Found problems: 54
2008 iTest Tournament of Champions, 1
Find the remainder when $712!$ is divided by $719$.
2008 iTest Tournament of Champions, 2
Note that there are exactly three ways to write the integer $4$ as a sum of positive odd integers where the order of the summands matters:
\begin{align*}
1+1+1+1&=4,\\
1+3&=4,\\
3+1&=4.
\end{align*}
Let $f(n)$ be the number of ways to write a natural number $n$ as a sum of positive odd integers where the order of the summands matters. Find the remainder when $f(2008)$ is divided by $100$.
2007 iTest Tournament of Champions, 4
Bobby Fisherman played a tournament in which he played $2009$ players. He either won or lost every game. He lost his first two games, but won $2002$ total games. At the conclusion of each game, he computed his exact winning percentage at that moment. Let $w_1,w_2,\ldots, w_{2009}$ be his winning percentages after games $1$, $2$, $\ldots$, $2009$ respectively. There are some real numbers, such as $0$, which are necessarily members of the set $W = \{w_1,w_2,\ldots, w_{2009}\}$. How many positive real numbers are necessarily elements of set $W$, regardless of the order in which he won or lost his games?
2007 iTest Tournament of Champions, 4
Let $x_1,x_2,\ldots, x_{2007}$ be real numbers such that $-1\leq x_i\leq 1$ for $1\leq i\leq 2007$, and \[\sum_{i=1}^{2007}x_i^3 = 0.\] Find the maximum possible value of $\Big\lfloor\sum_{i=1}^{2007}x_i\Big\rfloor$.
2008 iTest Tournament of Champions, 3
The $260$ volumes of the [i]Encyclopedia Galactica[/i] are out of order in the library. Fortunately for the librarian, the books are numbered. Due to his religion, which holds both encyclopedias and the concept of parity in high esteem, the librarian can only rearrange the books two at a time, and then only by switching the position of an even numbered volume with that of an odd numbered volume. Find the minimum number of such transpositions sufficient to get the books back into ordinary sequential order, regardless of the starting positions of the books. (Find the minimum number of transpositions in the worst-case scenario.)
2007 iTest Tournament of Champions, 3
Find the largest natural number $n$ such that \[2^n + 2^{11} + 2^8\] is a perfect square.
2008 iTest Tournament of Champions, 1
Let \[X = \cos\frac{2\pi}7 + \cos\frac{4\pi}7 + \cos\frac{6\pi}7 + \cdots + \cos\frac{2006\pi}7 + \cos\frac{2008\pi}7.\] Compute $\Big|\lfloor 2008 X\rfloor\Big|$.
2007 iTest Tournament of Champions, 4
In triangle $ABC$, points $A'$, $B'$, and $C'$ are chosen with $A'$ on segment $AB$, $B'$ on segment $BC$, and $C'$ on segment $CA$ so that triangle $A'B'C'$ is directly similar to $ABC$. The incenters of $ABC$ and $A'B'C'$ are $I$ and $I'$ respectively. Lines $BC$, $A'C'$, and $II'$ are parallel. If $AB=100$ and $AC=120$, what is the length of $BC$?
2007 iTest Tournament of Champions, 3
A sequence $a_1,a_2,a_3,\ldots$ is defined as follows: $a_1 = 2007$, and $a_n = a_{n-1} + n\pmod k$, where $0\leq a_n< k$. For how many values of $k$, where $2007 < k < 10^{12}$, does the sequence assume all $k$ possible values (modulo $k$ residues)?
2008 iTest Tournament of Champions, 5
It is well-known that the $n^{\text{th}}$ triangular number can be given by the formula $n(n+1)/2$. A Pythagorean triple of $\textit{square numbers}$ is an ordered triple $(a,b,c)$ such that $a^2+b^2=c^2$. Let a Pythagorean triple of $\textit{triangular numbers}$ (a PTTN) be an ordered triple of positive integers $(a,b,c)$ such that $a\leq b<c$ and
\[\dfrac{a(a+1)}2+\dfrac{b(b+1)}2=\dfrac{c(c+1)}2.\]
For instance, $(3,5,6)$ is a PTTN ($6+15=21$). Here we call both $a$ and $b$ $\textit{legs}$ of the PTTN. Find the smallest natural number $n$ such that $n$ is a leg of $\textit{at least}$ six distinct PTTNs.
2008 iTest Tournament of Champions, 5
For positive integers $m,n\geq 3$, let $h(m,n)$ be the maximum (finite) number of intersection points between a simple $m$-gon and a simple $n$-gon. (Note: a polygon is simple if it does not intersect itself.) Evaluate \[\sum_{m=3}^6\sum_{n=3}^{12}h(m,n).\]
2007 iTest Tournament of Champions, 5
Let $c$ be the number of ways to choose three vertices of an $6$-dimensional cube that form an equilateral triangle. Find the remainder when $c$ is divided by $2007$.
2007 iTest Tournament of Champions, 2
The area of triangle $ABC$ is $2007$. One of its sides has length $18$, and the tangent of the angle opposite that side is $2007/24832$. When the altitude is dropped to the side of length $18$, it cuts that side into two segments. Find the sum of the squares of those two segments.
