Found problems: 54
2007 iTest Tournament of Champions, 5
Acute triangle $ABC$ has altitudes $AD$, $BE$, and $CF$. Point $D$ is projected onto $AB$ and $AC$ to points $D_c$ and $D_b$ respectively. Likewise, $E$ is projected to $E_a$ on $BC$ and $E_c$ on $AB$, and $F$ is projected to $F_a$ on $BC$ and $F_b$ on $AC$. Lines $D_bD_c$, $E_cE_a$, $F_aF_b$ bound a triangle of area $T_1$, and lines $E_cF_b$, $D_bE_a$, $F_aD_c$ bound a triangle of area $T_2$. What is the smallest possible value of the ratio $T_2/T_1$?
2007 iTest Tournament of Champions, 2
Let $m$ be the maximum possible value of $x^{16} + \frac{1}{x^{16}}$, where \[x^6 - 4x^4 - 6x^3 - 4x^2 + 1=0.\] Find the remainder when $m$ is divided by $2007$.
2007 iTest Tournament of Champions, 1
Find the remainder when $3^{2007}$ is divided by $2007$.
2008 iTest Tournament of Champions, 4
Each of the $24$ students in Mr. Friedman's class cut up a $7\times 7$ grid of squares while he read them short stories by Mark Twain. While not all of the students cut their squares up in the same way, each of them cut their $7\times 7$ square into at most the three following types (shapes) of pieces.
[asy]
size(350);
defaultpen(linewidth(0.8));
real r = 4.5, s = 9;
filldraw(origin--(2,0)--(2,1)--(1,1)--(1,2)--(0,2)--cycle,blue);
draw((0,1)--(1,1)--(1,0));
filldraw((r,0)--(r+2,0)--(r+2,2)--(r,2)--cycle,green);
draw((r+1,0)--(r+1,2)^^(r,1)--(r+2,1));
filldraw((s,0)--(s+2,0)--(s+2,1)--(s+3,1)--(s+3,2)--(s+1,2)--(s+1,1)--(s,1)--cycle,red);
draw((s+1,0)--(s+1,1)--(s+2,1)--(s+2,2));
[/asy]
Let $a$, $b$, and $c$ be the number of total pieces of each type from left to right respectively after all $24$ $7\times 7$ squares are cut up. How many ordered triples $(a,b,c)$ are possible?
2008 iTest Tournament of Champions, 2
Let $A$ be the number of $12$-digit words that can be formed by from the alphabet $\{0,1,2,3,4,5,6\}$ if each pair of neighboring digits must differ by exactly $1$. Find the remainder when $A$ is divided by $2008$.
2007 iTest Tournament of Champions, 5
Convex quadrilateral $ABCD$ has the property that the circles with diameters $AB$ and $CD$ are tangent at point $X$ inside the quadrilateral, and likewise, the circles with diameters $BC$ and $DA$ are tangent at a point $Y$ inside the quadrilateral. Given that the perimeter of $ABCD$ is $96$, and the maximum possible length of $XY$ is $m$, find $\lfloor 2007m\rfloor$.
2008 iTest Tournament of Champions, 4
Let \[f(n) = \sum_{k=0}^{\lfloor n/2\rfloor}(-1)^k\frac{1}{n-k}\binom{n-k}k,\] for each positive integer $n$. If $|f(2007) + f(2008)| = a/b$ for relatively prime positive integers $a$ and $b$, find the remainder when $a$ is divded by $1000$.
2008 iTest Tournament of Champions, 1
Find the smallest positive integer $n$ such that there are at least three distinct ordered pairs $(x,y)$ of positive integers such that \[x^2-y^2=n.\]
2008 iTest Tournament of Champions, 4
Euclid places a morsel of food at the point $(0,0)$ and an ant at the point $(1,2)$. Every second, the ant walks one unit in one of the four coordinate directions. However, whenever the ant moves to $(x,\pm 3)$, Euclid's malicious brother Mobius picks it up and puts it at $(-x,\mp 2)$, and whenever it moves to $(\pm 2,y)$, his cousin Klein puts it at $(\mp 1,y)$. If $p$ and $q$ are relatively prime positive integers such that $\tfrac pq$ is the expected number of steps the ant takes before reaching the food, find $p+q$.
2007 iTest Tournament of Champions, 2
In the game of [i]Winners Make Zeros[/i], a pair of positive integers $(m,n)$ is written on a sheet of paper. Then the game begins, as the players make the following legal moves:
[list]
[*] If $m\geq n$, the player choose a positive integer $c$ such that $m-cn\geq 0$, and replaces $(m,n)$ with $(m-cn,n)$.
[*] If $m<n$, the player choose a positive integer $c$ such that $n-cm\geq 0$, and replaces $(m,n)$ with $(m,n-cm)$.
[/list]
When $m$ or $n$ becomes $0$, the game ends, and the last player to have moved is declared the winner. If $m$ and $n$ are originally $2007777$ and $2007$, find the largest choice the first player can make for $c$ (on his first move) such that the first player has a winning strategy after that first move.
