Found problems: 109
2011 Math Prize For Girls Problems, 9
Let $ABC$ be a triangle. Let $D$ be the midpoint of $\overline{BC}$, let $E$ be the midpoint of $\overline{AD}$, and let $F$ be the midpoint of $\overline{BE}$. Let $G$ be the point where the lines $AB$ and $CF$ intersect. What is the value of $\frac{AG}{AB}$?
2021 Math Prize for Girls Problems, 15
There are 300 points in space. Four planes $A$, $B$, $C$, and $D$ each have the property that they split the 300 points into two equal sets. (No plane contains one of the 300 points.) What is the maximum number of points that can be found inside the tetrahedron whose faces are on $A$, $B$, $C$, and $D$?
2021 Math Prize for Girls Problems, 14
Let $S$ be the set of monic polynomials in $x$ of degree 6 all of whose roots are members of the set $\{ -1, 0, 1\}$. Let $P$ be the sum of the polynomials in $S$. What is the coefficient of $x^4$ in $P(x)$?
2016 Math Prize for Girls Olympiad, 3
Let $n$ be a positive integer. Let $x_1$, $x_2$, $\ldots\,$, $x_n$ be a sequence of $n$ real numbers. Say that a sequence $a_1$, $a_2$, $\ldots\,$, $a_n$ is [i]unimodular[/i] if each $a_i$ is $\pm 1$. Prove that
\[
\sum a_1 a_2 \ldots a_n (a_1x_1 + a_2x_2 + \cdots + a_nx_n)^n = 2^{n} n!\, x_1 x_2 \ldots x_n ,
\]
where the sum is over all $2^{n}$ unimodular sequences $a_1$, $a_2$, $\ldots\,$, $a_n$.
2016 Math Prize for Girls Problems, 3
Compute the least possible value of $ABCD - AB \times CD$, where $ABCD$ is a 4-digit positive integer, and $AB$ and $CD$ are 2-digit positive integers. (Here $A$, $B$, $C$, and $D$ are digits, possibly equal. Neither $A$ nor $C$ can be zero.)
2021 Math Prize for Girls Problems, 6
The number $734{,}851{,}474{,}594{,}578{,}436{,}096$ is equal to $n^6$ for some positive integer $n$. What is the value of $n$?
2021 Math Prize for Girls Problems, 2
Let $m$ and $n$ be positive integers such that $m^4 - n^4 = 3439$. What is the value of $mn$?
2018 Math Prize for Girls Problems, 15
In the $xy$-coordinate plane, the $x$-axis and the line $y=x$ are mirrors. If you shoot a laser beam from the point $(126, 21)$ toward a point on the positive $x$-axis, there are $3$ places you can aim at where the beam will bounce off the mirrors and eventually return to $(126, 21)$. They are $(126, 0)$, $(105, 0)$, and a third point $(d, 0)$. What is $d$? (Recall that when light bounces off a mirror, the angle of incidence has the same measure as the angle of reflection.)
2018 Math Prize for Girls Problems, 1
If $x$ is a real number such that $(x - 3)(x - 1)(x + 1)(x + 3) + 16 = 116^2$, what is the largest possible value of $x$?
2016 Math Prize for Girls Problems, 8
A [i]strip[/i] is the region between two parallel lines. Let $A$ and $B$ be two strips in a plane. The intersection of strips $A$ and $B$ is a parallelogram $P$. Let $A'$ be a rotation of $A$ in the plane by $60^\circ$. The intersection of strips $A'$ and $B$ is a parallelogram with the same area as $P$. Let $x^\circ$ be the measure (in degrees) of one interior angle of $P$. What is the greatest possible value of the number $x$?
2018 Math Prize for Girls Problems, 17
Let $ABC$ be a triangle with $AB=5$, $BC=4$, and $CA=3$. On each side of $ABC$, externally erect a semicircle whose diameter is the corresponding side. Let $X$ be on the semicircular arc erected on side $\overline{BC}$ such that $\angle CBX$ has measure $15^\circ$. Let $Y$ be on the semicircular arc erected on side $\overline{CA}$ such that $\angle ACY$ has measure $15^\circ$. Similarly, let $Z$ be on the semicircular arc erected on side $\overline{AB}$ such that $\angle BAZ$ has measure $15^\circ$. What is the area of triangle $XYZ$?
2017 Math Prize for Girls Problems, 15
A restricted rook (RR) is a fictional chess piece that can move horizontally or vertically (like a rook), except that each move is restricted to a neighboring square (cell). If RR can only (with at most one exception) move up and to the right, how many possible distinct paths are there to move RR from the bottom left square to the top right square of a standard 8-by-8 chess board? Note that RR may visit some squares more than once. A path is the sequence of squares visited by RR on its way.
