Difference between revisions of "2006 iTest Problems"

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<math>\mathrm{(A)}\, 8</math>
 
<math>\mathrm{(A)}\, 8</math>
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[[2006 iTest Problems/Problem 1|Solution]]
  
 
===Problem 2===
 
===Problem 2===
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<math>\mathrm{(A)}\, 15\quad\mathrm{(B)}\, \frac{40}{3}</math>
 
<math>\mathrm{(A)}\, 15\quad\mathrm{(B)}\, \frac{40}{3}</math>
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 +
[[2006 iTest Problems/Problem 2|Solution]]
  
 
===Problem 3===
 
===Problem 3===
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<math>\mathrm{(A)}\, 2086\quad\mathrm{(B)}\, 4012\quad\mathrm{(C)}\, 2144</math>
 
<math>\mathrm{(A)}\, 2086\quad\mathrm{(B)}\, 4012\quad\mathrm{(C)}\, 2144</math>
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 +
[[2006 iTest Problems/Problem 3|Solution]]
  
 
===Problem 4===
 
===Problem 4===
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<math>\mathrm{(A)}\, \text{Elizabeth} \quad\mathrm{(B)}\,\text{Jeanne}\quad\mathrm{(C)}\,\text{Mary}\quad\mathrm{(D)}\,\text{Anne}</math>
 
<math>\mathrm{(A)}\, \text{Elizabeth} \quad\mathrm{(B)}\,\text{Jeanne}\quad\mathrm{(C)}\,\text{Mary}\quad\mathrm{(D)}\,\text{Anne}</math>
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 +
[[2006 iTest Problems/Problem 4|Solution]]
  
 
===Problem 5===
 
===Problem 5===
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<math>\mathrm{(A)}\,(4,0)\quad\mathrm{(B)}\,(6,0)\quad\mathrm{(C)}\,(-4,0)\quad\mathrm{(D)}\,(-6,0)\quad\mathrm{(E)}\,\text{none of the above}</math>
 
<math>\mathrm{(A)}\,(4,0)\quad\mathrm{(B)}\,(6,0)\quad\mathrm{(C)}\,(-4,0)\quad\mathrm{(D)}\,(-6,0)\quad\mathrm{(E)}\,\text{none of the above}</math>
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[[2006 iTest Problems/Problem 5|Solution]]
  
 
===Problem 6===
 
===Problem 6===
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<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,2\quad\mathrm{(D)}\,3\quad\mathrm{(E)}\,4\quad\mathrm{(F)}\,5</math>
 
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,2\quad\mathrm{(D)}\,3\quad\mathrm{(E)}\,4\quad\mathrm{(F)}\,5</math>
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 +
[[2006 iTest Problems/Problem 6|Solution]]
  
 
===Problem 7===
 
===Problem 7===
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<math>\mathrm{(A)}\,17\quad\mathrm{(B)}\,72\quad\mathrm{(C)}\,95\quad\mathrm{(D)}\,101\quad\mathrm{(E)}\,102\quad\mathrm{(F)}\,111\quad\mathrm{(G)}\,125</math>
 
<math>\mathrm{(A)}\,17\quad\mathrm{(B)}\,72\quad\mathrm{(C)}\,95\quad\mathrm{(D)}\,101\quad\mathrm{(E)}\,102\quad\mathrm{(F)}\,111\quad\mathrm{(G)}\,125</math>
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 +
[[2006 iTest Problems/Problem 7|Solution]]
  
 
===Problem 8===
 
===Problem 8===
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<math>\mathrm{(A)}\,4\quad\mathrm{(B)}\,8\quad\mathrm{(C)}\,6\sqrt{3}\quad\mathrm{(D)}\,4\sqrt{3}\quad\mathrm{(E)}\,4\sqrt{2}\quad\mathrm{(F)}\,12\quad\mathrm{(G)}\,6\quad\mathrm{(H)}\,\text{none of the above}</math>
 
<math>\mathrm{(A)}\,4\quad\mathrm{(B)}\,8\quad\mathrm{(C)}\,6\sqrt{3}\quad\mathrm{(D)}\,4\sqrt{3}\quad\mathrm{(E)}\,4\sqrt{2}\quad\mathrm{(F)}\,12\quad\mathrm{(G)}\,6\quad\mathrm{(H)}\,\text{none of the above}</math>
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[[2006 iTest Problems/Problem 8|Solution]]
  
 
===Problem 9===
 
===Problem 9===
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<math>\mathrm{(A)}\,\frac{\sqrt{3}}{3}\quad\mathrm{(B)}\,\frac{2\sqrt{3}}{3}\quad\mathrm{(C)}\,\frac{6}{13}\quad\mathrm{(D)}\,\frac{5}{13}\quad\mathrm{(E)}\,-\frac{5}{13} \\
 
<math>\mathrm{(A)}\,\frac{\sqrt{3}}{3}\quad\mathrm{(B)}\,\frac{2\sqrt{3}}{3}\quad\mathrm{(C)}\,\frac{6}{13}\quad\mathrm{(D)}\,\frac{5}{13}\quad\mathrm{(E)}\,-\frac{5}{13} \\
 
\quad\mathrm{(F)}\,\frac{\sqrt{26}}{26}\quad\mathrm{(G)}\,-\frac{\sqrt{26}}{26}\quad\mathrm{(H)}\,\frac{\sqrt{2}}{2}\quad\mathrm{(I)}\,\text{none of the above}</math>
 
\quad\mathrm{(F)}\,\frac{\sqrt{26}}{26}\quad\mathrm{(G)}\,-\frac{\sqrt{26}}{26}\quad\mathrm{(H)}\,\frac{\sqrt{2}}{2}\quad\mathrm{(I)}\,\text{none of the above}</math>
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[[2006 iTest Problems/Problem 9|Solution]]
  
 
===Problem 10===
 
===Problem 10===
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<math>\mathrm{(A)}\,256\quad\mathrm{(B)}\,496\quad\mathrm{(C)}\,512\quad\mathrm{(D)}\,640\quad\mathrm{(E)}\,796 \\
 
<math>\mathrm{(A)}\,256\quad\mathrm{(B)}\,496\quad\mathrm{(C)}\,512\quad\mathrm{(D)}\,640\quad\mathrm{(E)}\,796 \\
 
