Found problems: 18
2024 Pan-American Girls’ Mathematical Olympiad, 2
Danielle has an $m \times n$ board and wants to fill it with pieces composed of two or more diagonally connected squares as shown, without overlapping or leaving gaps:
a) Find all values of $(m,n)$ for which it is possible to fill the board.
b) If it is possible to fill an $m \times n$ board, find the minimum number of pieces Danielle can use to fill it.
[i]Note: The pieces can be rotated[/i].
2022 Pan-American Girls' Math Olympiad, 2
Find all ordered triplets $(p,q,r)$ of positive integers such that $p$ and $q$ are two (not necessarily distinct) primes, $r$ is even, and
\[p^3+q^2=4r^2+45r+103.\]
2024 Pan-American Girls’ Mathematical Olympiad, 6
Let $ABC$ be a triangle, and let $a$, $b$, and $c$ be the lengths of the sides opposite vertices $A$, $B$, and $C$, respectively. Let $R$ be its circumradius and $r$ its inradius. Suppose that $b + c = 2a$ and $R = 3r$.
The excircle relative to vertex $A$ intersects the circumcircle of $ABC$ at points $P$ and $Q$. Let $U$ be the midpoint of side $BC$, and let $I$ be the incenter of $ABC$.
Prove that $U$ is the centroid of triangle $QIP$.
2022 Pan-American Girls' Math Olympiad, 4
Let $ABC$ be a triangle, with $AB\neq AC$. Let $O_1$ and $O_2$ denote the centers of circles $\omega_1$ and $\omega_2$ with diameters $AB$ and $BC$, respectively. A point $P$ on segment $BC$ is chosen such that $AP$ intersects $\omega_1$ in point $Q$, with $Q\neq A$. Prove that $O_1$, $O_2$, and $Q$ are collinear if and only if $AP$ is the angle bisector of $\angle BAC$.
2021 Pan-American Girls' Math Olympiad, Problem 1
There are $n \geq 2$ coins numbered from $1$ to $n$. These coins are placed around a circle, not necesarily in order.
In each turn, if we are on the coin numbered $i$, we will jump to the one $i$ places from it, always in a clockwise order, beginning with coin number 1. For an example, see the figure below.
Find all values of $n$ for which there exists an arrangement of the coins in which every coin will be visited.
2022 Pan-American Girls' Math Olympiad, 3
Let $ABC$ be an acute triangle with $AB< AC$. Denote by $P$ and $Q$ points on the segment $BC$ such that $\angle BAP = \angle CAQ < \frac{\angle BAC}{2}$. $B_1$ is a point on segment $AC$. $BB_1$ intersects $AP$ and $AQ$ at $P_1$ and $Q_1$, respectively. The angle bisectors of $\angle BAC$ and $\angle CBB_1$ intersect at $M$. If $PQ_1\perp AC$ and $QP_1\perp AB$, prove that $AQ_1MPB$ is cyclic.
2022 Pan-American Girls' Math Olympiad, 6
Ana and Bety play a game alternating turns. Initially, Ana chooses an odd possitive integer and composite $n$ such that $2^j<n<2^{j+1}$ with $2<j$. In her first turn Bety chooses an odd composite integer $n_1$ such that
\[n_1\leq \frac{1^n+2^n+\dots+(n-1)^n}{2(n-1)^{n-1}}.\]
Then, on her other turn, Ana chooses a prime number $p_1$ that divides $n_1$. If the prime that Ana chooses is $3$, $5$ or $7$, the Ana wins; otherwise Bety chooses an odd composite positive integer $n_2$ such that \[n_2\leq \frac{1^{p_1}+2^{p_1}+\dots+(p_1-1)^{p_1}}{2(p_1-1)^{p_1-1}}.\]
After that, on her turn, Ana chooses a prime $p_2$ that divides $n_2,$, if $p_2$ is $3$, $5$, or $7$, Ana wins, otherwise the process repeats. Also, Ana wins if at any time Bety cannot choose an odd composite positive integer in the corresponding range. Bety wins if she manages to play at least $j-1$ turns. Find which of the two players has a winning strategy.
2024 Pan-American Girls’ Mathematical Olympiad, 5
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that
$f(f(x+y) - f(x)) + f(x)f(y) = f(x^2) - f(x+y),$
for all real numbers $x, y$.
2023 Pan-American Girls’ Mathematical Olympiad, 1
An integer \(n \geq 2\) is said to be [i]tuanis[/i] if, when you add the smallest prime divisor of \(n\) and the largest prime divisor of \(n\) (these divisors can be the same), you obtain an odd result. Calculate the sum of all [i]tuanis[/i] numbers that are less or equal to \(2023\).
