Found problems: 12
2005 iTest, 1
[b]1A.[/b] The iTest, by virtue of being the first national internet-based high school math competition, saves a lot of paper every year. The quantity of trees saved (“$a$”) is determined by the following formula: $a = x^2 + 3x + 9$, where $x$ is the number of participating students in the competition. If $x$ is the correct answer from short answer [hide=problem 22]x=20[/hide], then find $a$. [i](1 point)[/i]
[b]1B.[/b] Let $q$ be the sum of the digits of $a$. If $q = b! - (b-1)! + (b-2)! - (b-3)!$, find $b$. [i](2 points)[/i]
[b]1C.[/b] Find the number of the following statements that are false: [i] (4 points)[/i]
1. $q$ is the first prime number resulting from the sum of cubes of distinct fractions, where both the numerator and denominator are primes.
2. $q$ is composite.
3. $q$ is composite and is the sum of the first four prime numbers and $1$.
4. $q$ is the smallest prime equal to the difference of cubes of two consecutive primes.
5. $q$ is not the smallest prime equal to the product of twin primes plus their arithmetic mean.
6. The sum of $q$ consecutive Fibonacci numbers, starting from the $q^{th}$ Fibonacci number, is prime.
7. $q$ is the largest prime factor of $1bbb$.
8. $q$ is the $8^{th}$ largest prime number.
9. $a$ is composite.
10. $a + q + b = q^2$.
11. The decimal expansion of $q^q$ begins with $q$.
12. $q$ is the smallest prime equal to the sum of three distinct primes.
13. $q^5 + q^2 + q^1 + q^3 + q^5 + q^6 + q^4 + q^0 = 52135640$.
14. $q$ is not the smallest prime such that $q$ and $q^2$ have the same sum of their digits.
15. $q$ is the smallest prime such that $q$ = (the product of its digits + the sum of its digits).
[hide=ANSWER KEY]1A. 469
1B. 4
1C. 6[/hide]
2005 iTest, 15
Kathryn has a crush on Joe. Dressed as Catwoman, she attends the same school Halloween party as Joe, hoping he will be there. If Joe gets beat up, Kathryn will be able to help Joe, and will be able to tell him how much she likes him. Otherwise, Kathryn will need to get her hipster friend, Max, who is DJing the event, to play Joe’s favorite song, “Pieces of Me” by Ashlee Simpson, to get him out on the dance floor, where she’ll also be able to tell him how much she likes him. Since playing the song would be in flagrant violation of Max’s musical integrity as a DJ, Kathryn will have to bribe him to play the song. For every $\$10$ she gives Max, the probability of him playing the song goes up $10\%$ (from $0\%$ to $10\%$ for the first $\$10$, from $10\%$ to $20\%$ for the next $\$10$, all the way up to $100\%$ if she gives him $\$100$). Max only accepts money in increments of $\$10$. How much money should Kathryn give to Max to give herself at least a $65\%$ chance of securing enough time to tell Joe how much she likes him?
2007 ITest, 21
James writes down fifteen 1's in a row and randomly writes $+$ or $-$ between each pair of consecutive 1's. One such example is \[1+1+1-1-1+1-1+1-1+1-1-1-1+1+1.\] What is the probability that the value of the expression James wrote down is $7$?
$\begin{array}{@{\hspace{-1em}}l@{\hspace{14em}}l@{\hspace{14em}}l}
\textbf{(A) }0&\textbf{(B) }\dfrac{6435}{2^{14}}&\textbf{(C) }\dfrac{6435}{2^{13}}\\\\
\textbf{(D) }\dfrac{429}{2^{12}}&\textbf{(E) }\dfrac{429}{2^{11}}&\textbf{(F) }\dfrac{429}{2^{10}}\\\\
\textbf{(G) }\dfrac1{15}&\textbf{(H) } \dfrac1{31}&\textbf{(I) }\dfrac1{30}\\\\
\textbf{(J) }\dfrac1{29}&\textbf{(K) }\dfrac{1001}{2^{15}}&\textbf{(L) }\dfrac{1001}{2^{14}}\\\\
\textbf{(M) }\dfrac{1001}{2^{13}}&\textbf{(N) }\dfrac1{2^7}&\textbf{(O) }\dfrac1{2^{14}}\\\\
\textbf{(P) }\dfrac1{2^{15}}&\textbf{(Q) }\dfrac{2007}{2^{14}}&\textbf{(R) }\dfrac{2007}{2^{15}}\\\\
\textbf{(S) }\dfrac{2007}{2^{2007}}&\textbf{(T) }\dfrac1{2007}&\textbf{(U) }\dfrac{-2007}{2^{14}}\end{array}$
2005 iTest, 24
SQUARING OFF: Master Chief and Samus Aran take turns firing rockets at one another from across the Cartesian plane. Master Chief’s movement is restricted to lattice points within the $10\times 10$ square with vertices $(0,0)$, $(0,10)$, $(10,0)$, and $(10,10)$, while Samus Aran’s movement is restricted to lattice points inside the $10\times 10$ square with vertices $(0,0)$, $(-10,0)$, $(0,-10)$, and $(-10,-10)$. Neither player can be located on or beyond the border of his or her square. Both players randomly choose a lattice point at which they begin the game, and do not move the rest of the game (until either they are killed or kill the other player).