\quad\mathrm{(F)}\,946\quad\mathrm{(G)}\,1024\quad\mathrm{(H)}\,1134\quad\mathrm{(I)}\,1256\quad\mathrm{(J)}\,\text{none of the above}</math>
 
\quad\mathrm{(F)}\,946\quad\mathrm{(G)}\,1024\quad\mathrm{(H)}\,1134\quad\mathrm{(I)}\,1256\quad\mathrm{(J)}\,\text{none of the above}</math>
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[[2006 iTest Problems/Problem 10|Solution]]
  
 
===Problem 11===
 
===Problem 11===
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\text{(J) }1\qquad
 
\text{(J) }1\qquad
 
\text{(K) }\text{no triangle exists}\qquad</math>
 
\text{(K) }\text{no triangle exists}\qquad</math>
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 +
[[2006 iTest Problems/Problem 11|Solution]]
  
 
===Problem 12===
 
===Problem 12===
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\text{(K) }\frac{1}{20!}\qquad
 
\text{(K) }\frac{1}{20!}\qquad
 
\text{(L) }\text{none of the above}\qquad</math>
 
\text{(L) }\text{none of the above}\qquad</math>
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 +
[[2006 iTest Problems/Problem 12|Solution]]
  
 
===Problem 13===
 
===Problem 13===
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\text{(L) } 1715 \quad
 
\text{(L) } 1715 \quad
 
\text{(M) } \text{none of the above} \quad </math>
 
\text{(M) } \text{none of the above} \quad </math>
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 +
[[2006 iTest Problems/Problem 13|Solution]]
  
 
===Problem 14===
 
===Problem 14===
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\text{(M) } 80 \quad
 
\text{(M) } 80 \quad
 
\text{(N) } \text{none of the above}\quad </math>
 
\text{(N) } \text{none of the above}\quad </math>
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 +
[[2006 iTest Problems/Problem 14|Solution]]
  
 
===Problem 15===
 
===Problem 15===
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\text{(N) }50\qquad
 
\text{(N) }50\qquad
 
\text{(O) }\text{none of the above}\qquad</math>
 
\text{(O) }\text{none of the above}\qquad</math>
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[[2006 iTest Problems/Problem 15|Solution]]
  
 
===Problem 16===
 
===Problem 16===
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\text{(O) }\frac{11}{32}\qquad
 
\text{(O) }\frac{11}{32}\qquad
 
\text{(P) }\text{none of the above}</math>
 
\text{(P) }\text{none of the above}</math>
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 +
[[2006 iTest Problems/Problem 16|Solution]]
  
 
===Problem 17===
 
===Problem 17===
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\text{(P) }1\qquad  
 
\text{(P) }1\qquad  
 
\text{(Q) }2\qquad</math>
 
\text{(Q) }2\qquad</math>
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[[2006 iTest Problems/Problem 17|Solution]]
  
 
===Problem 18===
 
===Problem 18===
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\text{(Q)  Goldbach;  Hilbert}\qquad
 
\text{(Q)  Goldbach;  Hilbert}\qquad
 
\text{(R) none of the above}\qquad</math>
 
\text{(R) none of the above}\qquad</math>
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 +
[[2006 iTest Problems/Problem 18|Solution]]
  
 
===Problem 19===
 
===Problem 19===
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\textbf{(R) }24\qquad
 
\textbf{(R) }24\qquad
 
\textbf{(S) }25</math>
 
\textbf{(S) }25</math>
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 +
[[2006 iTest Problems/Problem 19|Solution]]
  
 
===Problem 20===
 
===Problem 20===
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\textbf{(S) }28\qquad
 
\textbf{(S) }28\qquad
 
\textbf{(T) }30\qquad</math>
 
\textbf{(T) }30\qquad</math>
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[[2006 iTest Problems/Problem 20|Solution]]
  
 
==Short Answer Section==
 
==Short Answer Section==
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What is the last (rightmost) digit of <math>3^{2006}</math>?
 
What is the last (rightmost) digit of <math>3^{2006}</math>?
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[[2006 iTest Problems/Problem 21|Solution]]
  
 
===Problem 22===
 
===Problem 22===
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label("$E$",E,NE);
 
label("$E$",E,NE);
 
</asy>
 
</asy>
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 +
[[2006 iTest Problems/Problem 22|Solution]]
  
 
===Problem 23===
 
===Problem 23===
  
 
Jack and Jill are playing a chance game.  They take turns alternately rolling a fair six sided die labeled with the integers 1 through 6 as usual (fair meaning the numbers appear with equal probability.) Jack wins if a prime number appears when he rolls, while Jill wins if when she rolls a number greater than 1 appears. The game terminates as soon as one of them has won. If Jack rolls first in a game, then the probability of that Jill wins the game can be expressed as <math>\tfrac mn</math> where <math>m</math> and <math>n</math> are relatively prime positive integers.  Compute <math>m+n</math>.
 
Jack and Jill are playing a chance game.  They take turns alternately rolling a fair six sided die labeled with the integers 1 through 6 as usual (fair meaning the numbers appear with equal probability.) Jack wins if a prime number appears when he rolls, while Jill wins if when she rolls a number greater than 1 appears. The game terminates as soon as one of them has won. If Jack rolls first in a game, then the probability of that Jill wins the game can be expressed as <math>\tfrac mn</math> where <math>m</math> and <math>n</math> are relatively prime positive integers.  Compute <math>m+n</math>.
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 +
[[2006 iTest Problems/Problem 23|Solution]]
  
 
===Problem 24===
 
===Problem 24===
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label("$E$",E,S);
 
label("$E$",E,S);
 
</asy>
 
</asy>
 +
 +
[[2006 iTest Problems/Problem 24|Solution]]
  
 
===Problem 25===
 
===Problem 25===
  
 
The expression <cmath>\dfrac{(1+2+\cdots + 10)(1^3+2^3+\cdots + 10^3)}{(1^2+2^2+\cdots + 10^2)^2}</cmath> reduces to <math>\tfrac mn</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers.  Find <math>m+n</math>.
 
