id stringclasses 6
values | problem stringclasses 6
values | answer stringclasses 1
value |
|---|---|---|
1 | A line in the plane is called *sunny* if it is not parallel to any of the $x$-axis, the $y$-axis, and the line $x+y=0$.
Let $n\ge3$ be a given integer. Determine all nonnegative integers $k$ such that there exist $n$ distinct lines in the plane satisfying both the following:
* for all positive integers $a$ and $b$ w... | |
2 | Let $\Omega$ and $\Gamma$ be circles with centers $M$ and $N$, respectively, such that the radius of $\Omega$ is less than the radius of $\Gamma$. Suppose circles $\Omega$ and $\Gamma$ intersect at two distinct points $A$ and $B$. Let $MN$ intersects $\Omega$ at $C$ and $\Gamma$ at $D$, such that points $C$, $M$, $N$, ... | |
3 | Let $\mathbb N$ denote the set of positive integers. A function $f:\mathbb N\to\mathbb N$ is said to be bonza if $f(a)$ divides $b^a-f(b)^{f(a)}$ for all positive integers $a$ and $b$.
Determine the smallest real constant $c$ such that $f(n)\le cn$ for all bonza functions $f$ and all positive integers $n$. | |
4 | A proper divisor of a positive integer $N$ is a positive divisor of $N$ other than $N$ itself.
The infinite sequence $a_1,a_2,\ldots$ consists of positive integers, each of which has at least three proper divisors. For each $n\ge1$, the integer $a_{n+1}$ is the sum of three largest proper divisors of $a_n$.
Determine... | |
5 | Alice and Bazza are playing the inekoalaty game, a two-player game whose rules depend on a positive real number $\lambda$ which is known to both players. On the $n$th turn of the game (starting with $n=1$) the following happens:
* If $n$ is odd, Alice chooses a nonnegative real number $x_n$ such that
$$x_1+x_2+\cdots... | |
6 | Consider a $2025\times2025$ grid of unit squares. Matilda wishes to place on the grid some rectangular tiles, possibly of difference sizes, such that each side of every tile lies on a grid line and every unit square is covered by at most one tile.
Determine the minimum number of tiles Matilda needs to place so that ea... |
README.md exists but content is empty.
- Downloads last month
- 12