Found problems: 109
2019 Math Prize for Girls Problems, 13
Each side of a unit square (side length 1) is also one side of an equilateral triangle that lies in the square. Compute the area of the intersection of (the interiors of) all four triangles.
2021 Math Prize for Girls Problems, 12
Let $P_1$, $P_2$, $P_3$, $P_4$, $P_5$, and $P_6$ be six parabolas in the plane, each congruent to the parabola $y = x^2/16$. The vertices of the six parabolas are evenly spaced around a circle. The parabolas open outward with their axes being extensions of six of the circle's radii. Parabola $P_1$ is tangent to $P_2$, which is tangent to $P_3$, which is tangent to $P_4$, which is tangent to $P_5$, which is tangent to $P_6$, which is tangent to $P_1$. What is the diameter of the circle?
2019 Math Prize for Girls Problems, 1
In the USA, standard letter-size paper is 8.5 inches wide and 11 inches long. What is the largest integer that cannot be written as a sum of a whole number (possibly zero) of 8.5's and a whole number (possibly zero) of 11's?
2017 Math Prize for Girls Problems, 20
Compute the value of the sum
\[
\sum_{k = 1}^{11} \frac{\sin(2^{k + 4} \pi / 89)}{\sin(2^k \pi / 89)} \, .
\]
2016 Math Prize for Girls Problems, 13
Alice, Beth, Carla, Dana, and Eden play a game in which each of them simultaneously closes her eyes and randomly chooses two of the others to point at (one with each hand). A participant loses if she points at someone who points back at her; otherwise, she wins. Find the probability that all five girls win.
2018 Math Prize for Girls Problems, 6
Martha writes down a random mathematical expression consisting of 3 single-digit positive integers with an addition sign "$+$" or a multiplication sign "$\times$" between each pair of adjacent digits. (For example, her expression could be $4 + 3\times 3$, with value 13.) Each positive digit is equally likely, each arithmetic sign ("$+$" or "$\times$") is equally likely, and all choices are independent. What is the expected value (average value) of her expression?
2016 Math Prize for Girls Problems, 4
Compute the smallest positive integer $n$ such that $2016^n$ does not divide $2016!$.
2017 Math Prize for Girls Problems, 7
Let $a_1$, $a_2$, ... be an infinite sequence of integers such that $0 \le a_k \le k$ for every positive integer $k$ and such that
\[
2017 = \sum_{k = 1}^\infty a_k \cdot k! \, .
\]
What is the value of the infinite series $\sum_{k = 1}^\infty a_k$?
2021 Math Prize for Girls Problems, 17
In the coordinate plane, let $A = (-8, 0)$, $B = (8, 0)$, and $C = (t, 6)$. What is the maximum value of $\sin m\angle CAB \cdot \sin m\angle CBA$, over all real numbers $t$?
2016 Math Prize for Girls Problems, 16
Let $A < B < C < D$ be positive integers such that every three of them form the side lengths of an obtuse triangle. Compute the least possible value of $D$.
2016 Math Prize for Girls Problems, 17
We define the weight $W$ of a positive integer as follows: $W(1) = 0$, $W(2) = 1$, $W(p) = 1 + W(p + 1)$ for every odd prime $p$, $W(c) = 1 + W(d)$ for every composite $c$, where $d$ is the greatest proper factor of $c$. Compute the greatest possible weight of a positive integer less than 100.
2017 Math Prize for Girls Problems, 18
Let $x$, $y$, and $z$ be nonnegative integers that are less than or equal to 100. Suppose that $x + y + z$, $xy + z$, $x + yz$, and $xyz$ are (in some order) four consecutive terms of an arithmetic sequence. Compute the number of such ordered triples $(x, y, z)$.
2016 Math Prize for Girls Problems, 2
Katrine has a bag containing 4 buttons with distinct letters M, P, F, G on them (one letter per button). She picks buttons randomly, one at a time, without replacement, until she picks the button with letter G. What is the probability that she has at least three picks and her third pick is the button with letter M?
2017 Math Prize for Girls Problems, 4
If $\mathrm{MATH} + \mathrm{WITH} = \mathrm{GIRLS}$, compute the smallest possible value of $\mathrm{GIRLS}$. Here $\mathrm{MATH}$ and $\mathrm{WITH}$ are 4-digit numbers and $\mathrm{GIRLS}$ is a 5-digit number (all with nonzero leading digits). Different letters represent different digits.
2018 Math Prize for Girls Problems, 13
A circle overlaps an equilateral triangle of side length $100\sqrt{3}$. The three chords in the circle formed by the three sides of the triangle have lengths 6, 36, and 60, respectively. What is the area of the circle?