2022 Pan-American Girls' Math Olympiad, 5
Find all positive integers $k$ for which there exist $a$, $b$, and $c$ positive integers such that \[\lvert (a-b)^3+(b-c)^3+(c-a)^3\rvert=3\cdot2^k.\]
2021 Pan-American Girls' Math Olympiad, Problem 5
Celeste has an unlimited amount of each type of $n$ types of candy, numerated type 1, type 2, ... type n. Initially she takes $m>0$ candy pieces and places them in a row on a table. Then, she chooses one of the following operations (if available) and executes it:
$1.$ She eats a candy of type $k$, and in its position in the row she places one candy type $k-1$ followed by one candy type $k+1$ (we consider type $n+1$ to be type 1, and type 0 to be type $n$).
$2.$ She chooses two consecutive candies which are the same type, and eats them.
Find all positive integers $n$ for which Celeste can leave the table empty for any value of $m$ and any configuration of candies on the table.
$\textit{Proposed by Federico Bach and Santiago Rodriguez, Colombia}$
2024 Pan-American Girls’ Mathematical Olympiad, 3
Let $M$ be a non-empty set of positive integers and let $S_M$ be the sum of all the elements of $M$. We define the [i]tlacoyo[/i] of $M$ as the sum of the digits of $S_M$. For example, if $M=\{2,7,34\}$, then $S_M=2+7+34=43$ and the tlacoyo of the set $M$ is $4+3=7$. \\
Prove that for every positive integer $n$, there exists a set $M$ of $n$ distinct positive integers, such that all its non-empty subsets have the same tlacoyo.
2022 Pan-American Girls' Math Olympiad, 1
Leticia has a $9\times 9$ board. She says that two squares are [i]friends[/i] is they share a side, if they are at opposite ends of the same row or if they are at opposite ends of the same column. Every square has $4$ friends on the board. Leticia will paint every square one of three colors: green, blue or red. In each square a number will be written based on the following rules:
- If the square is green, write the number of red friends plus twice the number of blue friends.
- If the square is red, write the number of blue friends plus twice the number of green friends.
- If the square is blue, write the number of green friends plus twice the number of red friends.
Considering that Leticia can choose the coloring of the squares on the board, find the maximum possible value she can obtain when she sums the numbers in all the squares.
2024 Pan-American Girls’ Mathematical Olympiad, 4
The $n$-factorial of a positive integer $x$ is the product of all positive integers less than or equal to $z$ that are congruent to $z$ modulo $n$.
For example, for the number 16, its 2-factorial is $16 \times 14 \times 12 \times 10 \times 8 \times 6 \times 4 \times 2$, its 3-factorial is $16 \times 13 \times 10 \times 7 \times 4 \times 1$ and its 18-factorial is 16.
A positive integer is called [i]olympic[/i] if it has $n$ digits, all different than zero, and if it is equal to the sum of the $n$-factorials of its digits. Find all positive olympic integers.
2021 Pan-American Girls' Math Olympiad, Problem 4
Lucía multiplies some positive one-digit numbers (not necessarily distinct) and obtains a number $n$ greater than 10. Then, she multiplies all the digits of $n$ and obtains an odd number. Find all possible values of the units digit of $n$.
$\textit{Proposed by Pablo Serrano, Ecuador}$
2021 Pan-American Girls' Math Olympiad, Problem 6
Let $ABC$ be a triangle with incenter $I$, and $A$-excenter $\Gamma$. Let $A_1,B_1,C_1$ be the points of tangency of $\Gamma$ with $BC,AC$ and $AB$, respectively. Suppose $IA_1, IB_1$ and $IC_1$ intersect $\Gamma$ for the second time at points $A_2,B_2,C_2$, respectively. $M$ is the midpoint of segment $AA_1$. If the intersection of $A_1B_1$ and $A_2B_2$ is $X$, and the intersection of $A_1C_1$ and $A_2C_2$ is $Y$, prove that $MX=MY$.
2024 Pan-American Girls’ Mathematical Olympiad, 1
Let $ABC$ be an acute triangle with $AB < AC$, let $\Gamma$ be its circumcircle and let $D$ be the foot of the altitude from $A$ to $BC$. Take a point $E$ on the segment $BC$ such that $CE=BD$. Let $P$ be the point on $\Gamma$ diametrically opposite to vertex $A$. Prove that $PE$ is perpendicular to $BC$.
2021 Pan-American Girls' Math Olympiad, Problem 2
Consider the isosceles right triangle $ABC$ with $\angle BAC = 90^\circ$. Let $\ell$ be the line passing through $B$ and the midpoint of side $AC$. Let $\Gamma$ be the circumference with diameter $AB$. The line $\ell$ and the circumference $\Gamma$ meet at point $P$, different from $B$. Show that the circumference passing through $A,\ C$ and $P$ is tangent to line $BC$ at $C$.