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

(UQ from 2007 iTest, but with T-values substituted)
 
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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.
 
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.
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===Problem 51===
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Find the highest point (largest possible <math>y</math>-coordinate) on the parabola
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<cmath>y=-2x^2+ 28x+ 418</cmath>
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[[2007 iTest Problems/Problem 51|Solution]]
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===Problem 52===
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Let <math>R</math> be the region consisting of points <math>(x,y)</math> of the Cartesian plane satisfying both
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<math>|x|-|y|\le 16</math> and <math>|y|\le 16</math>. Find the area of region <math>R</math>.
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[[2007 iTest Problems/Problem 52|Solution]]
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===Problem 53===
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Three distinct positive Fibonacci numbers, all greater than <math>1536</math>, are in arithmetic progression. Let <math>N</math> be the smallest possible value of their sum. Find the remainder when <math>N</math> is divided by <math>2007</math>.
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[[2007 iTest Problems/Problem 53|Solution]]
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===Problem 54===
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Consider the sequence <math>(1, 2007)</math>. Inserting the difference between <math>1</math> and <math>2007</math> between them, we get the sequence <math>(1, 2006, 2007)</math>. Repeating the process of inserting differences between numbers, we get the sequence <math>(1, 2005, 2006, 1, 2007)</math>. A third iteration of this process results in <math>(1, 2004, 2005, 1, 2006, 2005, 1, 2006, 2007)</math>. A total of <math>2007</math> iterations produces a sequence with <math>2^{2007}+1</math> terms. If the integer <math>2004</math> appears a total of <math>N</math> times among these <math>2^{2007}+1</math> terms, find the remainder when <math>N</math> gets divided by <math>2007</math>.
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[[2007 iTest Problems/Problem 54|Solution]]
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===Problem 55===
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Let <math>R=675</math>. Let <math>x</math> be the smallest real solution of <math>3x^2+Rx+R=90x\sqrt{x+1}</math>. Find the value of <math>\lfloor x\rfloor</math>.
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[[2007 iTest Problems/Problem 55|Solution]]
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===Problem 56===
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In the binary expansion of <math>\dfrac{2^{2007}-1}{2^{225}-1}</math>, how many of the first <math>10,000</math> digits to the right of the radix point are <math>0</math>'s?
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[[2007 iTest Problems/Problem 56|Solution]]
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===Problem 57===
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How many positive integers are within <math>810</math> of exactly <math>\lfloor \sqrt{810} \rfloor</math> perfect squares? (Note: <math>0^2=0</math> is considered a perfect square.)
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[[2007 iTest Problems/Problem 57|Solution]]
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===Problem 58===
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For natural numbers <math>k,n\geq 2</math>, we define
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<cmath>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</cmath>
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Compute the value of <math>S(10,112)-S(10,55)+S(10,2)</math>.
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[[2007 iTest Problems/Problem 58|Solution]]
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===Problem 59===
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Fermi and Feynman play the game <math>\textit{Probabicloneme}</math> in which Fermi wins with probability <math>a/b</math>, where <math>a</math> and <math>b</math> are relatively prime positive integers such that <math>a/b<1/2</math>. 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 <math>\textit{Probabicloneme}</math> 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 <math>1/11</math>. Find the value of <math>a</math>.
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[[2007 iTest Problems/Problem 59|Solution]]
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===Problem 60===
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Triangle <math>ABC</math> has <math>AB=99</math> and <math>AC=120</math>. Point <math>D</math> is on <math>BC</math> so that <math>AD</math> bisects angle <math>BAC</math>. The circle through <math>A, B</math>, and <math>D</math> has center <math>O_1</math> and intersects line <math>AC</math> again at <math>B'</math>, and likewise the circle through <math>A, C</math>, and <math>D</math> has center <math>O_2</math> and intersects line <math>AB</math> again at <math>C'</math>. If the four points <math>B', C', O_1</math>, and <math>O_2</math> lie on a circle, find the length of <math>BC</math>.
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[[2007 iTest Problems/Problem 60|Solution]]

Latest revision as of 23:41, 24 June 2018

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 \[y=-2x^2+ 28x+ 418\]

Solution

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 $R$.

Solution

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 $2007$.

Solution

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 $2007$.

Solution

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 $\lfloor x\rfloor$.

Solution

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?

Solution

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.)

Solution

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 $S(10,112)-S(10,55)+S(10,2)$.

Solution

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 $a$.

Solution

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 $BC$.

Solution

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