Olympiad problems

2024 UMD Math Competition Part I, #22

For how many angles $x$, in radians, satisfying $0\le x<2\pi$ do we have $\sin(14x)=\cos(68x)$? \[\rm a. ~128\qquad \mathrm b. ~130\qquad \mathrm c. ~132 \qquad\mathrm d. ~134\qquad\mathrm e. ~136\]

2023 UMD Math Competition Part I, #24

Tags: number theory , UMD

Bob is practicing addition in base $2.$ Each time he adds two numbers in base $2,$ he counts the number of carries. For example, when summing the numbers $1001$ and $1011$ in base $2,$ \[\begin{array}{ccccc} \overset{1}{}&& \overset {1}{}&\overset {1}{} \\ 0&1&0&0&1\\0&1&0&1&1 \\ \hline 1&0&1&0&0 \end{array}\] there are three carries (shown on the top row). Suppose that Bob starts with the number $0,$ and adds $111~($i.e. $7$ in base $2)$ to it one hundred times to obtain the number $1010111100~($i.e. $700$ in base $2).$ How many carries occur (in total) in these one hundred calculations? \[\mathrm a. ~ 280\qquad \mathrm b.~289\qquad \mathrm c. ~291 \qquad \mathrm d. ~294 \qquad \mathrm e. ~297\]

2023 UMD Math Competition Part I, #2

Tags: UMD , number theory

Peter Rabbit is hopping along the number line, always jumping in the positive $x$ direction. For his first jump, he starts at $0$ and jumps $1$ unit to get to the number $1.$ For his second jump, he jumps $4$ units to get to the number $5.$ He continues jumping by jumping $1$ unit whenever he is on a multiple of $3$ and by jumping $4$ units whenever he is on a number that is not a multiple of $3.$ What number does he land on at the end of his $100$th jump? $$ \mathrm a. ~ 297\qquad \mathrm b.~298\qquad \mathrm c. ~299 \qquad \mathrm d. ~300 \qquad \mathrm e. ~301 $$

Maryland University HSMC part II, 2023.2

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.

2024 UMD Math Competition Part I, #25

Tags: UMD , geometry

An equilateral triangle $T$ and a circle $C$ are on the same plane. Suppose each side length of $T$ is $6\sqrt3$ and the radius of $C$ is $2.$ The distance between the centers of $T$ and $C$ is $15.$ For every two points $X$ on $T$ and $Y$ on $C,$ let $M(X, Y)$ be the midpoint of segment $\overline{XY}.$ The points $M(X, Y)$ as $X$ varies on $T$ and $Y$ varies on $C$ create a region whose area is $A.$ Find $A.$ \[\mathrm a. ~\pi + 14\sqrt3 \qquad \mathrm b. ~3\pi + 10\sqrt3 \qquad \mathrm c. ~4\pi+9\sqrt3 \qquad\mathrm d. ~\pi + 15\sqrt3 \qquad\mathrm e. ~4\pi+6\sqrt3\]

2023 UMD Math Competition Part I, #25

Tags: algebra , UMD

Suppose that $S$ is a series of real numbers between $2$ and $8$ inclusive, and that for any two elements $y > x$ in $S,$ $$ 98y - 102x - xy \ge 4. $$ What is the maximum possible size for the set $S?$ $$ \mathrm a. ~ 12\qquad \mathrm b.~14\qquad \mathrm c. ~16 \qquad \mathrm d. ~18 \qquad \mathrm e. 20 $$

2023 UMD Math Competition Part I, #17

Tags: UMD , geometry

The lengths of the sides of triangle $A'B'C'$ are equal to the lengths of the three medians of triangle $ABC.$ Then the ratio $\mathrm{Area} (A'B'C') / \mathrm{Area} (ABC)$ equals $$ \mathrm a. ~ \frac 12\qquad \mathrm b.~\frac 23\qquad \mathrm c. ~\frac34 \qquad \mathrm d. ~\frac56 \qquad \mathrm e. ~\text{Cannot be determined from the information given.} $$

2023 UMD Math Competition Part I, #18

Tags: UMD , algebra

How many ordered triples of integers $(a, b, c)$ satisfy the following system? $$ \begin{cases} ab + c &= 17 \\ a + bc &= 19 \end{cases} $$ $$ \mathrm a. ~ 2\qquad \mathrm b.~3\qquad \mathrm c. ~4 \qquad \mathrm d. ~5 \qquad \mathrm e. ~6 $$