The expression <cmath>\dfrac{(1+2+\cdots + 10)(1^3+2^3+\cdots + 10^3)}{(1^2+2^2+\cdots + 10^2)^2}</cmath> reduces to <math>\tfrac mn</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers.  Find <math>m+n</math>.
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[[2006 iTest Problems/Problem 25|Solution]]
  
 
===Problem 26===
 
===Problem 26===
  
 
A rectangle has area <math>A</math> and perimeter <math>P</math>.  The largest possible value of <math>\tfrac A{P^2}</math> can be expressed as <math>\tfrac mn</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers.  Compute <math>m+n</math>.
 
A rectangle has area <math>A</math> and perimeter <math>P</math>.  The largest possible value of <math>\tfrac A{P^2}</math> can be expressed as <math>\tfrac mn</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers.  Compute <math>m+n</math>.
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 +
[[2006 iTest Problems/Problem 26|Solution]]
  
 
===Problem 27===
 
===Problem 27===
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label("$E$",E,dir(C--E));
 
label("$E$",E,dir(C--E));
 
</asy>
 
</asy>
 +
 +
[[2006 iTest Problems/Problem 27|Solution]]
  
 
===Problem 28===
 
===Problem 28===
  
 
The largest prime factor of <math>999999999999</math> is greater than <math>2006</math>. Determine the remainder obtained when this prime factor is divided by <math>2006</math>.
 
The largest prime factor of <math>999999999999</math> is greater than <math>2006</math>. Determine the remainder obtained when this prime factor is divided by <math>2006</math>.
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 +
[[2006 iTest Problems/Problem 28|Solution]]
  
 
===Problem 29===
 
===Problem 29===
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label("$C$",C,SE);
 
label("$C$",C,SE);
 
</asy>
 
</asy>
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 +
[[2006 iTest Problems/Problem 29|Solution]]
  
 
===Problem 30===
 
===Problem 30===
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label("$P$",P,N);
 
label("$P$",P,N);
 
</asy>
 
</asy>
 +
 +
[[2006 iTest Problems/Problem 30|Solution]]
  
 
===Problem 31===
 
===Problem 31===
  
 
The value of the infinite series <cmath>\sum_{n=2}^\infty\dfrac{n^4+n^3+n^2-n+1}{n^6-1}</cmath> can be expressed as <math>\tfrac pq</math> where <math>p</math> and <math>q</math> are relatively prime positive numbers.  Compute <math>p+q</math>.
 
The value of the infinite series <cmath>\sum_{n=2}^\infty\dfrac{n^4+n^3+n^2-n+1}{n^6-1}</cmath> can be expressed as <math>\tfrac pq</math> where <math>p</math> and <math>q</math> are relatively prime positive numbers.  Compute <math>p+q</math>.
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 +
[[2006 iTest Problems/Problem 31|Solution]]
  
 
===Problem 32===
 
===Problem 32===
  
 
Triangle <math>ABC</math> is scalene.  Points <math>P</math> and <math>Q</math> are on segment <math>BC</math> with <math>P</math> between <math>B</math> and <math>Q</math> such that <math>BP=21</math>, <math>PQ=35</math>, and <math>QC=100</math>.  If <math>AP</math> and <math>AQ</math> trisect <math>\angle A</math>, then <math>\tfrac{AB}{AC}</math> can be written uniquely as <math>\tfrac{p\sqrt q}r</math>, where <math>p</math> and <math>r</math> are relatively prime positive integers and <math>q</math> is a positive integer not divisible by the square of any prime.  Determine <math>p+q+r</math>.
 
Triangle <math>ABC</math> is scalene.  Points <math>P</math> and <math>Q</math> are on segment <math>BC</math> with <math>P</math> between <math>B</math> and <math>Q</math> such that <math>BP=21</math>, <math>PQ=35</math>, and <math>QC=100</math>.  If <math>AP</math> and <math>AQ</math> trisect <math>\angle A</math>, then <math>\tfrac{AB}{AC}</math> can be written uniquely as <math>\tfrac{p\sqrt q}r</math>, where <math>p</math> and <math>r</math> are relatively prime positive integers and <math>q</math> is a positive integer not divisible by the square of any prime.  Determine <math>p+q+r</math>.
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 +
[[2006 iTest Problems/Problem 32|Solution]]
  
 
===Problem 33===
 
===Problem 33===
  
 
Six students sit in a group and chat during a complicated mathematical lecture. The professor, annoyed by the chatter, splits the group into two or more smaller groups. However, the smaller groups with at least two members continue to produce chatter, so the professor again chooses one noisy group and splits it into smaller groups. This process continues until the professor achieves the silence he needs to teach Algebraic Combinatorics. Suppose the procedure can be carried out in <math>N</math> ways, where the order of group breaking matters (if A and B are disjoint groups, then breaking up group A and then B is considered different form breaking up group B and then A even if the resulting partitions are identical) and where a group of students is treated as an unordered set of people. Compute the remainder obtained when <math>N</math> is divided by <math>2006</math>.
 
Six students sit in a group and chat during a complicated mathematical lecture. The professor, annoyed by the chatter, splits the group into two or more smaller groups. However, the smaller groups with at least two members continue to produce chatter, so the professor again chooses one noisy group and splits it into smaller groups. This process continues until the professor achieves the silence he needs to teach Algebraic Combinatorics. Suppose the procedure can be carried out in <math>N</math> ways, where the order of group breaking matters (if A and B are disjoint groups, then breaking up group A and then B is considered different form breaking up group B and then A even if the resulting partitions are identical) and where a group of students is treated as an unordered set of people. Compute the remainder obtained when <math>N</math> is divided by <math>2006</math>.
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 +
[[2006 iTest Problems/Problem 33|Solution]]
  
 
===Problem 34===
 
===Problem 34===
  
 
For each positive integer <math>n</math> let <math>S_n</math> denote the set of positive integers <math>k</math> such that <math>n^k-1</math> is divisible by <math>2006</math>.  Define the function <math>P(n)</math> by the rule <cmath>P(n):=\begin{cases}\min(s)_{s\in S_n}&\text{if }S_n\neq\emptyset,\\0&\text{otherwise}.\end{cases}</cmath> Let <math>d</math> be the least upper bound of <math>\{P(1),P(2),P(3),\ldots\}</math> and let <math>m</math> be the number of integers <math>i</math> such that <math>1\leq i\leq 2006</math> and <math>P(i) = d</math>.  Compute the value of <math>d+m</math>.
 