Each player’s turn consists of firing a rocket, targeted at a specific undestroyed lattice point inside the border of the opponent’s movement square, which hits immediately. When a rocket hits its intended lattice point, it explodes, destroying the surrounding $3\times 3$ square ($8$ additional adjacent lattice points).
The game ends when one player is hit by a rocket (when the player is located within the $3\times 3$ grid hit by a rocket). If the highest possible probability that Samus Aran wins the game in three turns or less, assuming Master Chief goes first, is expressed as $a/b$, where $a$ and $b$ are relatively prime integers, find $a+b$.
2007 ITest, 25
Ted's favorite number is equal to \[1\cdot\binom{2007}1+2\cdot\binom{2007}2+3\cdot\binom{2007}3+\cdots+2007\cdot\binom{2007}{2007}.\] Find the remainder when Ted's favorite number is divided by $25$.
$\begin{array}{@{\hspace{-1em}}l@{\hspace{14em}}l@{\hspace{14em}}l}
\textbf{(A) }0&\textbf{(B) }1&\textbf{(C) }2\\\\
\textbf{(D) }3&\textbf{(E) }4&\textbf{(F) }5\\\\
\textbf{(G) }6&\textbf{(H) }7&\textbf{(I) }8\\\\
\textbf{(J) }9&\textbf{(K) }10&\textbf{(L) }11\\\\
\textbf{(M) }12&\textbf{(N) }13&\textbf{(O) }14\\\\
\textbf{(P) }15&\textbf{(Q) }16&\textbf{(R) }17\\\\
\textbf{(S) }18&\textbf{(T) }19&\textbf{(U) }20\\\\
\textbf{(V) }21&\textbf{(W) }22 & \textbf{(X) }23\\\\
\textbf{(Y) }24 \end{array}$
2005 iTest, 2
[b]2A. [/b] Two triangles $ABC$ and $ABD$ share a common side. $ABC$ is drawn such that its entire area lies inside the larger triangle $ABD$. If $AB = 20$, side $AD$ meets side $AB$ at a right angle, and point $C$ is between points $A$ and $D$, then find the area outside of triangle $ABC$ but within $ABD$, given that both triangles have integral side lengths and $AB$ is the smallest side of either triangle. $ABC$ and $ABD$ are both primitive right triangles. [i] (1 point)[/i]
[b]2B.[/b] Find the sum of all positive integral factors of the correct answer to [b]2A[/b]. [i](2 points)[/i]
[b]2C.[/b] Let $B$ be the sum of the digits of the correct answer to [b]2B[/b] above. If the solution to the functional equation $21*f(x) - 7*f(1/x) = Bx$ is of the form $(Ax^2 + C) / Dx$, find $C$, given that $A$, $C$, and $D$ are relatively prime (they don’t share a common prime factor). [i](3 points)[/i]
[hide=ANSWER KEY]2A.780
2B. 2352
2C. 3[/hide]
2007 ITest, 22
Find the value of $c$ such that the system of equations \begin{align*}|x+y|&=2007,\\|x-y|&=c\end{align*} has exactly two solutions $(x,y)$ in real numbers.
$\begin{array}{@{\hspace{-1em}}l@{\hspace{14em}}l@{\hspace{14em}}l}
\textbf{(A) }0&\textbf{(B) }1&\textbf{(C) }2\\\\
\textbf{(D) }3&\textbf{(E) }4&\textbf{(F) }5\\\\
\textbf{(G) }6&\textbf{(H) }7&\textbf{(I) }8\\\\
\textbf{(J) }9&\textbf{(K) }10&\textbf{(L) }11\\\\
\textbf{(M) }12&\textbf{(N) }13&\textbf{(O) }14\\\\
\textbf{(P) }15&\textbf{(Q) }16&\textbf{(R) }17\\\\
\textbf{(S) }18&\textbf{(T) }223&\textbf{(U) }678\\\\
\textbf{(V) }2007 & &\end{array}$
2007 ITest, 24
Let $N$ be the smallest positive integer $N$ such that $2008N$ is a perfect square and $2007N$ is a perfect cube. Find the remainder when $N$ is divided by $25$.
