Found problems: 109
2017 Math Prize for Girls Olympiad, 3
Let $ABCD$ be a cyclic quadrilateral such that $\angle BAD \le \angle ADC$. Prove that $AC + CD \le AB + BD$.
2024 Moldova EGMO TST, 6
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.
2018 Math Prize for Girls Problems, 5
Consider the following system of 7 linear equations with 7 unknowns:
\[
\begin{split}
a+b+c+d+e & = 1 \\
b+c+d+e+f & = 2 \\
c+d+e+f+g & = 3 \\
d+e+f+g+a & = 4 \\
e+f+g+a+b & = 5 \\
f+g+a+b+c & = 6 \\
g+a+b+c+d & = 7 .
\end{split}
\]
What is $g$?
2021 Math Prize for Girls Problems, 18
Let $N$ be the set of square-free positive integers less than or equal to 50. (A [i]square-free[/i] number is an integer that is not divisible by a perfect square bigger than 1.) How many 3-element subsets $S$ of $N$ are there such that the greatest common divisor of all 3 numbers in $S$ is 1, but no pair of numbers in $S$ is relatively prime?
2018 Math Prize for Girls Problems, 4
Let $ABCDEF$ be a regular hexagon. Let $P$ be the intersection point of $\overline{AC}$ and $\overline{BD}$. Suppose that the area of triangle $EFP$ is 25. What is the area of the hexagon?
2021 Math Prize for Girls Problems, 16
Let $G$ be the set of points $(x, y)$ such that $x$ and $y$ are positive integers less than or equal to 20. Say that a ray in the coordinate plane is [i]ocular[/i] if it starts at $(0, 0)$ and passes through at least one point in $G$. Let $A$ be the set of angle measures of acute angles formed by two distinct ocular rays. Determine
\[
\min_{a \in A} \tan a.
\]
2017 Math Prize for Girls Problems, 2
In the figure below, $BDEF$ is a square inscribed in $\triangle ABC$. If $\frac{AB}{BC} = \frac{4}{5}$, what is the area of $BDEF$ divided by the area of $\triangle ABC$?
[asy]
unitsize(20);
pair A = (0, 3);
pair B = (0, 0);
pair C = (4, 0);
draw(A -- B -- C -- cycle);
real w = 12.0 / 7;
pair D = (w, 0);
pair E = (w, w);
pair F = (0, w);
draw(D -- E -- F);
dot(Label("$A$", A, NW), A);
dot(Label("$B$", B, SW), B);
dot(Label("$C$", C, SE), C);
dot(Label("$D$", D, S), D);
dot(Label("$E$", E, NE), E);
dot(Label("$F$", F, W), F);
[/asy]
2021 Math Prize for Girls Problems, 20
Let $G$ be the set of points $(x, y)$ such that $x$ and $y$ are positive integers less than or equal to 6. A [i]magic grid[/i] is an assignment of an integer to each point in $G$ such that, for every square with horizontal and vertical sides and all four vertices in $G$, the sum of the integers assigned to the four vertices is the same as the corresponding sum for any other such square. A magic grid is formed so that the product of all 36 integers is the smallest possible value greater than 1. What is this product?
2017 Math Prize for Girls Problems, 16
Samantha is about to celebrate her sweet 16th birthday. To celebrate, she chooses a five-digit positive integer of the form SWEET, in which the two E's represent the same digit but otherwise the digits are distinct. (The leading digit S can't be 0.) How many such integers are divisible by 16?