Found problems: 66
2012 NIMO Problems, 6
In rhombus $NIMO$, $MN = 150\sqrt{3}$ and $\measuredangle MON = 60^{\circ}$. Denote by $S$ the locus of points $P$ in the interior of $NIMO$ such that $\angle MPO \cong \angle NPO$. Find the greatest integer not exceeding the perimeter of $S$.
[i]Proposed by Evan Chen[/i]
2016 NIMO Problems, 4
In rhombus $ABCD$, let $M$ be the midpoint of $AB$ and $N$ be the midpoint of $AD$. If $CN = 7$ and $DM = 24$, compute $AB^2$.
[i]Proposed by Andy Liu[/i]
2015 NIMO Problems, 1
A function $f$ from the positive integers to the nonnegative integers is defined recursively by $f(1) = 0$ and $f(n+1) = 2^{f(n)}$ for every positive integer $n$. What is the smallest $n$ such that $f(n)$ exceeds the number of atoms in the observable universe (approximately $10^{80}$)?
[i]Proposed by Evan Chen[/i]
2015 NIMO Summer Contest, 12
Let $ABC$ be a triangle whose angles measure $A$, $B$, $C$, respectively. Suppose $\tan A$, $\tan B$, $\tan C$ form a geometric sequence in that order. If $1\le \tan A+\tan B+\tan C\le 2015$, find the number of possible integer values for $\tan B$. (The values of $\tan A$ and $\tan C$ need not be integers.)
[i] Proposed by Justin Stevens [/i]
2017 NIMO Problems, 8
For each nonnegative integer $n$, we define a set $H_n$ of points in the plane as follows:
[list]
[*]$H_0$ is the unit square $\{(x,y) \mid 0 \le x, y \le 1\}$.
[*]For each $n \ge 1$, we construct $H_n$ from $H_{n-1}$ as follows. Note that $H_{n-1}$ is the union of finitely many square regions $R_1, \ldots, R_k$. For each $i$, divide $R_i$ into four congruent square quadrants. If $n$ is odd, then the upper-right and lower-left quadrants of each $R_i$ make up $H_n$. If $n$ is even, then the upper-left and lower-right quadrants of each $R_i$ make up $H_n$.
[/list]
The figures $H_0$, $H_1$, $H_2$, and $H_3$ are shown below.
[asy]
pair[]sq(int n){pair[]a;
if(n == 0)a.push((.5,.5));
else for(pair k:sq(n-1)) { pair l=1/2^(n+1)*(1,(-1)^(1+(n%2)));a.push(k+l);a.push(k-l); }
return a;}
void hh(int n,real k){
pair[] S=sq(n);real r=1/2^(n+1);
for(pair p:S)filldraw(shift(p+(k,0))*((r,r)--(r,-r)--(-r,-r)--(-r,r)--cycle));
label("$H_"+string(n)+"$",(k+.5,-.3));}
size(7cm);
for(int i=0;i<=3;++i)hh(i,1.6*i);
[/asy]
Suppose that the point $P = (x,y)$ lies in $H_n$ for all $n \ge 0$. The greatest possible value of $xy$ is $\tfrac{m}{n}$, for relatively prime positive integers $m, n$. Compute $100m+n$.
[i]Proposed by Michael Tang[/i]
2016 NIMO Problems, 8
Justin the robot is on a mission to rescue abandoned treasure from a minefield. To do this, he must travel from the point $(0, 0, 0)$ to $(4, 4, 4)$ in three-dimensional space, only taking one-unit steps in the positive $x, y,$ or $z$-directions. However, the evil David anticipated Justin's arrival, and so he has surreptitiously placed a mine at the point $(2,2,2)$. If at any point Justin is at most one unit away from this mine (in any direction), the mine detects his presence and explodes, thwarting Justin.
How many paths can Justin take to reach his destination safely?
