD Math Competition Part I, #5

Tags: UMD , algebra

You shoot an arrow in the air. It falls to earth, you know not where. But you do know that the arrow’s height in feet after ${t}$ seconds is $-16t^2 + 80t + 96.$ After how many seconds does the arrow hit the ground? (the ground has height 0) $$ \mathrm a. ~ 2\qquad \mathrm b.~3\qquad \mathrm c. ~4 \qquad \mathrm d. ~5 \qquad \mathrm e. ~6 $$

2023 UMD Math Competition Part I, #21

Tags: algebra , UMD

Let $a, b, c, d, e$ be real numbers such that $a<b<c<d<e.$ The least possible value of the function $f: \mathbb R \to \mathbb R$ with $f(x) = |x-a| + |x - b|+ |x - c| + |x - d|+ |x - e|$ is $$ \mathrm a. ~ e+d+c+b+a\qquad \mathrm b.~e+d+c-b-a\qquad \mathrm c. ~e+d+|c|-b-a \qquad \mathrm d. ~e+d+b-a \qquad \mathrm e. ~e+d-b-a $$