Found problems: 190
2016 ASDAN Math Tournament, 10
Compute the radius of the sphere inscribed in the tetrahedron with coordinates $(2,0,0)$, $(4,0,0)$, $(0,1,0)$, and $(0,0,3)$.
2016 Kazakhstan National Olympiad, 1
Prove that one can arrange all positive divisors of any given positive integer around a circle so that for any two neighboring numbers one is divisible by another.
2016 CMIMC, 6
Define a $\textit{tasty residue}$ of $n$ to be an integer $1<a<n$ such that there exists an integer $m>1$ satisfying \[a^m\equiv a\pmod n.\] Find the number of tasty residues of $2016$.
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 CMIMC, 1
In a race, people rode either bicycles with blue wheels or tricycles with tan wheels. Given that 15 more people rode bicycles than tricycles and there were 15 more tan wheels than blue wheels, what is the total number of people who rode in the race?
2016 ASDAN Math Tournament, 8
Let $ABC$ be a triangle with $AB=24$, $BC=30$, and $AC=36$. Point $M$ lies on $BC$ such that $BM=12$, and point $N$ lies on $AC$ such that $CN=20$. Let $X$ be the intersection of $AM$ and $BN$ and let line $CX$ intersect $AB$ at point $L$. Compute
$$\frac{AX}{XM}+\frac{BX}{XN}+\frac{CX}{XL}.$$
2017 NIMO Problems, 1
In how many ways can Eve fill each of the six squares of a $2 \times 3$ grid with either a $0$ or a $1$, such that Anne can then divide the grid into three congruent rectangles: one containing two $0$s, one containing two $1$s, and one containing a $0$ and a $1$?
[i]Proposed by Michael Tang[/i]
2016 ASDAN Math Tournament, 5
Given that $x$ and $y$ are real numbers, compute the minimum value of
$$x^4+4x^3+8x^2+4xy+6x+4y^2+10.$$
2016 CMIMC, 10
Let $\triangle ABC$ be a triangle with circumcircle $\Omega$ and let $N$ be the midpoint of the major arc $\widehat{BC}$. The incircle $\omega$ of $\triangle ABC$ is tangent to $AC$ and $AB$ at points $E$ and $F$ respectively. Suppose point $X$ is placed on the same side of $EF$ as $A$ such that $\triangle XEF\sim\triangle ABC$. Let $NX$ intersect $BC$ at a point $P$. Given that $AB=15$, $BC=16$, and $CA=17$, compute $\tfrac{PX}{XN}$.
2016 ASDAN Math Tournament, 17
Consider triangle $ABC$ with sides $AB=4$, $BC=11$, and $CA=9$. The triangle is spun around a line that passes through $B$ and the interior of the triangle (including the edges $BC$ and $BA$). Of all possible lines with these constraints, what is the largest possible volume of the resulting solid?
2016 CMIMC, 2
In concurrent computing, two processes may have their steps interwoven in an unknown order, as long as the steps of each process occur in order. Consider the following two processes:
\begin{tabular}{c|cc}
Process & $A$ & $B$\\
\hline
Step 1 & $x\leftarrow x-4$ & $x\leftarrow x-5$\\
Step 2 & $x\leftarrow x\cdot3$ & $x\leftarrow x\cdot4$\\
Step 3 & $x\leftarrow x-4$ & $x\leftarrow x-5$\\
Step 4 & $x\leftarrow x\cdot3$ & $x\leftarrow x\cdot4$
\end{tabular}
One such interweaving is $A1$, $B1$, $A2$, $B2$, $A3$, $B3$, $B4$, $A4$, but $A1$, $A3$, $A2$, $A4$, $B1$, $B2$, $B3$, $B4$ is not since the steps of $A$ do not occur in order. We run $A$ and $B$ concurrently with $x$ initially valued at $6$. Find the minimal possible value of $x$ among all interweavings.
2016 CMIMC, 7
Let $ABC$ be a triangle with incenter $I$ and incircle $\omega$. It is given that there exist points $X$ and $Y$ on the circumference of $\omega$ such that $\angle BXC=\angle BYC=90^\circ$. Suppose further that $X$, $I$, and $Y$ are collinear. If $AB=80$ and $AC=97$, compute the length of $BC$.
2017 NIMO Problems, 3
A circle $C_0$ is inscribed in an equilateral triangle $XYZ$ of side length 112. Then, for each positive integer $n$, circle $C_n$ is inscribed in the region bounded by $XY$, $XZ$, and an arc of circle $C_{n-1}$, forming an infinite sequence of circles tangent to sides $XY$ and $XZ$ and approaching vertex $X$. If these circles collectively have area $m\pi$, find $m$.
[i]Proposed by Michael Tang[/i]
2016 CMIMC, 7
Determine the smallest positive prime $p$ which satisfies the congruence \[p+p^{-1}\equiv 25\pmod{143}.\] Here, $p^{-1}$ as usual denotes multiplicative inverse.
2016 CMIMC, 1
Construction Mayhem University has been on a mission to expand and improve its campus! The university has recently adopted a new construction schedule where a new project begins every two days. Each project will take exactly one more day than the previous one to complete (so the first project takes 3, the second takes 4, and so on.)
