Olympiad problems

2019 Math Prize for Girls Problems, 19

Tags: Math Prize for Girls

Consider the base 27 number \[ n = ABCDEFGHIJKLMNOPQRSTUVWXYZ , \] where each letter has the value of its position in the alphabet. What remainder do you get when you divide $n$ by 100? (The remainder is an integer between 0 and 99, inclusive.)

2018 Math Prize for Girls Problems, 16

Tags: Math Prize for Girls

Define a function $f$ on the unit interval $0 \le x \le 1$ by the rule \[ f(x) = \begin{cases} 1-3x & \text{if } 0 \le x < 1/3 \, ; \\ 3x-1 & \text{if } 1/3 \le x < 2/3 \, ; \\ 3-3x & \text{if } 2/3 \le x \le 1 \, . \end{cases} \] Determine $f^{(2018)}(1/730)$. Recall that $f^{(n)}$ denotes the $n$th iterate of $f$; for example, $f^{(3)}(1/730) = f(f(f(1/730)))$.

2021 Math Prize for Girls Problems, 7

Tags: Math Prize for Girls

Compute the value of the infinite series \[ \sum_{k=0}^{\infty} \frac{\cos(k \pi / 4)}{2^k} \, . \]

2021 Math Prize for Girls Problems, 9

Tags: Math Prize for Girls , geometry

Let $H$ be a regular hexagon with area 360. Three distinct vertices $X$, $Y$, and $Z$ are picked randomly, with all possible triples of distinct vertices equally likely. Let $A$, $B$, and $C$ be the unpicked vertices. What is the expected value (average value) of the area of the intersection of $\triangle ABC$ and $\triangle XYZ$?

2017 Math Prize for Girls Problems, 12

Tags: Math Prize for Girls

Let $S$ be the set of all real values of $x$ with $0 < x < \pi/2$ such that $\sin x$, $\cos x$, and $\tan x$ form the side lengths (in some order) of a right triangle. Compute the sum of $\tan^2 x$ over all $x$ in $S$.

2019 Math Prize for Girls Problems, 12

Tags: Math Prize for Girls

Say that a positive integer is MPR (Math Prize Resolvable) if it can be represented as the sum of a 4-digit number MATH and a 5-digit number PRIZE. (Different letters correspond to different digits. The leading digits M and P can't be zero.) Say that a positive integer is MPRUUD (Math Prize Resolvable with Unique Units Digits) if it is MPR and the set of units digits $\{ \mathrm{H}, \mathrm{E} \}$ in the definition of MPR can be uniquely identified. Find the smallest positive integer that is MPR but not MPRUUD.

2019 Math Prize for Girls Problems, 4

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A paper equilateral triangle with area 2019 is folded over a line parallel to one of its sides. What is the greatest possible area of the overlap of folded and unfolded parts of the triangle?

2017 Math Prize for Girls Problems, 13

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A polynomial whose roots are all equal to each other is called a [i]unicorn[/i]. Compute the number of distinct ordered triples $(M, P, G)$, where $M$, $P$, $G$ are complex numbers, such that the polynomials $z^3 + M z^2 + Pz + G$ and $z^3 + G z^2 + Pz + M$ are both unicorns.

2021 Math Prize for Girls Problems, 19

Tags: Math Prize for Girls , geometry , 3D geometry , tetrahedron

Let $T$ be a regular tetrahedron. Let $t$ be the regular tetrahedron whose vertices are the centers of the faces of $T$. Let $O$ be the circumcenter of either tetrahedron. Given a point $P$ different from $O$, let $m(P)$ be the midpoint of the points of intersection of the ray $\overrightarrow{OP}$ with $t$ and $T$. Let $S$ be the set of eight points $m(P)$ where $P$ is a vertex of either $t$ or $T$. What is the volume of the convex hull of $S$ divided by the volume of $t$?

2017 Math Prize for Girls Problems, 3

Tags: Math Prize for Girls

If $A$ and $B$ are numbers such that the polynomial $x^{2017} + Ax + B$ is divisible by $(x + 1)^2$, what is the value of $B$?

2018 Math Prize for Girls Problems, 8

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A mustache is created by taking the set of points $(x, y)$ in the $xy$-coordinate plane that satisfy $4 + 4 \cos(\pi x/24) \le y \le 6 + 6\cos(\pi x/24)$ and $-24 \le x \le 24$. What is the area of the mustache?

2016 Math Prize for Girls Problems, 1

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Let $T$ be a triangle with side lengths 3, 4, and 5. If $P$ is a point in or on $T$, what is the greatest possible sum of the distances from $P$ to each of the three sides of $T$?

2018 Math Prize for Girls Problems, 3

Tags: Math Prize for Girls

Let $S$ be the set of all positive integers from 1 through 1000 that are not perfect squares. What is the length of the longest, non-constant, arithmetic sequence that consists of elements of $S$?