2021 Math Prize for Girls Problems, 10
Let $P$ be the product of all the entries in row 2021 of Pascal's triangle (the row that begins 1, 2021, $\ldots$). What is the largest integer $j$ such that $P$ is divisible by $101^j$?
2016 Math Prize for Girls Problems, 5
A permutation of a finite set $S$ is a one-to-one function from $S$ to $S$. A permutation $P$ of the set $\{ 1, 2, 3, 4, 5 \}$ is called a W-permutation if $P(1) > P(2) < P(3) > P(4) < P(5)$. A permutation of the set $\{1, 2, 3, 4, 5 \}$ is selected at random. Compute the probability that it is a W-permutation.
2019 Math Prize for Girls Problems, 3
The degree measures of the six interior angles of a convex hexagon form an arithmetic sequence (not necessarily in cyclic order). The common difference of this arithmetic sequence can be any real number in the open interval $(-D, D)$. Compute the greatest possible value of $D$.
2019 Math Prize for Girls Problems, 9
Find the least real number $K$ such that for all real numbers $x$ and $y$, we have $(1 + 20 x^2)(1 + 19 y^2) \ge K xy$.
2018 Math Prize for Girls Problems, 12
You own a calculator that computes exactly. It has all the standard buttons, including a button that replaces the number currently displayed with its arctangent, and a button that replaces whatever is currently displayed with its cosine. You turn on the calculator and it reads 0. You create a sequence by alternately clicking on the arctangent button and the cosine button. (The calculator is in radian mode.) Let $a_n$ be the value displayed after you've pressed the cosine button for the $n$th time. What is $\prod_{k=1}^{11} a_k$?
2016 Math Prize for Girls Problems, 9
How many distinct lines pass through the point $(0, 2016)$ and intersect the parabola $y = x^2$ at two lattice points? (A lattice point is a point whose coordinates are integers.)
2017 Math Prize for Girls Problems, 11
Let $S(N)$ be the number of 1's in the binary representation of an integer $N$, and let $D(N) = S(N + 1) - S(N)$. Compute the sum of $D(N)$ over all $N$ such that $1 \le N \le 2017$ and $D(N) < 0$.
2018 Math Prize for Girls Problems, 10
Let $T_1$ be an isosceles triangle with sides of length 8, 11, and 11. Let $T_2$ be an isosceles triangle with sides of length $b$, 1, and 1. Suppose that the radius of the incircle of $T_1$ divided by the radius of the circumcircle of $T_1$ is equal to the radius of the incircle of $T_2$ divided by the radius of the circumcircle of $T_2$. Determine the largest possible value of $b$.
2016 Math Prize for Girls Problems, 14
We call a set $X$ of real numbers [i]three-averaging[/i] if for every two distinct elements $a$ and $b$ of $X$, there exists an element $c$ in $X$ (different from both $a$ and $b$) such that the number $(a + b + c)/3$ also belongs to $X$. For instance, the set $\{ 0, 1008, 2016 \}$ is three-averaging. What is the least possible number of elements in a three-averaging set with more than 3 elements?
2018 Math Prize for Girls Problems, 2
How many ordered pairs of integers $(x, y)$ satisfy $2 |y| \le x \le 40\,$?
2018 Math Prize for Girls Olympiad, 2
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 function $F$ from $\mathbb{N}$ to $\mathbb{N}$ is [i]divisor-respecting[/i] if $d(F(mn)) = d(F(m)) d(F(n))$ for all positive integers $m$ and $n$, and $d(F(n)) \le d(n)$ for all positive integers $n$. Find all divisor-respecting functions. Justify your answer.
2016 Math Prize for Girls Problems, 6
The largest term in the binomial expansion of $(1 + \tfrac{1}{2})^{31}$ is of the form $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. What is the value of $b$? As an example of a binomial expansion, the binomial expansion of an expression of the form $(x + y)^3$ is the sum of four terms
\[
x^3 + 3x^2y + 3xy^2 + y^3.
\]
2021 Math Prize for Girls Problems, 8
In $\triangle ABC$, let point $D$ be on $\overline{BC}$ such that the perimeters of $\triangle ADB$ and $\triangle ADC$ are equal. Let point $E$ be on $\overline{AC}$ such that the perimeters of $\triangle BEA$ and $\triangle BEC$ are equal. Let point $F$ be the intersection of $\overline{AB}$ with the line that passes through $C$ and the intersection of $\overline{AD}$ and $\overline{BE}$. Given that $BD = 10$, $CD = 2$, and $BF/FA = 3$,
what is the perimeter of $\triangle ABC$?