Difference between revisions of "2007 iTest Problems/Ultimate Question"

The following questions are from the "Ultimate Question" of the 2007 iTest, but with the T-values substituted. This is for people who want to work on the problems without having to solve the previous problems.

Problem 51

Find the highest point (largest possible $y$-coordinate) on the parabola $$y=-2x^2+ 28x+ 418$$

Problem 52

Let $R$ be the region consisting of points $(x,y)$ of the Cartesian plane satisfying both $|x|-|y|\le 16$ and $|y|\le 16$. Find the area of region $R$.

Problem 53

Three distinct positive Fibonacci numbers, all greater than $1536$, are in arithmetic progression. Let $N$ be the smallest possible value of their sum. Find the remainder when $N$ is divided by $2007$.

Problem 54

Consider the sequence $(1, 2007)$. Inserting the difference between $1$ and $2007$ between them, we get the sequence $(1, 2006, 2007)$. Repeating the process of inserting differences between numbers, we get the sequence $(1, 2005, 2006, 1, 2007)$. A third iteration of this process results in $(1, 2004, 2005, 1, 2006, 2005, 1, 2006, 2007)$. A total of $2007$ iterations produces a sequence with $2^{2007}+1$ terms. If the integer $2004$ appears a total of $N$ times among these $2^{2007}+1$ terms, find the remainder when $N$ gets divided by $2007$.

Problem 55

Let $R=675$. Let $x$ be the smallest real solution of $3x^2+Rx+R=90x\sqrt{x+1}$. Find the value of $\lfloor x\rfloor$.

Problem 56

In the binary expansion of $\dfrac{2^{2007}-1}{2^225-1}$, how many of the first $10,000$ digits to the right of the radix point are $0$'s?

Problem 57

How many positive integers are within $810$ of exactly $\lfloor \sqrt{810} \rfloor$ perfect squares? (Note: $0^2=0$ is considered a perfect square.)

Problem 58

For natural numbers $k,n\geq 2$, we define $$S(k,n)=\left\lfloor\frac{2^{n+1}+1}{2^{n-1}+1}\right\rfloor+\left\lfloor\frac{3^{n+1}+1}{3^{n-1}+1}\right\rfloor+\cdots+\left\lfloor\frac{k^{n+1}+1}{k^{n-1}+1}\right\rfloor$$ Compute the value of $S(10,112)-S(10,55)+S(10,2)$.

Problem 59

Fermi and Feynman play the game $\textit{Probabicloneme}$ in which Fermi wins with probability $a/b$, where $a$ and $b$ are relatively prime positive integers such that $a/b<1/2$. The rest of the time Feynman wins (there are no ties or incomplete games). It takes a negligible amount of time for the two geniuses to play $\textit{Probabicloneme}$ so they play many many times. Assuming they can play infinitely many games (eh, they're in Physicist Heaven, we can bend the rules), the probability that they are ever tied in total wins after they start (they have the same positive win totals) is $1/11$. Find the value of $a$.

Problem 60

Triangle $ABC$ has $AB=99$ and $AC=120$. Point $D$ is on $BC$ so that $AD$ bisects angle $BAC$. The circle through $A, B$, and $D$ has center $O_1$ and intersects line $AC$ again at $B'$, and likewise the circle through $A, C$, and $D$ has center $O_2$ and intersects line $AB$ again at $C'$. If the four points $B', C', O_1$, and $O_2$ lie on a circle, find the length of $BC$.