For each positive integer <math>n</math> let <math>S_n</math> denote the set of positive integers <math>k</math> such that <math>n^k-1</math> is divisible by <math>2006</math>.  Define the function <math>P(n)</math> by the rule <cmath>P(n):=\begin{cases}\min(s)_{s\in S_n}&\text{if }S_n\neq\emptyset,\\0&\text{otherwise}.\end{cases}</cmath> Let <math>d</math> be the least upper bound of <math>\{P(1),P(2),P(3),\ldots\}</math> and let <math>m</math> be the number of integers <math>i</math> such that <math>1\leq i\leq 2006</math> and <math>P(i) = d</math>.  Compute the value of <math>d+m</math>.
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[[2006 iTest Problems/Problem 34|Solution]]
  
 
===Problem 35===
 
===Problem 35===
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\end{align*}
 
\end{align*}
 
</cmath>
 
</cmath>
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[[2006 iTest Problems/Problem 35|Solution]]
  
 
===Problem 36===
 
===Problem 36===
  
 
Let <math>\alpha</math> denote <math>\cos^{-1}(\tfrac 23)</math>.  The recursive sequence <math>a_0,a_1,a_2,\ldots</math> satisfies <math>a_0 = 1</math> and, for all positive integers <math>n</math>, <cmath>a_n = \dfrac{\cos(n\alpha) - (a_1a_{n-1} + \cdots + a_{n-1}a_1)}{2a_0}.</cmath> Suppose that the series <cmath>\sum_{k=0}^\infty\dfrac{a_k}{2^k}</cmath> can be expressed uniquely as <math>\tfrac{p\sqrt q}r</math>, where <math>p</math> and <math>r</math> are coprime positive integers and <math>q</math> is not divisible by the square of any prime.  Find the value of <math>p+q+r</math>.
 
Let <math>\alpha</math> denote <math>\cos^{-1}(\tfrac 23)</math>.  The recursive sequence <math>a_0,a_1,a_2,\ldots</math> satisfies <math>a_0 = 1</math> and, for all positive integers <math>n</math>, <cmath>a_n = \dfrac{\cos(n\alpha) - (a_1a_{n-1} + \cdots + a_{n-1}a_1)}{2a_0}.</cmath> Suppose that the series <cmath>\sum_{k=0}^\infty\dfrac{a_k}{2^k}</cmath> can be expressed uniquely as <math>\tfrac{p\sqrt q}r</math>, where <math>p</math> and <math>r</math> are coprime positive integers and <math>q</math> is not divisible by the square of any prime.  Find the value of <math>p+q+r</math>.
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[[2006 iTest Problems/Problem 36|Solution]]
  
 
===Problem 37===
 
===Problem 37===
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\end{align*}</cmath>
 
\end{align*}</cmath>
 
The value of <math>y^2</math> can be expressed uniquely as <math>\tfrac{m-n\sqrt p}q</math>, where <math>m</math>, <math>n</math>, <math>p</math>, <math>q</math> are positive integers such that <math>p</math> is not divisible by the square of any prime and no prime dividing <math>q</math> divides both <math>m</math> and <math>n</math>.  Compute <math>m+n+p+q</math>.
 
The value of <math>y^2</math> can be expressed uniquely as <math>\tfrac{m-n\sqrt p}q</math>, where <math>m</math>, <math>n</math>, <math>p</math>, <math>q</math> are positive integers such that <math>p</math> is not divisible by the square of any prime and no prime dividing <math>q</math> divides both <math>m</math> and <math>n</math>.  Compute <math>m+n+p+q</math>.
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 +
[[2006 iTest Problems/Problem 37|Solution]]
 +
  
 
===Problem 38===
 
===Problem 38===
  
 
Segment <math>AB</math> is a diameter of circle <math>\Gamma_1</math>.  Point <math>C</math> lies in the interior of segment <math>AB</math> such that <math>BC=7</math>, and <math>D</math> is a point on <math>\Gamma_1</math> such that <math>BD=CD=10</math>.  Segment <math>AC</math> is a diameter of the circle <math>\Gamma_2</math>.  A third circle, <math>\omega</math>, is drawn internally tangent to <math>\Gamma_1</math>, externally tangent to <math>\Gamma_2</math>, and tangent to segment <math>CD</math>.  If <math>\omega</math> is centered on the opposite side of <math>CD</math> as <math>B</math>, then the radius of <math>\omega</math> can be expressed as <math>\tfrac mn</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers.  Compute <math>m+n</math>.
 
Segment <math>AB</math> is a diameter of circle <math>\Gamma_1</math>.  Point <math>C</math> lies in the interior of segment <math>AB</math> such that <math>BC=7</math>, and <math>D</math> is a point on <math>\Gamma_1</math> such that <math>BD=CD=10</math>.  Segment <math>AC</math> is a diameter of the circle <math>\Gamma_2</math>.  A third circle, <math>\omega</math>, is drawn internally tangent to <math>\Gamma_1</math>, externally tangent to <math>\Gamma_2</math>, and tangent to segment <math>CD</math>.  If <math>\omega</math> is centered on the opposite side of <math>CD</math> as <math>B</math>, then the radius of <math>\omega</math> can be expressed as <math>\tfrac mn</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers.  Compute <math>m+n</math>.
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 +
[[2006 iTest Problems/Problem 38|Solution]]
  
 
===Problem 39===
 
===Problem 39===
  
 
<math>ABCDEFGHIJKL</math> is a regular dodecagon. The number 1 is written at the vertex A, and 0's are written at each of the other vertices. Suddenly and simultaneously, the number at each vertex is replaced by the arithmetic mean of the two numbers appearing at the adjacent vertices. If this procedure is repeated a total of <math>2006</math> times, then the resulting number at A can be expressed as <math>m/n</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Compute the remainder obtained when <math>m + n</math> is divided by <math>2006</math>.
 
