Found problems: 27
2016 NIMO Summer Contest, 12
Let $p$ be a prime. It is given that there exists a unique nonconstant function $\chi:\{1,2,\ldots, p-1\}\to\{-1,1\}$ such that $\chi(1) = 1$ and $\chi(mn) = \chi(m)\chi(n)$ for all $m, n \not\equiv 0 \pmod p$ (here the product $mn$ is taken mod $p$). For how many positive primes $p$ less than $100$ is it true that \[\sum_{a=1}^{p-1}a^{\chi(a)}\equiv 0\pmod p?\] Here as usual $a^{-1}$ denotes multiplicative inverse.
[i]Proposed by David Altizio[/i]
2015 NIMO Summer Contest, 9
On a blackboard lies $50$ magnets in a line numbered from $1$ to $50$, with different magnets containing different numbers. David walks up to the blackboard and rearranges the magnets into some arbitrary order. He then writes underneath each pair of consecutive magnets the positive difference between the numbers on the magnets. If the expected number of times he writes the number $1$ can be written in the form $\tfrac mn$ for relatively prime positive integers $m$ and $n$, compute $100m+n$.
[i] Proposed by David Altizio [/i]
2015 NIMO Summer Contest, 7
The NIMO problem writers have invented a new chess piece called the [i]Oriented Knight[/i]. This new chess piece has a limited number of moves: it can either move two squares to the right and one square upward or two squares upward and one square to the right. How many ways can the knight move from the bottom-left square to the top-right square of a $16\times 16$ chess board?
[i] Proposed by Tony Kim and David Altizio [/i]
2015 NIMO Summer Contest, 1
For all real numbers $a$ and $b$, let \[a\Join b=\dfrac{a+b}{a-b}.\] Compute $1008\Join 1007$.
[i] Proposed by David Altizio [/i]
2015 NIMO Summer Contest, 8
It is given that the number $4^{11}+1$ is divisible by some prime greater than $1000$. Determine this prime.
[i] Proposed by David Altizio [/i]
2017 NIMO Summer Contest, 2
Joy has $33$ thin rods, one each of every integer length from $1$ cm through $30$ cm, and also three more rods with lengths $3$ cm, $7$ cm, and $15$ cm. She places those three rods on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How many of the remaining rods can she choose as the fourth rod?
[i]Proposed by Michael Tang[/i]
2015 NIMO Summer Contest, 2
On a 30 question test, Question 1 is worth one point, Question 2 is worth two points, and so on up to Question 30. David takes the test and afterward finds out he answered nine of the questions incorrectly. However, he was not told which nine were incorrect. What is the highest possible score he could have attained?
[i] Proposed by David Altizio [/i]
2017 NIMO Summer Contest, 3
If $p$, $q$, and $r$ are nonzero integers satisfying \[p^2+q^2 = r^2,\] compute the smallest possible value of $(p+q+r)^2$.
[i]Proposed by David Altizio[/i]
2017 NIMO Summer Contest, 6
Let $P = (-2, 0)$. Points $P$, $Q$, $R$ lie on the graph of the function $y = x^3 - 3x + 2$ such that $Q$ is the midpoint of segment $PR$. Compute $PR^2$.
[i]Proposed by David Altizio[/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 Summer Contest, 4
The square $BCDE$ is inscribed in circle $\omega$ with center $O$. Point $A$ is the reflection of $O$ over $B$. A "hook" is drawn consisting of segment $AB$ and the major arc $\widehat{BE}$ of $\omega$ (passing through $C$ and $D$). Assume $BCDE$ has area $200$. To the nearest integer, what is the length of the hook?
[i]Proposed by Evan Chen[/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]
2017 NIMO Summer Contest, 8
Konsistent Karl is taking this contest. He can solve the first five problems in one minute each, the next five in two minutes each, and the last five in three minutes each. What is the maximum possible score Karl can earn? (Recall that this contest is $15$ minutes long, there are $15$ problems, and the $n$th problem is worth $n$ points. Assume that entering answers and moving between or skipping problems takes no time.)
[i]Proposed by Michael Tang[/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 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 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]
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]
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]
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]
2017 NIMO Summer Contest, 5
Find the smallest positive integer $n$ for which the number \[ A_n = \prod_{k=1}^n \binom{k^2}{k} = \binom{1}{1} \binom{4}{2} \cdots \binom{n^2}{n} \] ends in the digit $0$ when written in base ten.
[i]Proposed by Evan Chen[/i]
2017 NIMO Summer Contest, 14
Let $x, y, z$ be real numbers such that $x+y+z=-2$ and
\[\begin{aligned}
& (x^2+xy+y^2)(y^2+yz+z^2) \\
&+ (y^2+yz+z^2)(z^2+zx+x^2) \\
&+ (z^2+zx+x^2)(x^2+xy+y^2) \\
& = 625+ \tfrac34(xy+yz+zx)^2. \end{aligned}\]
Compute $|xy+yz+zx|$.
[i]Proposed by Michael Tang[/i]
2015 NIMO Summer Contest, 14
We say that an integer $a$ is a quadratic, cubic, or quintic residue modulo $n$ if there exists an integer $x$ such that $x^2\equiv a \pmod n$, $x^3 \equiv a \pmod n$, or $x^5 \equiv a \pmod n$, respectively. Further, an integer $a$ is a primitive residue modulo $n$ if it is exactly one of these three types of residues modulo $n$.
How many integers $1 \le a \le 2015$ are primitive residues modulo $2015$?
[i] Proposed by Michael Ren [/i]
2017 NIMO Summer Contest, 7
Let $S$ be the maximum possible value of \[\frac{a}{b^3+4}+\frac{b}{c^3+4}+\frac{c}{d^3+4}+\frac{d}{a^3+4} \] given that $a$, $b$, $c$, $d$ are nonnegative real numbers such that $a+b+c+d=4$. Given that $S$ can be written in the form $m/n$ where $m,n$ are coprime positive integers, find $100m+n$.
[i]Proposed by Kaan Dokmeci[/i]
2017 NIMO Summer Contest, 10
In triangle $ABC$ we have $AB=36$, $BC=48$, $CA=60$. The incircle of $ABC$ is centered at $I$ and touches $AB$, $AC$, $BC$ at $M$, $N$, $D$, respectively. Ray $AI$ meets $BC$ at $K$. The radical axis of the circumcircles of triangles $MAN$ and $KID$ intersects lines $AB$ and $AC$ at $L_1$ and $L_2$, respectively. If $L_1L_2 = x$, compute $x^2$.
[i]Proposed by Evan Chen[/i]