2023 UMD Math Competition Part I, #14

Tags: quadratics , UMD , algebra

Let $m \neq -1$ be a real number. Consider the quadratic equation $$ (m + 1)x^2 + 4mx + m - 3 =0. $$ Which of the following must be true? $\quad\rm(I)$ Both roots of this equation must be real. $\quad\rm(II)$ If both roots are real, then one of the roots must be less than $-1.$ $\quad\rm(III)$ If both roots are real, then one of the roots must be larger than $1.$ $$ \mathrm a. ~ \text{Only} ~(\mathrm I)\rm \qquad \mathrm b. ~(I)~and~(II)\qquad \mathrm c. ~Only~(III) \qquad \mathrm d. ~Both~(I)~and~(III) \qquad \mathrm e. ~(I), (II),~and~(III) $$

2024 UMD Math Competition Part I, #23

Tags: UMD , algebra

For how many pairs of integers $(m, n)$ with $0 < m \le n \le 50$ do there exist precisely four triples of integers $(x, y, z)$ satisfying the following system? \[\begin{cases} x^2 + y+ z = m \\ x + y^2 + z = n\end{cases}\] \[\rm a. ~180\qquad \mathrm b. ~182\qquad \mathrm c. ~186 \qquad\mathrm d. ~188\qquad\mathrm e. ~190\]

2023 UMD Math Competition Part I, #13

Tags: UMD , geometry

The orthocenter of triangle $ABC$ lies on its circumcircle. One of the angles of $ABC$ must equal: (The orthocenter of a triangle is the point where all three altitudes intersect.) $$ \mathrm a. ~ 30^\circ\qquad \mathrm b.~60^\circ\qquad \mathrm c. ~90^\circ \qquad \mathrm d. ~120^\circ \qquad \mathrm e. ~\text{It cannot be deduced from the given information.} $$

2024 UMD Math Competition Part I, #12

Tags: UMD , combinatorics

A square has $2$ diagonals. A regular pentagon has $5$ diagonals. $n$ is the smallest positive integer such that a regular $n$-gon has greater than or equal to $2024$ diagonals. What is the sum of the digits of $n$? \[\mathrm a. ~10\qquad \mathrm b. ~11 \qquad \mathrm c. ~12 \qquad\mathrm d. ~13\qquad\mathrm e. ~14\]

2024 UMD Math Competition Part I, #15

Tags: UMD , algebra

How many real numbers $a$ are there for which both solutions to the equation \[x^2 + (a - 2024)x + a = 0\] are integers? \[\mathrm a. ~15\qquad \mathrm b. ~16 \qquad \mathrm c. ~18 \qquad\mathrm d. ~20\qquad\mathrm e. ~24\qquad\]

2023 UMD Math Competition Part I, #19

Tags: UMD , algebra

Three positive real numbers $a, b, c$ satisfy $a^b = 343, b^c = 10, a^c = 7.$ Find $b^b.$ $$ \mathrm a. ~ 1000\qquad \mathrm b.~900\qquad \mathrm c. ~1200 \qquad \mathrm d. ~4000 \qquad \mathrm e. ~100 $$

2024 UMD Math Competition Part II, #4

Tags: calculus , UMD , algebra

Prove for every positive integer $n{:}$ \[ \frac {1 \cdot 3 \cdots (2n - 1)}{2 \cdot 4 \cdots (2n)} < \frac 1{\sqrt{3n}}\]

2023 UMD Math Competition Part I, #10

Tags: UMD , algebra

There are $100$ people in a room. Some are [i]wise[/i] and some are [i]optimists[/i]. $\quad \bullet~$ A [i]wise[/i] person can look at someone and know if they are wise or if they are an optimist. $\quad \bullet~$ An [i]optimist[/i] thinks everyone is wise (including themselves). Everyone in the room writes down what they think is the number of wise people in the room. What is the smallest possible value for the average? $$ \mathrm a. ~ 10\qquad \mathrm b.~25\qquad \mathrm c. ~50 \qquad \mathrm d. ~75 \qquad \mathrm e. ~100 $$

2023 UMD Math Competition Part I, #23

Tags: geometry , UMD

Assume a triangle $ABC$ satisfies $|AB| = 1, |AC| = 2$ and $\angle ABC = \angle ACB + 90^\circ.$ What is the area of $ABC?$ \[ \mathrm a. ~ 6/7\qquad \mathrm b.~5/7\qquad \mathrm c. ~1/2 \qquad \mathrm d. ~4/5 \qquad \mathrm e. ~3/5 \]

2023 UMD Math Competition Part I, #11

Tags: UMD , geometry

Let $S_1$ be a square with side $s$ and $C_1$ be the circle inscribed in it. Let $C_2$ be a circle with radius $r$ and $S_2$ be a square inscribed in it. We are told that the area of $S_1 - C_1$ is the same as the area of $C_2 - S_2.$ Which of the following numbers is closest to $s/r?$ $$ \mathrm a. ~ 1\qquad \mathrm b.~2\qquad \mathrm c. ~3 \qquad \mathrm d. ~4 \qquad \mathrm e. ~5 $$