<math>ABCDEFGHIJKL</math> is a regular dodecagon. The number 1 is written at the vertex A, and 0's are written at each of the other vertices. Suddenly and simultaneously, the number at each vertex is replaced by the arithmetic mean of the two numbers appearing at the adjacent vertices. If this procedure is repeated a total of <math>2006</math> times, then the resulting number at A can be expressed as <math>m/n</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Compute the remainder obtained when <math>m + n</math> is divided by <math>2006</math>.
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 +
[[2006 iTest Problems/Problem 39|Solution]]
  
 
===Problem 40===
 
===Problem 40===
  
 
Acute triangle <math>ABC</math> satisfies <math>AB=2AC</math> and <math>AB^4+BC^4+CA^4 = 2006\cdot 10^{10}</math>.  Tetrahedron <math>DEFP</math> is formed by choosing points <math>D</math>, <math>E</math>, and <math>F</math> on the segments <math>BC</math>, <math>CA</math>, and <math>AB</math> (respectively) and folding <math>A</math>, <math>B</math>, <math>C</math>, over <math>EF</math>, <math>FD</math>, and <math>DE</math> (respectively) to the common point <math>P</math>.  Let <math>R</math> denote the circumradius of <math>DEFP</math>.  Compute the smallest positive integer <math>N</math> for which we can be certain that <math>n\geq R</math>.  It may be helpful to use <math>\sqrt[4]{1239} = 5.9329109\ldots</math>.
 
Acute triangle <math>ABC</math> satisfies <math>AB=2AC</math> and <math>AB^4+BC^4+CA^4 = 2006\cdot 10^{10}</math>.  Tetrahedron <math>DEFP</math> is formed by choosing points <math>D</math>, <math>E</math>, and <math>F</math> on the segments <math>BC</math>, <math>CA</math>, and <math>AB</math> (respectively) and folding <math>A</math>, <math>B</math>, <math>C</math>, over <math>EF</math>, <math>FD</math>, and <math>DE</math> (respectively) to the common point <math>P</math>.  Let <math>R</math> denote the circumradius of <math>DEFP</math>.  Compute the smallest positive integer <math>N</math> for which we can be certain that <math>n\geq R</math>.  It may be helpful to use <math>\sqrt[4]{1239} = 5.9329109\ldots</math>.
 +
 +
[[2006 iTest Problems/Problem 40|Solution]]
  
 
==Ultimate Question==
 
==Ultimate Question==
Line 487: Line 568:
 
Find the real number <math>x</math> such that
 
Find the real number <math>x</math> such that
 
<cmath>\sqrt{x-9} + \sqrt{x-6} = \sqrt{x-1}.</cmath>
 
<cmath>\sqrt{x-9} + \sqrt{x-6} = \sqrt{x-1}.</cmath>
 +
 +
[[2006 iTest Problems/Problem U1|Solution]]
  
 
====Problem U2====
 
====Problem U2====
Line 502: Line 585:
 
label("B",(10,0),E);
 
label("B",(10,0),E);
 
</asy>
 
</asy>
 +
 +
[[2006 iTest Problems/Problem U2|Solution]]
  
 
====Problem U3====
 
====Problem U3====
  
 
Let <math>T  =  TNFTPP</math>.  When  properly  sorted,  <math>T  -  35</math>  math  books  on  a  shelf  are  arranged  in  alphabetical order  from  left  to  right.  An  eager  student  checked  out  and  read  all  of  them.  Unfortunately,  the student  did  not  realize  how  the  books  were  sorted,  and  so  after  finishing  the  student  put  the  books  back  on  the  shelf  in  a  random  order.  If  all  arrangements  are  equally  likely,  the  probability  that exactly  <math>6</math>  of  the  books  were  returned  to  their  correct  (original)  position  can  be  expressed  as  <math>\frac{m}{n}</math>,  where  <math>m</math>  and  <math>n</math>  are  relatively  prime  positive  integers.  Compute  <math>m  +  n</math>.
 
Let <math>T  =  TNFTPP</math>.  When  properly  sorted,  <math>T  -  35</math>  math  books  on  a  shelf  are  arranged  in  alphabetical order  from  left  to  right.  An  eager  student  checked  out  and  read  all  of  them.  Unfortunately,  the student  did  not  realize  how  the  books  were  sorted,  and  so  after  finishing  the  student  put  the  books  back  on  the  shelf  in  a  random  order.  If  all  arrangements  are  equally  likely,  the  probability  that exactly  <math>6</math>  of  the  books  were  returned  to  their  correct  (original)  position  can  be  expressed  as  <math>\frac{m}{n}</math>,  where  <math>m</math>  and  <math>n</math>  are  relatively  prime  positive  integers.  Compute  <math>m  +  n</math>.
 +
 +
[[2006 iTest Problems/Problem U3|Solution]]
  
 
===Problem 42===
 
===Problem 42===
Line 512: Line 599:
  
 
Let  <math>T  =  TNFTPP</math>.  As  <math>n</math>  ranges  over  the  integers,  the  expression  <math>n^4  -  898n^2  +  T  -  2160</math>  evaluates  to just  one  prime  number.  Find  this  prime.
 
Let  <math>T  =  TNFTPP</math>.  As  <math>n</math>  ranges  over  the  integers,  the  expression  <math>n^4  -  898n^2  +  T  -  2160</math>  evaluates  to just  one  prime  number.  Find  this  prime.
 +
 +
[[2006 iTest Problems/Problem U4|Solution]]
  
 
====Problem U5====
 
====Problem U5====
  
 
Let  <math>T  =  TNFTPP</math>,  and  let  <math>S</math>  be  the  sum  of  the  digits  of  <math>T</math>.  In  triangle  <math>ABC</math>,  points  <math>D</math>,  <math>E</math>,  and  <math>F</math>  are  the feet  of  the  angle  bisectors  of  <math>\angle A</math>,  <math>\angle B</math>,  <math>\angle C</math>  respectively.  Let  point  <math>P</math>  be  the  intersection  of segments  <math>AD</math>  and  <math>BE</math>,  and  let  <math>p</math>  denote  the  perimeter  of  <math>ABC</math>.  If  <math>AP  =  3PD</math>,  <math>BE  =  S  -  1</math>,  and  <math>CF  =  9</math>, then  the  value  of  <math>\frac{AD}{p}</math>  can  be  expressed  uniquely  as <math>\frac{\sqrt{m}}{n}</math>  where  <math>m</math>  and  <math>n</math>  are  positive  integers such  that  <math>m</math>  is  not  divisible  by  the  square  of  any  prime.  Find  <math>m  +  n</math>.
 