[i]Proposed by Justin Stevens[/i]
2016 NIMO Problems, 7
Let $A$ and $B$ be points with $AB=12$. A point $P$ in the plane of $A$ and $B$ is $\textit{special}$ if there exist points $X, Y$ such that
[list]
[*]$P$ lies on segment $XY$,
[*]$PX : PY = 4 : 7$, and
[*]the circumcircles of $AXY$ and $BXY$ are both tangent to line $AB$.
[/list]
A point $P$ that is not special is called $\textit{boring}$.
Compute the smallest integer $n$ such that any two boring points have distance less than $\sqrt{n/10}$ from each other.
[i]Proposed by Michael Ren[/i]
2015 NIMO Summer Contest, 13
Let $\triangle ABC$ be a triangle with $AB=85$, $BC=125$, $CA=140$, and incircle $\omega$. Let $D$, $E$, $F$ be the points of tangency of $\omega$ with $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ respectively, and furthermore denote by $X$, $Y$, and $Z$ the incenters of $\triangle AEF$, $\triangle BFD$, and $\triangle CDE$, also respectively. Find the circumradius of $\triangle XYZ$.
[i] Proposed by David Altizio [/i]
2015 NIMO Summer Contest, 3
A list of integers with average $89$ is split into two disjoint groups. The average of the integers in the first group is $73$ while the average of the integers in the second group is $111$. What is the smallest possible number of integers in the original list?
[i] Proposed by David Altizio [/i]
2016 NIMO Problems, 6
Consider a sequence $a_0$, $a_1$, $\ldots$, $a_9$ of distinct positive integers such that $a_0=1$, $a_i < 512$ for all $i$, and for every $1 \le k \le 9$ there exists $0 \le m \le k-1$ such that \[(a_k-2a_m)(a_k-2a_m-1) = 0.\] Let $N$ be the number of these sequences. Find the remainder when $N$ is divided by $1000$.
[i]Based on a proposal by Gyumin Roh[/i]
2016 NIMO Summer Contest, 9
Compute the number of real numbers $t$ such that \[t = 50 \sin(t - \lfloor t \rfloor).\] Here $\lfloor \cdot\rfloor$ denotes the greatest integer function.
[i]Proposed by David Altizio[/i]
2015 NIMO Problems, 5
Let $a$, $b$, $c$ be positive integers and $p$ be a prime number. Assume that \[ a^n(b+c)+b^n(a+c)+c^n(a+b)\equiv 8\pmod{p} \] for each nonnegative integer $n$. Let $m$ be the remainder when $a^p+b^p+c^p$ is divided by $p$, and $k$ the remainder when $m^p$ is divided by $p^4$. Find the maximum possible value of $k$.
[i]Proposed by Justin Stevens and Evan Chen[/i]
2015 NIMO Summer Contest, 4
Let $P$ be a function defined by $P(t)=a^t+b^t$, where $a$ and $b$ are complex numbers. If $P(1)=7$ and $P(3)=28$, compute $P(2)$.
[i] Proposed by Justin Stevens [/i]
2016 NIMO Problems, 7
Determine the number of odd integers $1 \le n \le 100$ with the property that
\[
\sum_{\substack{1 \le k \le n \\ \gcd(k,n) = 1}} \cos\left(\frac{2\pi k}{n} \right) = 1
\quad\text{and}\quad
\sum_{\substack{1 \le k \le n \\ \gcd(k,n) = 1}} \sin\left(\frac{2\pi k}{n} \right) = 0.
\]
[i]Based on a proposal by Mayank Pandey[/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]
2015 NIMO Problems, 3
Let $O$, $A$, $B$, and $C$ be points in space such that $\angle AOB=60^{\circ}$, $\angle BOC=90^{\circ}$, and $\angle COA=120^{\circ}$. Let $\theta$ be the acute angle between planes $AOB$ and $AOC$. Given that $\cos^2\theta=\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $100m+n$.
