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057f8a

Three airline companies operate flights from Dodola island. Each company has a different schedule of departures. The first company departs every 100 days, the second every 120 days and the third every 150 days. What is the greatest positive integer $d$ for which it is true that there will be $d$ consecutive days without a flight from Dodola island, regardless of the departure times of the various airlines?

192e23

Fred and George take part in a tennis tournament with $4046$ other players. In each round, the players are paired into $2024$ matches. How many ways are there to arrange the first round such that Fred and George do not have to play each other? (Two arrangements for the first round are \textit{different} if there is a player with a different opponent in the two arrangements.)

1acac0

Triangle $ABC$ has side length $AB = 120$ and circumradius $R = 100$. Let $D$ be the foot of the perpendicular from $C$ to the line $AB$. What is the greatest possible length of segment $CD$?

1fce4b

Find the three-digit number $n$ such that writing any other three-digit number $10^{2024}$ times in a row and $10^{2024}+2$ times in a row results in two numbers divisible by $n$.

349493

We call a sequence $a_1, a_2, \ldots$ of non-negative integers \textit{delightful} if there exists a positive integer $N$ such that for all $n > N$, $a_n = 0$, and for all $i \geq 1$, $a_i$ counts the number of multiples of $i$ in $a_1, a_2, \ldots, a_N$. How many delightful sequences of non-negative integers are there?

480182

Let $ABC$ be a triangle with $BC=108$, $CA=126$, and $AB=39$. Point $X$ lies on segment $AC$ such that $BX$ bisects $\angle CBA$. Let $\omega$ be the circumcircle of triangle $ABX$. Let $Y$ be a point on $\omega$ different from $X$ such that $CX=CY$. Line $XY$ meets $BC$ at $E$. The length of the segment $BE$ can be written as $\frac{m}{n}$, where $m$ and $n$ are coprime positive integers. Find $m+n$.

71beb6

For a positive integer $n$, let $S(n)$ denote the sum of the digits of $n$ in base 10. Compute $S(S(1)+S(2)+\cdots+S(N))$ with $N=10^{100}-2$.

88c219

For positive integers $x_1,\ldots, x_n$ define $G(x_1, \ldots, x_n)$ to be the sum of their $\frac{n(n-1)}{2}$ pairwise greatest common divisors. We say that an integer $n \geq 2$ is \emph{artificial} if there exist $n$ different positive integers $a_1, ..., a_n$ such that \[a_1 + \cdots + a_n = G(a_1, \ldots, a_n) +1.\] Find the sum of all artificial integers $m$ in the range $2 \leq m \leq 40$.

a1d40b

The Fibonacci numbers are defined as follows: $F_0 = 0$, $F_1 = 1$, and $F_{n+1} = F_n + F_{n-1}$ for $n \geq 1$. There are $N$ positive integers $n$ strictly less than $10^{101}$ such that $n^2 + (n+1)^2$ is a multiple of 5 but $F_{n-1}^2 + F_n^2$ is not. How many prime factors does $N$ have, counted with multiplicity?

bbd91e

Alice writes all positive integers from $1$ to $n$ on the board for some positive integer $n \geq 11$. Bob then erases ten of them. The mean of the remaining numbers is $3000/37$. The sum of the numbers Bob erased is $S$. What is the remainder when $n \times S$ is divided by $997$?