Let  <math>T  =  TNFTPP</math>,  and  let  <math>S</math>  be  the  sum  of  the  digits  of  <math>T</math>.  In  triangle  <math>ABC</math>,  points  <math>D</math>,  <math>E</math>,  and  <math>F</math>  are  the feet  of  the  angle  bisectors  of  <math>\angle A</math>,  <math>\angle B</math>,  <math>\angle C</math>  respectively.  Let  point  <math>P</math>  be  the  intersection  of segments  <math>AD</math>  and  <math>BE</math>,  and  let  <math>p</math>  denote  the  perimeter  of  <math>ABC</math>.  If  <math>AP  =  3PD</math>,  <math>BE  =  S  -  1</math>,  and  <math>CF  =  9</math>, then  the  value  of  <math>\frac{AD}{p}</math>  can  be  expressed  uniquely  as <math>\frac{\sqrt{m}}{n}</math>  where  <math>m</math>  and  <math>n</math>  are  positive  integers such  that  <math>m</math>  is  not  divisible  by  the  square  of  any  prime.  Find  <math>m  +  n</math>.
 +
 +
[[2006 iTest Problems/Problem U5|Solution]]
  
 
====Problem U6====
 
====Problem U6====
Line 524: Line 615:
  
 
The smallest possible value of <math>\frac{y}{x}</math> is equal to <math>\frac{m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive integers.  Find <math>m+n</math>.
 
The smallest possible value of <math>\frac{y}{x}</math> is equal to <math>\frac{m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive integers.  Find <math>m+n</math>.
 +
 +
[[2006 iTest Problems/Problem U6|Solution]]
  
 
====Problem U7====
 
====Problem U7====
  
 
Let  <math>T  =  TNFTPP</math>.  Triangle  <math>ABC</math>  has  integer  side  lengths,  including  <math>BC  =  100T  -  4</math>,  and  a  right angle,  <math>\angle ABC</math>.  Let  <math>r</math>  and  <math>s</math>  denote  the  inradius  and  semiperimeter  of  <math>ABC</math>  respectively.  Find  the ''perimeter''  of  the  triangle  ABC  which  minimizes  <math>\frac{s}{r}</math>.
 
Let  <math>T  =  TNFTPP</math>.  Triangle  <math>ABC</math>  has  integer  side  lengths,  including  <math>BC  =  100T  -  4</math>,  and  a  right angle,  <math>\angle ABC</math>.  Let  <math>r</math>  and  <math>s</math>  denote  the  inradius  and  semiperimeter  of  <math>ABC</math>  respectively.  Find  the ''perimeter''  of  the  triangle  ABC  which  minimizes  <math>\frac{s}{r}</math>.
 +
 +
[[2006 iTest Problems/Problem U7|Solution]]
  
 
===Problem 43===
 
===Problem 43===
Line 534: Line 629:
  
 
Let <math>T  =  TNFTPP</math>,  and  let  <math>S</math>  be  the  sum  of  the  digits  of  <math>T</math>.  Cyclic  quadrilateral  <math>ABCD</math>  has  side lengths  <math>AB  =  S  -  11</math>,  <math>BC  =  2</math>,  <math>CD  =  3</math>,  and  <math>DA  =  10</math>.  Let  <math>M</math>  and  <math>N</math>  be  the  midpoints  of  sides  <math>AD</math>  and <math>BC</math>.  The  diagonals  <math>AC</math>  and  <math>BD</math>  intersect  <math>MN</math>  at  <math>P</math>  and  <math>Q</math>  respectively.  <math>\frac{PQ}{MN}</math>  can  be  expressed  as <math>\frac{m}{n}</math>  where  <math>m</math>  and  <math>n</math>  are  relatively  prime  positive  integers.  Determine  <math>m  +  n</math>.
 
Let <math>T  =  TNFTPP</math>,  and  let  <math>S</math>  be  the  sum  of  the  digits  of  <math>T</math>.  Cyclic  quadrilateral  <math>ABCD</math>  has  side lengths  <math>AB  =  S  -  11</math>,  <math>BC  =  2</math>,  <math>CD  =  3</math>,  and  <math>DA  =  10</math>.  Let  <math>M</math>  and  <math>N</math>  be  the  midpoints  of  sides  <math>AD</math>  and <math>BC</math>.  The  diagonals  <math>AC</math>  and  <math>BD</math>  intersect  <math>MN</math>  at  <math>P</math>  and  <math>Q</math>  respectively.  <math>\frac{PQ}{MN}</math>  can  be  expressed  as <math>\frac{m}{n}</math>  where  <math>m</math>  and  <math>n</math>  are  relatively  prime  positive  integers.  Determine  <math>m  +  n</math>.
 +
 +
[[2006 iTest Problems/Problem U8|Solution]]
  
 
====Problem U9====
 
====Problem U9====
  
 
Let  <math>T  =  TNFTPP</math>.  Determine  the  number  of  5  element  subsets  <math>S</math>  of  <math>\{1,2,3, \cdots ,T  +  100\}</math>  such  that  the  sum  of  the  elements  of  <math>S</math>  is  divisible  by  5.
 
Let  <math>T  =  TNFTPP</math>.  Determine  the  number  of  5  element  subsets  <math>S</math>  of  <math>\{1,2,3, \cdots ,T  +  100\}</math>  such  that  the  sum  of  the  elements  of  <math>S</math>  is  divisible  by  5.
 +
 +
[[2006 iTest Problems/Problem U9|Solution]]
  
 
====Problem U10====
 
====Problem U10====
Line 548: Line 647:
  
 
'''Recall that you are turning in the sum of all ten answers, NOT the answer to this problem.
 