[i]Proposed by Michael Ren[/i]
2015 NIMO Summer Contest, 6
Let $S_0 = \varnothing$ denote the empty set, and define $S_n = \{ S_0, S_1, \dots, S_{n-1} \}$ for every positive integer $n$. Find the number of elements in the set
\[ (S_{10} \cap S_{20}) \cup (S_{30} \cap S_{40}). \]
[i] Proposed by Evan Chen [/i]
2012 NIMO Problems, 8
Bob has invented the Very Normal Coin (VNC). When the VNC is flipped, it shows heads $\textstyle\frac{1}{2}$ of the time and tails $\textstyle\frac{1}{2}$ of the time - unless it has yielded the same result five times in a row, in which case it is guaranteed to yield the opposite result. For example, if Bob flips five heads in a row, then the next flip is guaranteed to be tails.
Bob flips the VNC an infinite number of times. On the $n$th flip, Bob bets $2^{-n}$ dollars that the VNC will show heads (so if the second flip shows heads, Bob wins $\$0.25$, and if the third flip shows tails, Bob loses $\$0.125$).
Assume that dollars are infinitely divisible. Given that the first flip is heads, the expected number of dollars Bob is expected to win can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a, b$. Compute $100a + b$.
[i]Proposed by Lewis Chen[/i]
2014 NIMO Problems, 3
In triangle $ABC$, we have $AB=AC=20$ and $BC=14$. Consider points $M$ on $\overline{AB}$ and $N$ on $\overline{AC}$. If the minimum value of the sum $BN + MN + MC$ is $x$, compute $100x$.
[i]Proposed by Lewis Chen[/i]
2016 NIMO Problems, 5
In a chemistry experiment, a tube contains 100 particles, 68 on the right and 32 on the left. Each second, if there are $a$ particles on the left side of the tube, some number $n$ of these particles move to the right side, where $n \in \{0,1,\dots,a\}$ is chosen uniformly at random. In a similar manner, some number of the particles from the right side of the tube move to the left, at the same time. The experiment ends at the moment when all particles are on the same side of the tube. The probability that all particles end on the left side is $\tfrac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100a+b$.
[i]Proposed by Alvin Zou[/i]
2016 NIMO Problems, 3
Right triangle $ABC$ has hypotenuse $AB = 26$, and the inscribed circle of $ABC$ has radius $5$. The largest possible value of $BC$ can be expressed as $m + \sqrt{n}$, where $m$ and $n$ are both positive integers. Find $100m + n$.
[i]Proposed by Jason Xia[/i]
2015 NIMO Summer Contest, 10
Let $ABCD$ be a tetrahedron with $AB=CD=1300$, $BC=AD=1400$, and $CA=BD=1500$. Let $O$ and $I$ be the centers of the circumscribed sphere and inscribed sphere of $ABCD$, respectively. Compute the smallest integer greater than the length of $OI$.
[i] Proposed by Michael Ren [/i]
2015 NIMO Summer Contest, 11
We say positive integer $n$ is $\emph{metallic}$ if there is no prime of the form $m^2-n$. What is the sum of the three smallest metallic integers?
[i] Proposed by Lewis Chen [/i]
2017 NIMO Problems, 2
Let $\{a_n\}$ be a sequence of integers such that $a_1=2016$ and \[\dfrac{a_{n-1}+a_n}2=n^2-n+1\] for all $n\geq 1$. Compute $a_{100}$.
[i]Proposed by David Altizio[/i]
2015 NIMO Summer Contest, 5
Let $\triangle ABC$ have $AB=3$, $AC=5$, and $\angle A=90^\circ$. Point $D$ is the foot of the altitude from $A$ to $\overline{BC}$, and $X$ and $Y$ are the feet of the altitudes from $D$ to $\overline{AB}$ and $\overline{AC}$ respectively. If $XY^2$ can be written in the form $\tfrac mn$ where $m$ and $n$ are positive relatively prime integers, what is $100m+n$?
[i] Proposed by David Altizio [/i]