2018 Math Prize for Girls Problems, 14
Let $f(x)$ be the polynomial $\prod_{k=1}^{50} \bigl( x - (2k-1) \bigr)$. Let $c$ be the coefficient of $x^{48}$ in $f(x)$. When $c$ is divided by 101, what is the remainder? (The remainder is an integer between 0 and 100.)
2021 Math Prize for Girls Problems, 3
Let $O$ be the center of an equilateral triangle $ABC$ of area $1/\pi$. As shown in the diagram below, a circle centered at $O$ meets the triangle at points $D$, $E$, $F$, $G$, $H$, and $I$, which trisect each of the triangle's sides. Compute the total area of all six shaded regions.
[asy]
unitsize(90);
pair A = dir(0);
pair B = dir(120);
pair C = dir(240);
draw(A -- B -- C -- cycle);
pair D = (2*A + B)/3;
pair E = (A + 2*B)/3;
pair F = (2*B + C)/3;
pair G = (B + 2*C)/3;
pair H = (2*C + A)/3;
pair I = (C + 2*A)/3;
draw(E -- F);
draw(G -- H);
draw(I -- D);
draw(D -- G);
draw(E -- H);
draw(F -- I);
pair O = (0, 0);
real r = 1/sqrt(3);
draw(circle(O, r));
fill(O -- D -- E -- cycle, gray);
fill(O -- F -- G -- cycle, gray);
fill(O -- H -- I -- cycle, gray);
fill(arc(O, r, -30, 30) -- cycle, gray);
fill(arc(0, r, 90, 150) -- cycle, gray);
fill(arc(0, r, 210, 270) -- cycle, gray);
label("$A$", A, A);
label("$B$", B, B);
label("$C$", C, C);
label("$D$", D, unit(D));
label("$E$", E, unit(E));
label("$F$", F, unit(F));
label("$G$", G, unit(G));
label("$H$", H, unit(H));
label("$I$", I, unit(I));
label("$O$", O, C);
[/asy]
2019 Math Prize for Girls Problems, 7
Mr. Jones teaches algebra. He has a whiteboard with a pre-drawn coordinate grid that runs from $-10$ to $10$ in both the $x$ and $y$ coordinates. Consequently, when he illustrates the graph of a quadratic, he likes to use a quadratic with the following properties:
I. The quadratic sends integers to integers.
II. The quadratic has distinct integer roots both between $-10$ and $10$, inclusive.
III. The vertex of the quadratic has integer $x$ and $y$ coordinates both between $-10$ and $10$, inclusive.
How many quadratics are there with these properties?
2015 Math Prize for Girls Olympiad, 3
Let $f$ be the cubic polynomial
\[
f(x) = x^3 + bx^2 + cx + d,
\]
where $b$, $c$, and $d$ are real numbers. Let $x_1$, $x_2$, $\ldots\,$, $x_n$ be nonnegative numbers, and let $m$ be their average. Suppose that $m \ge - \dfrac{b}{2}\,$. Prove that
\[
\sum_{i = 1}^n f(x_i) \ge n f(m).
\]
2018 Math Prize for Girls Problems, 7
For every positive integer $n$, let $T_n = \frac{n(n+1)}{2}$ be the $n^{\text{th}}$ triangular number. What is the $2018^{\text{th}}$ smallest positive integer $n$ such that $T_n$ is a multiple of 1000?
2019 Math Prize for Girls Problems, 8
How many positive integers less than 4000 are not divisible by 2, not divisible by 3, not divisible by 5, and not divisible by 7?
2018 Math Prize for Girls Olympiad, 1
Let $P$ be a point in the plane. Suppose that $P$ is inside (or on) each of 6 circles $\omega_1$, $\omega_2$, ..., $\omega_6$ in the plane. Prove that there exist distinct $i$ and $j$ so that the center of circle $\omega_i$ is inside (or on) circle $\omega_j$.
2017 Math Prize for Girls Problems, 1
A bag contains 4 red marbles, 5 yellow marbles, and 6 blue marbles. Three marbles are to be picked out randomly (without replacement). What is the probability that exactly two of them have the same color?
2015 Math Prize for Girls Olympiad, 2
A tetrahedron $T$ is inside a cube $C$. Prove that the volume of $T$ is at most one-third the volume of $C$.
2017 Math Prize for Girls Problems, 17
Circle $\omega_1$ with radius 3 is inscribed in a strip $S$ having border lines $a$ and $b$. Circle $\omega_2$ within $S$ with radius 2 is tangent externally to circle $\omega_1$ and is also tangent to line $a$. Circle $\omega_3$ within $S$ is tangent externally to both circles $\omega_1$ and $\omega_2$, and is also tangent to line $b$. Compute the radius of circle $\omega_3$.