Olympiad problems

2018 Math Prize for Girls Olympiad, 3

There is a wooden $3 \times 3 \times 3$ cube and 18 rectangular $3 \times 1$ paper strips. Each strip has two dotted lines dividing it into three unit squares. The full surface of the cube is covered with the given strips, flat or bent. Each flat strip is on one face of the cube. Each bent strip (bent at one of its dotted lines) is on two adjacent faces of the cube. What is the greatest possible number of bent strips? Justify your answer.

2017 Math Prize for Girls Problems, 5

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The New York Public Library requires patrons to choose a 4-digit Personal Identification Number (PIN) to access its online system. (Leading zeros are allowed.) The PIN is not allowed to contain either of the following two forbidden patterns: * A digit that is repeated 3 or more times in a row. For example, 0001 and 5555 are not PINs, but 0010 is a PIN. * A pair of digits that is duplicated. For example, 1212 and 6363 are not PINs, but 1221 and 6633 are PINs. How many distinct possible PINs are there?

2017 Math Prize for Girls Problems, 10

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

Let $C$ be a cube. Let $P$, $Q$, and $R$ be random vertices of $C$, chosen uniformly and independently from the set of vertices of $C$. (Note that $P$, $Q$, and $R$ might be equal.) Compute the probability that some face of $C$ contains $P$, $Q$, and $R$.

2016 Math Prize for Girls Problems, 7

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Let $S$ be the set of all real numbers $x$ such that $0 \le x \le 2016 \pi$ and $\sin x < 3 \sin(x/3)$. The set $S$ is the union of a finite number of disjoint intervals. Compute the total length of all these intervals.

2016 Math Prize for Girls Olympiad, 1

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Triangle $T_1$ has sides of length $a_1$, $b_1$, and $c_1$; its area is $K_1$. Triangle $T_2$ has sides of length $a_2$, $b_2$, and $c_2$; its area is $K_2$. Triangle $T_3$ has sides of length $a_1 + a_2$, $b_1 + b_2$, and $c_1 + c_2$; its area is $K_3$. (a) Prove that $K_1^2 + K_2^2 < K_3^2$. (b) Prove that $\sqrt{K_1} + \sqrt{K_2} \le \sqrt{K_3} \,$.

2015 Math Prize for Girls Olympiad, 1

Tags: Math Prize for Girls

Prove that every positive integer has a unique representation in the form \[ \sum_{i=0}^k d_i 2^i \, , \] where $k$ is a nonnegative integer and each $d_i$ is either 1 or 2. (This representation is similar to usual binary notation except that the digits are 1 and 2, not 0 and 1.)

2019 Math Prize for Girls Problems, 18

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How many ordered triples $(a, b, c)$ of integers with $-15 \le a, b, c \le 15$ are there such that the three equations $ax + by = c$, $bx + cy = a$, and $cx + ay = b$ correspond to lines that are distinct and concurrent?

2016 Math Prize for Girls Problems, 15

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Let $H$ be a convex, equilateral heptagon whose angles measure (in degrees) $168^\circ$, $108^\circ$, $108^\circ$, $168^\circ$, $x^\circ$, $y^\circ$, and $z^\circ$ in clockwise order. Compute the number $y$.

2019 Math Prize for Girls Olympiad, 3

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Say that a positive integer is [i]red[/i] if it is of the form $n^{2020}$, where $n$ is a positive integer. Say that a positive integer is [i]blue[/i] if it is not red and is of the form $n^{2019}$, where $n$ is a positive integer. True or false: Between every two different red positive integers greater than $10^{100{,}000{,}000}$, there are at least 2019 blue positive integers. Prove that your answer is correct.

2017 Math Prize for Girls Olympiad, 4

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A [i]lattice point[/i] is a point in the plane whose two coordinates are both integers. A [i]lattice line[/i] is a line in the plane that contains at least two lattice points. Is it possible to color every lattice point red or blue such that every lattice line contains exactly 2017 red lattice points? Prove that your answer is correct.

2016 Math Prize for Girls Olympiad, 4

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Let $d(n)$ be the number of positive divisors of a positive integer $n$. Let $\mathbb{N}$ be the set of all positive integers. Say that a bijection $F$ from $\mathbb{N}$ to $\mathbb{N}$ is [i]divisor-friendly[/i] if $d(F(mn)) = d(F(m)) d(F(n))$ for all positive integers $m$ and $n$. (Note: A bijection is a one-to-one, onto function.) Does there exist a divisor-friendly bijection? Prove or disprove.

2009 Math Prize For Girls Problems, 19

Tags: geometry , circumcircle , Math Prize for Girls

Let $ S$ be a set of $ 100$ points in the plane. The distance between every pair of points in $ S$ is different, with the largest distance being $ 30$. Let $ A$ be one of the points in $ S$, let $ B$ be the point in $ S$ farthest from $ A$, and let $ C$ be the point in $ S$ farthest from $ B$. Let $ d$ be the distance between $ B$ and $ C$ rounded to the nearest integer. What is the smallest possible value of $ d$?

