Found problems: 9
2016 NIMO Summer Contest, 14
Find the smallest positive integer $n$ such that $n^2+4$ has at least four distinct prime factors.
[i]Proposed by Michael Tang[/i]
2016 NIMO Summer Contest, 1
What is the value of \[\left(9+\dfrac{9}{9}\right)^{9-9/9} - \dfrac{9}{9}?\]
[i]Proposed by David Altizio[/i]
2016 NIMO Summer Contest, 11
A set $S$ of positive integers is $\textit{sum-complete}$ if there are positive integers $m$ and $n$ such that an integer $a$ is the sum of the elements of some nonempty subset of $S$ if and only if $m \le a \le n$.
Let $S$ be a sum-complete set such that $\{1, 3\} \subset S$ and $|S| = 8$. Find the greatest possible value of the sum of the elements of $S$.
[i]Proposed by Michael Tang[/i]
2017 NIMO Summer Contest, 1
Let $x$ be the answer to this question. Find the value of $2017 - 2016x$.
[i]Proposed by Michael Tang[/i]
2016 NIMO Summer Contest, 3
Consider all $1001$-element subsets of the set $\{1,2,3,...,2015\}$. From each such subset choose the median. Find the arithmetic mean of all these medians.
[i]Proposed by Michael Ren[/i]
2016 NIMO Summer Contest, 15
Let $ABC$ be a triangle with $AB=17$ and $AC=23$. Let $G$ be the centroid of $ABC$, and let $B_1$ and $C_1$ be on the circumcircle of $ABC$ with $BB_1\parallel AC$ and $CC_1\parallel AB$. Given that $G$ lies on $B_1C_1$, the value of $BC^2$ can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Determine $100m+n$.
[i]Proposed by Michael Ren[/i]
2016 NIMO Summer Contest, 13
The area of the region in the $xy$-plane satisfying the inequality \[\min_{1 \le n \le 10} \max\left(\frac{x^2+y^2}{4n^2}, \, 2 - \frac{x^2+y^2}{4n^2-4n+1}\right) \le 1\] is $k\pi$, for some integer $k$. Find $k$.
[i]Proposed by Michael Tang[/i]
2016 NIMO Summer Contest, 10
In rectangle $ABCD$, point $M$ is the midpoint of $AB$ and $P$ is a point on side $BC$. The perpendicular bisector of $MP$ intersects side $DA$ at point $X$. Given that $AB = 33$ and $BC = 56$, find the least possible value of $MX$.
[i]Proposed by Michael Tang[/i]
2016 NIMO Summer Contest, 2
Compute the number of permutations $(a,b,c,x,y,z)$ of $(1,2,3,4,5,6)$ which satisfy the five inequalities
\[ a < b < c, \quad x < y < z, \quad a < x, \quad b < y, \quad\text{and}\quad c < z. \]
[i]Proposed by Evan Chen[/i]