'''Recall that you are turning in the sum of all ten answers, NOT the answer to this problem.
 +
 +
[[2006 iTest Problems/Problem U10|Solution]]
 +
 +
==See Also==
 +
* [[iTest Problems and Solutions]]
 +
 +
{{iTest box|year=2006|before=[[2005 iTest]]|after=[[2007 iTest]]|ver=[[2006 iTest Problems/Problem U1|U1]] '''•''' [[2006 iTest Problems/Problem U2|U2]] '''•''' [[2006 iTest Problems/Problem U3|U3]] '''•''' [[2006 iTest Problems/Problem U4|U4]] '''•''' [[2006 iTest Problems/Problem U5|U5]] '''•''' [[2006 iTest Problems/Problem U6|U6]] '''•''' [[2006 iTest Problems/Problem U7|U7]] '''•''' [[2006 iTest Problems/Problem U8|U8]] '''•''' [[2006 iTest Problems/Problem U9|U9]] '''•''' [[2006 iTest Problems/Problem U10|U10]]}}

Latest revision as of 16:02, 13 January 2020

Multiple Choice Section

Problem 1

Find the number of positive integral divisors of 2006.

$\mathrm{(A)}\, 8$

Solution

Problem 2

Find the harmonic mean of 10 and 20.

$\mathrm{(A)}\, 15\quad\mathrm{(B)}\, \frac{40}{3}$

Solution

Problem 3

Let $I, T, E, S$ be distinct positive integers such that the product $ITEST  =  2006$. What is the largest possible value of the sum $I  +  T  +  E  +  S  +  T  +  2006$?

$\mathrm{(A)}\, 2086\quad\mathrm{(B)}\, 4012\quad\mathrm{(C)}\, 2144$

Solution

Problem 4

Four couples go ballroom dancing one evening. Their first names are Henry, Peter, Louis, Roger, Elizabeth, Jeanne, Mary, and Anne. If Henry's wife is not dancing with her husband (but with Elizabeth's husband), Roger and Anne are not dancing, Peter is playing the trumpet, and Mary is playing the piano, and Anne's husband is not Peter, who is Roger's wife?

$\mathrm{(A)}\, \text{Elizabeth} \quad\mathrm{(B)}\,\text{Jeanne}\quad\mathrm{(C)}\,\text{Mary}\quad\mathrm{(D)}\,\text{Anne}$

Solution

Problem 5

A line has y-intercept $(0,3)$ and forms a right angle to the line $2x  +  y  =  3$. Find the x-intercept of the line.

$\mathrm{(A)}\,(4,0)\quad\mathrm{(B)}\,(6,0)\quad\mathrm{(C)}\,(-4,0)\quad\mathrm{(D)}\,(-6,0)\quad\mathrm{(E)}\,\text{none of the above}$

Solution

Problem 6

What is the remainder when $2^{2006}$ is divided by 7?

$\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,2\quad\mathrm{(D)}\,3\quad\mathrm{(E)}\,4\quad\mathrm{(F)}\,5$

Solution

Problem 7

The sum of $17$ consecutive integers is $2006$. Find the second largest integer.

$\mathrm{(A)}\,17\quad\mathrm{(B)}\,72\quad\mathrm{(C)}\,95\quad\mathrm{(D)}\,101\quad\mathrm{(E)}\,102\quad\mathrm{(F)}\,111\quad\mathrm{(G)}\,125$

Solution

Problem 8

The point $P$ is a point on a circle with center $O$. Perpendicular lines are drawn from $P$ to perpendicular diameters, $AB$ and $CD$, meeting them at points $Y$ and $Z$, respectively. If the diameter of the circle is $16$, what is the length of $YZ$?

$\mathrm{(A)}\,4\quad\mathrm{(B)}\,8\quad\mathrm{(C)}\,6\sqrt{3}\quad\mathrm{(D)}\,4\sqrt{3}\quad\mathrm{(E)}\,4\sqrt{2}\quad\mathrm{(F)}\,12\quad\mathrm{(G)}\,6\quad\mathrm{(H)}\,\text{none of the above}$

Solution

Problem 9

If $\sin(x)  =  -\frac{5}{13}$ and $x$ is in the third quadrant, what is the absolute value of $\cos(\frac{x}{2})$?

$\mathrm{(A)}\,\frac{\sqrt{3}}{3}\quad\mathrm{(B)}\,\frac{2\sqrt{3}}{3}\quad\mathrm{(C)}\,\frac{6}{13}\quad\mathrm{(D)}\,\frac{5}{13}\quad\mathrm{(E)}\,-\frac{5}{13} \\ \quad\mathrm{(F)}\,\frac{\sqrt{26}}{26}\quad\mathrm{(G)}\,-\frac{\sqrt{26}}{26}\quad\mathrm{(H)}\,\frac{\sqrt{2}}{2}\quad\mathrm{(I)}\,\text{none of the above}$

Solution

Problem 10

Find the number of elements in the first $64$ rows of Pascal's Triangle that are divisible by $4$.

$\mathrm{(A)}\,256\quad\mathrm{(B)}\,496\quad\mathrm{(C)}\,512\quad\mathrm{(D)}\,640\quad\mathrm{(E)}\,796 \\ \quad\mathrm{(F)}\,946\quad\mathrm{(G)}\,1024\quad\mathrm{(H)}\,1134\quad\mathrm{(I)}\,1256\quad\mathrm{(J)}\,\text{none of the above}$

Solution

Problem 11

Find the radius of the inscribed circle of a triangle with sides of length $13$, $30$, and $37$.

$\text{(A) }\frac{9}{2}\qquad \text{(B) }\frac{7}{2}\qquad \text{(C) }4\qquad \text{(D) }-\sqrt{2}\qquad \text{(E) }4\sqrt{5}\qquad \text{(F) }6\qquad \\ \text{(G) }\frac{11}{2}\qquad \text{(H) }\frac{13}{2}\qquad \text{(I) }\text{none of the above}\qquad \text{(J) }1\qquad \text{(K) }\text{no triangle exists}\qquad$

Solution

Problem 12

What is the highest possible probability of getting $12$ of these $20$ multiple choice questions correct, given that you don't know how to work any of them and are forced to blindly guess on each one?

$\text{(A) }\frac{1}{6!}\qquad \text{(B) }\frac{1}{7!}\qquad \text{(C) }\frac{1}{8!}\qquad \text{(D) }\frac{1}{9!}\qquad \text{(E) }\frac{1}{10!}\qquad \text{(F) }\frac{1}{11!}\qquad\\ \\ \text{(G) }\frac{1}{12!}\qquad \text{(H) }\frac{2}{8!}\qquad \text{(I) }\frac{2}{10!}\qquad \text{(J) }\frac{2}{12!}\qquad \text{(K) }\frac{1}{20!}\qquad \text{(L) }\text{none of the above}\qquad$

Solution

Problem 13

Suppose that $x,  y,  z$ are three distinct prime numbers such that $x  +  y  +  z  =  49$. Find the maximum possible value for the product $xyz$.