2008 iTest Tournament of Champions, 3
Arthur stands on a circle drawn with chalk in a parking lot. It is sunrise and there are birds in the trees nearby. He stands on one of five triangular nodes that are spaced equally around the circle, wondering if and when the aliens will pick him up and carry him from the node he is standing on. He flips a fair coin $12$ times, each time chanting the name of a nearby star system. Each time he flips a head, he walks around the circle, in the direction he is facing, until he reaches the next node in that direction. Each time he flips a tail, he reverses direction, then walks around the circle until he reaches the next node in that new direction. After $12$ flips, Arthur finds himself on the node at which he started. He thinks this is fate, but Arthur is quite mistaken. If $a$ and $b$ are relatively prime positive integers such that $a/b$ is the probability that Arthur flipped exactly $6$ heads, find $a+b$.
2008 iTest Tournament of Champions, 1
Let $a$, $b$, $c$, and $d$ be positive real numbers such that $abcd=17$. Let $m$ be the minimum possible value of \[a^2+b^2+c^2+a(b+c+d) + b(c+d) + cd.\] Compute $\lfloor 17m\rfloor$.
2007 iTest Tournament of Champions, 1
Find the smallest positive integer $n$ such that a cube with sides of length $n$ can be divided up into exactly $2007$ smaller cubes, each of whose sides is of integer length.
2007 iTest Tournament of Champions, 2
Al and Bill play a game involving a fair six-sided die. The die is rolled until either there is a number less than $5$ rolled on consecutive tosses, or there is a number greater than $4$ on consecutive tosses. Al wins if the last roll is a $5$ or $6$. Bill wins if the last roll is a $2$ or lower. Let $m$ and $n$ be relatively prime positive integers such that $m/n$ is the probability that Bill wins. Find the value of $m+n$.
2008 iTest Tournament of Champions, 3
Let $\phi = \tfrac{1+\sqrt 5}2$ be the positive root of $x^2=x+1$. Define a function $f:\mathbb N\to\mathbb N$ by
\begin{align*}
f(0) &= 1\\
f(2x) &= \lfloor\phi f(x)\rfloor\\
f(2x+1) &= f(2x) + f(x).
\end{align*}
Find the remainder when $f(2007)$ is divided by $2008$.
2007 iTest Tournament of Champions, 3
Find the sum of all integers $n$ such that \[n^4+n^3+n^2+n+1\] is a perfect square.
2007 iTest Tournament of Champions, 1
A fair $20$-sided die has faces numbered $1$ through $20$. The die is rolled three times and the outcomes are recorded. If $a$ and $b$ are relatively prime integers such that $a/b$ is the probability that the three recorded outcomes can be the sides of a triangle with positive area, find $a+b$.
2007 iTest Tournament of Champions, 4
For each positive integer $n$, let $S_n = \sum_{k=1}^nk^3$, and let $d(n)$ be the number of positive divisors of $n$. For how many positive integers $m$, where $m\leq 25$, is there a solution $n$ to the equation $d(S_n) = m$?
2008 iTest Tournament of Champions, 2
Jon wrote the $n$ smallest perfect squares on one sheet of paper, and the $n$ smallest triangular numbers on another (note that $0$ is both square and triangular). Jon notices that there are the same number of triangular numbers on the first paper as there are squares on the second paper, but if $n$ had been one smaller, this would not have been true. If $n < 2008$, let $m$ be the greatest number Jon could have written on either paper. Find the remainder when $m$ is divided by $2008$.
2007 iTest Tournament of Champions, 5
Find the largest possible value of $a+b$ less than or equal to $2007$, for which $a$ and $b$ are relatively prime, and such that there is some positive integer $n$ for which \[\frac{2^3-1}{2^3+1}\cdot\frac{3^3-1}{3^3+1}\cdot\frac{4^3-1}{4^3+1}\cdots\frac{n^3-1}{n^3+1} = \frac ab.\]
2008 iTest Tournament of Champions, 5
Three circles with centers $V_0$, $V_1$, $V_2$ and radii $33$, $30$, $25$ respectively and mutually externally tangent: $P_i$ is the tangency point between circles $V_{i+1}$ and $V_{i+2}$, where indeces are taken modulo $3$. For $i=0,1,2$, line $P_{i+1}P_{i+2}$ intersects circle $V_{i+1}$ at $P_{i+2}$ and $Q_i$, and the same line intersects circle $V_{i+2}$ at $P_{i+1}$ and $R_i$. If $Q_0R_1$ intersects $Q_2R_0$ at $X$, then the distance from $X$ to line $R_1Q_2$ can be expressed as $\tfrac{a\sqrt b}c$, where the integer $b$ is not divisible by the square of any prime, and positive integers $a$ and $c$ are relatively prime. Find the value of $b+c$.
2007 iTest Tournament of Champions, 2
Let $a/b$ be the probability that a randomly chosen positive divisor of $12^{2007}$ is also a divisor of $12^{2000}$, where $a$ and $b$ are relatively prime positive integers. Find the remainder when $a+b$ is divided by $2007$.