$\begin{array}{@{\hspace{-1em}}l@{\hspace{14em}}l@{\hspace{14em}}l}
\textbf{(A) }0&\textbf{(B) }1&\textbf{(C) }2\\\\
\textbf{(D) }3&\textbf{(E) }4&\textbf{(F) }5\\\\
\textbf{(G) }6&\textbf{(H) }7&\textbf{(I) }8\\\\
\textbf{(J) }9&\textbf{(K) }10&\textbf{(L) }11\\\\
\textbf{(M) }12&\textbf{(N) }13&\textbf{(O) }14\\\\
\textbf{(P) }15&\textbf{(Q) }16&\textbf{(R) }17\\\\
\textbf{(S) }18&\textbf{(T) }19&\textbf{(U) }20\\\\
\textbf{(V) }21&\textbf{(W) }22 & \textbf{(X) }23 \end{array}$
2008 ITest, 4
The difference between two prime numbers is $11$. Find their sum.
2007 ITest, 23
Find the product of the non-real roots of the equation \[(x^2-3)^2+5(x^2-3)+6=0.\]
$\begin{array}{@{\hspace{0em}}l@{\hspace{13.7em}}l@{\hspace{13.7em}}l}
\textbf{(A) }0&\textbf{(B) }1&\textbf{(C) }-1\\\\
\textbf{(D) }2&\textbf{(E) }-2&\textbf{(F) }3\\\\
\textbf{(G) }-3&\textbf{(H) }4&\textbf{(I) }-4\\\\
\textbf{(J) }5&\textbf{(K) }-5&\textbf{(L) }6\\\\
\textbf{(M) }-6&\textbf{(N) }3+2i&\textbf{(O) }3-2i\\\\
\textbf{(P) }\dfrac{-3+i\sqrt3}2&\textbf{(Q) }8&\textbf{(R) }-8\\\\
\textbf{(S) }12&\textbf{(T) }-12&\textbf{(U) }42\\\\
\textbf{(V) }\text{Ying} & \textbf{(W) }2007 &\end{array}$
2005 iTest, 3
[b]3A.[/b] Sudoku, the popular math game that caught on internationally before making its way here to the United States, is a game of logic based on a grid of $9$ rows and $9$ columns. This grid is subdivided into $9$ squares (“subgrids”) of length $3$. A successfully completed Sudoku puzzle fills this grid with the numbers $1$ through $9$ such that each number appears only once in each row, column, and individual $3 \times 3$ subgrid. Each Sudoku puzzle has one and only one correct solution.
Complete the following Sudoku puzzle, and find the sum of the numbers represented by $X, Y$, and $Z$ in the grid. [i](1 point)[/i]
$\begin{tabular}{|l|l|l|l|l|l|l|l|l|}
\hline
& & 2 & 9 & 7 & 4 & & & \\ \hline
& Z & & & & & & 5 & 7 \\ \hline
& & & & & & Y & & \\ \hline
& & 4 & & 5 & & & & 2 \\ \hline
& & 9 & X & 1 & & 6 & & \\ \hline
8 & & & & 3 & & 4 & & \\ \hline
& & & & & & & & \\ \hline
1 & 3 & & & & & & & \\ \hline
& & & 6 & 8 & 2 & 9 & & \\ \hline
\end{tabular}$
[b]3B.[/b] Let $A$ equal the correct answer from [b]3A[/b]. In triangle $WXY$, $tan \angle YWX= (A + 8) / .5A$, and the altitude from $W$ divides $XY$ into segments of $3$ and $A + 3$. What is the sum of the digits of the square of the area of the triangle? [i](2 points)[/i]
[b]3C.[/b] Let $B$ equal the correct answer from [b]3B[/b]. If a student team taking the $2005$ iTest solves $B$ problems correctly, and the probability that this student team makes over a $18$ is $x/y$ where $x$ and $y$ are relatively prime, find $x + y$.
Assume that each chain reaction question – all $3$ parts it contains – counts as a single problem. Also assume that the student team does not attempt any tiebreakers. [i](4 points)[/i]
[i][Note for problem 3C beacuse you might not know how points are given at that iTest:
Part A (aka Short Answer), has 40 problems of 1 point each, total 40
Part B (aka Chain Reaction), has 3 problems of 7,6,7 points each, total 20
Part C (aka Long Answer), has 5 problems of 8 point each, total 40
all 3 parts add to 100 points totally ([url=https://artofproblemsolving.com/community/c3176431_itest_2005]here [/url] is that test)][/i]
[hide=ANSWER KEY]3A.14
3B. 4
3C. 6563 [/hide]
2005 iTest, 38
LeBron James and Carmelo Anthony play a game of one-on-one basketball where the first player to $3$ points or more wins. LeBron James has a $20\%$ chance of making a $3$-point shot; Carmelo has a $10\%$ chance of making a $3$-pointer. LeBron has a $40\%$ chance of making a $2$-point shot from anywhere inside the $3$-point line (excluding dunks, which are also worth $2$ points); Carmelo has a $52\%$ chance of making a $ 2$-point shot from anywhere inside the 3-point line (excluding dunks). LeBron has a $90\%$ chance of dunking on Carmelo; Carmelo has a $95\%$ chance of dunking on LeBron. If each player has $3$ possessions to try to win, LeBron James goes first, and both players follow a rational strategy to try to win, what is the probability that Carmelo Anthony wins the game?