Suppose the new schedule starts on Day 1. On which day will there first be at least $10$ projects in place at the same time?
2017 NIMO Problems, 7
Eve randomly chooses two $\textbf{distinct}$ points on the coordinate plane from the set of all $11^2$ lattice points $(x, y)$ with $0 \le x \le 10$, $0 \le y \le 10$. Then, Anne the ant walks from the point $(0,0)$ to the point $(10, 10)$ using a sequence of one-unit right steps and one-unit up steps. Let $P$ be the number of paths Anne could take that pass through both of the points that Eve chose. The expected value of $P$ is $\dbinom{20}{10} \cdot \dfrac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100a+b$.
[i]Proposed by Michael Tang[/i]
2016 ASDAN Math Tournament, 9
A cake in the shape of a rectangular prism has dimensions $6\text{ cm}\times14\text{ cm}\times21\text{ cm}$. It is cut into $1764$ equally sized cubes such that each cube is $1\text{ cm}^3$. Andy the ant starts at one corner of the cake and eats through the cake in a straight line to the opposite corner of the cake. How many of the $1\text{ cm}^3$ cubes does Andy bite through?
2017 NIMO Problems, 6
Triangle $\triangle ABC$ has circumcenter $O$ and incircle $\gamma$. Suppose that $\angle BAC =60^\circ$ and $O$ lies on $\gamma$. If \[ \tan B \tan C = a + \sqrt{b} \] for positive integers $a$ and $b$, compute $100a+b$.
[i]Proposed by Kaan Dokmeci[/i]
2016 ASDAN Math Tournament, 7
Eddy and Moor play a game with the following rules:
[list=a]
[*] The game begins with a pile of $N$ stones, where $N$ is a positive integer. [/*]
[*] The $2$ players alternate taking turns (e.g. Eddy moves, then Moor moves, then Eddy moves, and so on). [/*]
[*] During a player's turn, given $a$ stones remaining in the pile, the player may remove $b$ stones from the pile, where $\gcd(a,b)=1$ and $b\leq a$. [/*]
[*] If a player cannot make a move, they lose. [/*]
[/list]
For example, if Eddy goes first and $N=4$, then Eddy can remove $3$ stones from the pile (since $3\leq4$ and $\gcd(3,4)=1$), leaving $1$ stone in the pile. Moor can then remove $1$ stone from the pile (since $1\leq1$ and $\gcd(1,1)=1$), leaving $0$ stones in the pile. Since Eddy cannot remove stones from an empty pile, he cannot make a move, and therefore loses.
Both Eddy and Moor want to win, so they will both always make the best possible move. If Eddy moves first, for how many values of $N<2016$ can Eddy win no matter what moves Moor chooses?
2016 CMIMC, 1
A $\emph{planar}$ graph is a connected graph that can be drawn on a sphere without edge crossings. Such a drawing will divide the sphere into a number of faces. Let $G$ be a planar graph with $11$ vertices of degree $2$, $5$ vertices of degree $3$, and $1$ vertex of degree $7$. Find the number of faces into which $G$ divides the sphere.
2017 NIMO Problems, 5
Let $\{a_i\}_{i=0}^\infty$ be a sequence of real numbers such that \[\sum_{n=1}^\infty\dfrac {x^n}{1-x^n}=a_0+a_1x+a_2x^2+a_3x^3+\cdots\] for all $|x|<1$. Find $a_{1000}$.
[i]Proposed by David Altizio[/i]
2016 CMIMC, 3
Suppose $x$ and $y$ are real numbers which satisfy the system of equations \[x^2-3y^2=\frac{17}x\qquad\text{and}\qquad 3x^2-y^2=\frac{23}y.\] Then $x^2+y^2$ can be written in the form $\sqrt[m]{n}$, where $m$ and $n$ are positive integers and $m$ is as small as possible. Find $m+n$.
2016 ASDAN Math Tournament, 3
Julia adds up the numbers from $1$ to $2016$ in a calculator. However, every time she inputs a $2$, the calculator malfunctions and inputs a $3$ instead (for example, when Julia inputs $202$, the calculator inputs $303$ instead). How much larger is the total sum returned by the broken calculator? (No $2$s are replaced by $3$s in the output, and the calculator only malfunctions while Julia is inputting numbers.)
2016 CMIMC, 6
Aaron is trying to write a program to compute the terms of the sequence defined recursively by $a_0=0$, $a_1=1$, and \[a_n=\begin{cases}a_{n-1}-a_{n-2}&n\equiv0\pmod2\\2a_{n-1}-a_{n-2}&\text{else}\end{cases}\] However, Aaron makes a typo, accidentally computing the recurrence by \[a_n=\begin{cases}a_{n-1}-a_{n-2}&n\equiv0\pmod3\\2a_{n-1}-a_{n-2}&\text{else}\end{cases}\] For how many $0\le k\le2016$ did Aaron coincidentally compute the correct value of $a_k$?
2016 ASDAN Math Tournament, 8
It is possible to express the sum
$$\sum_{n=1}^{24}\frac{1}{\sqrt{n+\sqrt{n^2-1}}}$$
as $a\sqrt{2}+b\sqrt{3}$, for some integers $a$ and $b$. Compute the ordered pair $(a,b)$.