2017 Math Prize for Girls Problems, 14

Tags: Math Prize for Girls

A [i]permutation[/i] of a finite set $S$ is a one-to-one function from $S$ to $S$. Given a permutation $f$ of the set $\{ 1, 2, \ldots, 100 \}$, define the [i]displacement[/i] of $f$ to be the sum \[ \sum_{i = 1}^{100} \left\lvert f(i) - i \right\rvert . \] How many permutations of $\{ 1, 2, \ldots, 100 \}$ have displacement 4?

2019 Math Prize for Girls Problems, 20

Tags: Math Prize for Girls

Evaluate the infinite product \[ \prod_{k = 2}^{\infty} \left( 1 - 4 \sin^2 \frac{\pi}{3\cdot 2^{k}} \right) . \]

2018 Math Prize for Girls Olympiad, 4

Tags: Math Prize for Girls , inequalities

For all integers $x$ and $y$, let $a_{x, y}$ be a real number. Suppose that $a_{0, 0} = 0$. Suppose that only a finite number of the $a_{x, y}$ are nonzero. Prove that \[ \sum_{x = -\infty}^\infty \sum_{y = -\infty}^{\infty} a_{x,y} ( a_{x, 2x + y} + a_{x + 2y, y} ) \le \sqrt{3} \sum_{x = -\infty}^\infty \sum_{y = -\infty}^{\infty} a_{x, y}^2 \, . \]

2016 Math Prize for Girls Problems, 11

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Compute the number of ordered pairs of complex numbers $(u, v)$ such that $uv = 10$ and such that the real and imaginary parts of $u$ and $v$ are integers.

2017 Math Prize for Girls Olympiad, 1

Tags: Math Prize for Girls

Given positive integers $n$ and $k$, say that $n$ is $k$-[i]solvable[/i] if there are positive integers $a_1$, $a_2$, ..., $a_k$ (not necessarily distinct) such that \[ \frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_k} = 1 \] and \[ a_1 + a_2 + \cdots + a_k = n. \] Prove that if $n$ is $k$-solvable, then $42n + 12$ is $(k + 3)$-solvable.

2017 Math Prize for Girls Problems, 6

Tags: Math Prize for Girls

Let $b$ and $c$ be integers chosen randomly (uniformly and independently) from the set \[ \{ -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6 \} . \] (Note that $b$ and $c$ can be equal.) What is the probability that the two roots of the quadratic $x^2 + bx + c$ are consecutive integers?

2019 Math Prize for Girls Problems, 17

Tags: Math Prize for Girls

Let $P$ be a right prism whose two bases are equilateral triangles with side length 2. The height of $P$ is $2\sqrt{3}$. Let $l$ be the line connecting the centroids of the bases. Remove the solid, keeping only the bases. Rotate one of the bases $180^\circ$ about $l$. Let $T$ be the convex hull of the two current triangles. What is the volume of $T$?

2019 Math Prize for Girls Problems, 15

Tags: Math Prize for Girls

How many ordered pairs $(x, y)$ of real numbers $x$ and $y$ are there such that $-100 \pi \le x \le 100 \pi$, $-100 \pi \le y \le 100 \pi$, $x + y = 20.19$, and $\tan x + \tan y = 20.19$?

2021 Math Prize for Girls Problems, 11

Tags: Math Prize for Girls , number theory , relatively prime

Say that a sequence $a_1$, $a_2$, $a_3$, $a_4$, $a_5$, $a_6$, $a_7$, $a_8$ is [i]cool[/i] if * the sequence contains each of the integers 1 through 8 exactly once, and * every pair of consecutive terms in the sequence are relatively prime. In other words, $a_1$ and $a_2$ are relatively prime, $a_2$ and $a_3$ are relatively prime, $\ldots$, and $a_7$ and $a_8$ are relatively prime. How many cool sequences are there?

2018 Math Prize for Girls Problems, 9

Tags: Math Prize for Girls

How many 3-term geometric sequences $a$, $b$, $c$ are there where $a$, $b$, and $c$ are positive integers with $a < b < c$ and $c = 8000$?

2019 Math Prize for Girls Problems, 2

Tags: Math Prize for Girls

Let $a_1$, $a_2$, $\ldots\,$, $a_{2019}$ be a sequence of real numbers. For every five indices $i$, $j$, $k$, $\ell$, and $m$ from 1 through 2019, at least two of the numbers $a_i$, $a_j$, $a_k$, $a_\ell$, and $a_m$ have the same absolute value. What is the greatest possible number of distinct real numbers in the given sequence?

2018 Math Prize for Girls Problems, 11

Tags: Math Prize for Girls

Maryam has a fair tetrahedral die, with the four faces of the die labeled 1 through 4. At each step, she rolls the die and records which number is on the bottom face. She stops when the current number is greater than or equal to the previous number. (In particular, she takes at least two steps.) What is the expected number (average number) of steps that she takes?