2016 Math Prize for Girls Problems, 10

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How many solutions of the equation $\tan x = \tan \tan x$ are on the interval $0 \le x \le \tan^{-1} 942$? (Here $\tan^{-1}$ means the inverse tangent function, sometimes written $\arctan$.)

2017 Math Prize for Girls Problems, 9

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Say that a positive integer $n$ is [i]smooth[/i] if $\frac{1}{n}$ has a terminating decimal expansion. (Note that 1 is smooth.) Compute the value of the infinite series \[ \sum_n \frac{1}{n^3} \, , \] where $n$ ranges over all smooth positive integers.

2021 Math Prize for Girls Problems, 4

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For a positive integer $n$, let $v(n)$ denote the largest integer $j$ such that $n$ is divisible by $2^j$. Let $a$ and $b$ be chosen uniformly and independently at random from among the integers between 1 and 32, inclusive. What is the probability that $v(a) > v(b)$?

2016 Math Prize for Girls Problems, 19

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In the coordinate plane, consider points $A = (0, 0)$, $B = (11, 0)$, and $C = (18, 0)$. Line $\ell_A$ has slope 1 and passes through $A$. Line $\ell_B$ is vertical and passes through $B$. Line $\ell_C$ has slope $-1$ and passes through $C$. The three lines $\ell_A$, $\ell_B$, and $\ell_C$ begin rotating clockwise about points $A$, $B$, and $C$, respectively. They rotate at the same angular rate. At any given time, the three lines form a triangle. Determine the largest possible area of such a triangle.

2019 Math Prize for Girls Problems, 6

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For each integer from 1 through 2019, Tala calculated the product of its digits. Compute the sum of all 2019 of Tala's products.

2021 Math Prize for Girls Problems, 1

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A soccer coach named $C$ does a header drill with two players $A$ and $B$, but they all forgot to put sunscreen on their foreheads. They solve this issue by dunking the ball into a vat of sunscreen before starting the drill. Coach $C$ heads the ball to $A$, who heads the ball back to $C$, who then heads the ball to $B$, who heads the ball back to $C$; this pattern $CACBCACB\ldots\,$ repeats ad infinitum. Each time a person heads the ball, $1/10$ of the sunscreen left on the ball ends up on the person's forehead. In the limit, what fraction of the sunscreen originally on the ball will end up on the coach's forehead?

2019 Math Prize for Girls Problems, 11

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Twelve $1$'s and ten $-1$'s are written on a chalkboard. You select 10 of the numbers and compute their product, then add up these products for every way of choosing 10 numbers from the 22 that are written on the chalkboard. What sum do you get?

2021 Math Prize for Girls Problems, 5

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Among all fractions (whose numerator and denominator are positive integers) strictly between $\tfrac{6}{17}$ and $\tfrac{9}{25}$, which one has the smallest denominator?

2019 Math Prize for Girls Problems, 5

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Two ants sit at the vertex of the parabola $y = x^2$. One starts walking northeast (i.e., upward along the line $y = x$ and the other starts walking northwest (i.e., upward along the line $y = -x$). Each time they reach the parabola again, they swap directions and continue walking. Both ants walk at the same speed. When the ants meet for the eleventh time (including the time at the origin), their paths will enclose 10 squares. What is the total area of these squares?

2019 Math Prize for Girls Problems, 14

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Devah draws a row of 1000 equally spaced dots on a sheet of paper. She goes through the dots from left to right, one by one, checking if the midpoint between the current dot and some remaining dot to its left is also a remaining dot. If so, she erases the current dot. How many dots does Devah end up erasing?

2019 Math Prize for Girls Olympiad, 4

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Let $n$ be a positive integer. Let $d$ be an integer such that $d \ge n$ and $d$ is a divisor of $\frac{n(n + 1)}{2}$. Prove that the set $\{ 1, 2, \dots, n \}$ can be partitioned into disjoint subsets such that the sum of the numbers in each subset equals $d$.

2017 Math Prize for Girls Problems, 8

Tags: Math Prize for Girls , complex numbers

Let $c$ be a complex number. Suppose there exist distinct complex numbers $r$, $s$, and $t$ such that for every complex number $z$, we have \[ (z - r)(z - s)(z - t) = (z - cr)(z - cs)(z - ct). \] Compute the number of distinct possible values of $c$.

2021 Math Prize for Girls Problems, 13

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There are 2021 light bulbs in a row, labeled 1 through 2021, each with an on/off switch. They all start in the off position when 1011 people walk by. The first person flips the switch on every bulb; the second person flips the switch on every 3rd bulb (bulbs 3, 6, etc.); the third person flips the switch on every 5th bulb; and so on. In general, the $k$th person flips the switch on every $(2k - 1)$th light bulb, starting with bulb $2k - 1$. After all 1011 people have gone by, how many light bulbs are on?