2007 iTest Tournament of Champions, 1
Given that \begin{align*}x &= 1 - \frac 12 + \frac13 - \frac 14 + \cdots + \frac1{2007},\\ y &= \frac{1}{1005} + \frac{1}{1006} + \frac{1}{1007} + \cdots + \frac 1{2007},\end{align*}
find the value of $k$ such that \[x = y + \frac 1k.\]
2007 iTest Tournament of Champions, 1
Let $A$ be the area of the locus of points $z$ in the complex plane that satisfy $|z+12+9i| \leq 15$. Compute $\lfloor A\rfloor$.
2007 iTest Tournament of Champions, 2
Let \[S = 1 + \frac 18 + \frac{1\cdot 5}{8\cdot 16} + \frac{1\cdot 5\cdot 9}{8\cdot 16\cdot 24} + \cdots + \frac{1\cdot 5\cdot 9\cdots (4k+1)}{8\cdot 16\cdot 24\cdots(8k+8)} + \cdots.\] Find the positive integer $n$ such that $2^n < S^{2007} < 2^{n+1}$.
2007 iTest Tournament of Champions, 3
Find the smallest value of $n$ for which the series \[1\cdot 3^1 + 2\cdot 3^2 + 3\cdot 3^3 + \cdots + n\cdot 3^n\] exceeds $3^{2007}$.
2008 iTest Tournament of Champions, 4
If $m$ is a positive integer, let $S_m$ be the set of rational numbers in reduced form with denominator at most $m$. Let $f(m)$ be the sum of the numerator and denominator of the element of $S_m$ closest to $e$ (Euler's constant). Given that $f(2007) = 3722$, find the remainder when $f(1000)$ is divided by $2008$.
2008 iTest Tournament of Champions, 5
Two squares of side length $2$ are glued together along their boundary so that the four vertices of the first square are glued to the midpoints of the four sides of the other square, and vice versa. This gluing results in a convex polyhedron. If the square of the volume of this polyhedron is written in simplest form as $\tfrac{a+b\sqrt c}d$, what is the value of $a+b+c+d$?
2008 iTest Tournament of Champions, 1
Yatta and Yogi play a game in which they begin with a pile of $n$ stones. The players take turns removing $1$, $2$, $3$, $5$, $6$, $7$, or $8$ stones from the pile. That is, when it is a player's turn to remove stones, that player may remove from $1$ to $8$ stones, but [i]cannot[/i] remove exactly $4$ stones. The player who removes the last stone [i]loses[/i]. Yogi goes first and finds that he has a winning position, meaning that so long as he plays perfectly, Yatta cannot defeat him. For how many positive integers $n$ from $100$ to $2008$ inclusive is this the case?
2008 iTest Tournament of Champions, 5
While running from an unrealistically rendered zombie, Willy Smithers runs into a vacant lot in the shape of a square, $100$ meters on a side. Call the four corners of the lot corners $1$, $2$, $3$, and $4$, in clockwise order. For $k = 1, 2, 3, 4$, let $d_k$ be the distance between Willy and corner $k$. Let
(a) $d_1<d_2<d_4<d_3$,
(b) $d_2$ is the arithmetic mean of $d_1$ and $d_3$, and
(c) $d_4$ is the geometric mean of $d_2$ and $d_3$.
If $d_1^2$ can be written in the form $\dfrac{a-b\sqrt c}d$, where $a,b,c,$ and $d$ are positive integers, $c$ is square-free, and the greatest common divisor of $a$, $b$, and $d$ is $1,$ find the remainder when $a+b+c+d$ is divided by $2008$.
2008 iTest Tournament of Champions, 2
Find the value of $|xy|$ given that $x$ and $y$ are integers and \[6x^2y^2+5x^2-18y^2=17253.\]
2007 iTest Tournament of Champions, 5
A polynomial $p(x)$ of degree $1000$ is such that $p(n) = (n+1)2^n$ for all nonnegative integers $n$ such that $n\leq 1000$. Given that \[p(1001) = a\cdot 2^b - c,\] where $a$ is an odd integer, and $0 < c < 2007$, find $c-(a+b)$.
2007 iTest Tournament of Champions, 5
Let $s=a+b+c$, where $a$, $b$, and $c$ are integers that are lengths of the sides of a box. The volume of the box is numerically equal to the sum of the lengths of the twelve edges of the box plus its surface area. Find the sum of the possible values of $s$.
2008 iTest Tournament of Champions, 4
Find the maximum of $x+y$ given that $x$ and $y$ are positive real numbers that satisfy \[x^3+y^3+(x+y)^3+36xy=3456.\]
2007 iTest Tournament of Champions, 3
For each positive integer $n$, let $g(n)$ be the sum of the digits when $n$ is written in binary. For how many positive integers $n$, where $1\leq n\leq 2007$, is $g(n)\geq 3$?
2007 iTest Tournament of Champions, 3
Find the real number $k$ such that $a$, $b$, $c$, and $d$ are real numbers that satisfy the system of equations
\begin{align*}
abcd &= 2007,\\
a &= \sqrt{55 + \sqrt{k+a}},\\
b &= \sqrt{55 - \sqrt{k+b}},\\
c &= \sqrt{55 + \sqrt{k-c}},\\
d &= \sqrt{55 - \sqrt{k-d}}.
\end{align*}
2007 iTest Tournament of Champions, 4
Find the smallest positive integer $k$ such that \[(16a^2 + 36b^2 + 81c^2)(81a^2 + 36b^2 + 16c^2) < k(a^2 + b^2 + c^2)^2,\] for some ordered triple of positive integers $(a,b,c)$.