$\text{(A) } 615 \quad \text{(B) } 1295 \quad \text{(C) } 2387 \quad \text{(D) } 1772 \quad \text{(E) } 715 \quad \text{(F) } 442 \quad \text{(G) } 1479 \quad \\ \text{(H) } 2639 \quad \text{(I) } 3059 \quad \text{(J) } 3821 \quad \text{(K) } 3145 \quad \text{(L) } 1715 \quad \text{(M) } \text{none of the above} \quad$

Solution

Problem 14

Find $x$, where $x$ is the smallest positive integer such that $2^x$ leaves a remainder of $1$ when divided by $5$, $7$, and $31$.

$\text{(A) } 15 \quad \text{(B) } 20 \quad \text{(C) } 25 \quad \text{(D) } 30 \quad \text{(E) } 28 \quad \text{(F) } 32 \quad \text{(G) } 64 \quad \\ \text{(H) } 128 \quad \text{(I) } 45 \quad \text{(J) } 50 \quad \text{(K) } 60 \quad \text{(L) } 70 \quad \text{(M) } 80 \quad \text{(N) } \text{none of the above}\quad$

Solution

Problem 15

How many integers between $1$ and $2006$, inclusive, are perfect squares?

$\text{(A) }37\qquad \text{(B) }38\qquad \text{(C) }39\qquad \text{(D) }40\qquad \text{(E) }41\qquad \text{(F) }42\qquad \text{(G) }43\qquad \text{(H) }44\qquad$

$\text{(I) }45\qquad \text{(J) }46\qquad \text{(K) }47\qquad \text{(L) }48\qquad \text{(M) }49\qquad \text{(N) }50\qquad \text{(O) }\text{none of the above}\qquad$

Solution

Problem 16

The Minnesota Twins face the New York Mets in the 2006 World Series. Assuming the two teams are evenly matched (each has a $.5$ probability of winning any game) what is the probability that the World Series (a best of 7 series of games which lasts until one team wins four games) will require the full seven games to determine a winner?

$\text{(A) }\frac{1}{16}\qquad \text{(B) }\frac{1}{8}\qquad \text{(C) }\frac{3}{16}\qquad \text{(D) }\frac{1}{4}\qquad \text{(E) }\frac{5}{16}\qquad$

$\text{(F) }\frac{3}{8}\qquad \text{(G) }\frac{5}{32}\qquad \text{(H) }\frac{7}{32}\qquad \text{(I) }\frac{9}{32}\qquad \text{(J) }\frac{3}{64}\qquad \text{(K) }\frac{5}{64}\qquad$

$\text{(L) }\frac{7}{64}\qquad \text{(M) }\frac{1}{2}\qquad \text{(N) }\frac{13}{32}\qquad \text{(O) }\frac{11}{32}\qquad \text{(P) }\text{none of the above}$

Solution

Problem 17

Let $\sin(2x) = \frac{1}{7}$. Find the numerical value of $\sin(x)\sin(x)\sin(x)\sin(x) + \cos(x)\cos(x)\cos(x)\cos(x)$.

$\text{(A) }\frac{2305}{2401}\qquad \text{(B) }\frac{4610}{2401}\qquad \text{(C) }\frac{2400}{2401}\qquad \text{(D) }\frac{6915}{2401}\qquad \text{(E) }\frac{1}{2401}\qquad \text{(F) }0\qquad$

$\text{(G) }\frac{195}{196}\qquad \text{(H) }\frac{195}{98}\qquad \text{(I) }\frac{97}{98}\qquad \text{(J) }\frac{1}{49}\qquad \text{(K) }\frac{2}{49}\qquad \text{(L) }\frac{48}{49}\qquad$

$\text{(M) }\frac{96}{49}\qquad \text{(N) }\pi\qquad \text{(O) }\text{none of the above}\qquad \text{(P) }1\qquad  \text{(Q) }2\qquad$

Solution

Problem 18

Every even number greater than 2 can be expressed as the sum of two prime numbers.'

Name the mathematician for which this theorem was named, and then name the mathematician to whom he transmitted this theorem via letter in 1742.

$\text{(A) Ptolemy;  Archimedes}\qquad \text{(B)  Goldbach;  Newton}\qquad \text{(C)  Lagrange;  Goldbach}\qquad$

$\text{(D)  Euclid;  Plato}\qquad \text{(E)  Goldbach;  Bernoulli}\qquad \text{(F)  Goldbach;  Euler}\qquad$

$\text{(G)  L'Hopital;  Goldbach}\qquad \text{(H)  Goldbach;  L'Hopital}\qquad \text{(I)  Ramanujan;  Fermat}\qquad$

$\text{(J)  Fermat;  Ramanujan}\qquad \text{(K)  Goldbach;  Ramanujan}\qquad \text{(L)  Goldbach;  Fermat}\qquad$

$\text{(M)  De  Moivre;  Cauchy}\qquad \text{(N)  Cauchy;  De  Moivre}\qquad \text{(O)  Goldbach;  Cauchy}\qquad$

$\text{(P)  Goldbach;  Descartes}\qquad \text{(Q)  Goldbach;  Hilbert}\qquad \text{(R) none of the above}\qquad$

Solution

Problem 19

Questions 19 and 20 are Sudoku-related questions. Sudoku is a puzzle game that has one and only one solution for each puzzle. Digits from 1 to 9 must go into each space on the $9 \times 9$ grid such that every row, column, and $3 \times 3$ square contains one and only one of each digit.

Find the sum of $w + x + y + z$ by solving the Sudoku puzzle below.

1 _ _ | 3 5 8 | _ _ 6
4 _ _ | _ _ _ | _ x 8
_ _ 9 | _ 1 _ | 7 _ _
---------------------
_ z _ | 1 _ _ | _ 5 _
_ _ 3 | 2 _ 4 | 8 _ _
_ 2 _ | w _ 9 | _ _ _
---------------------
_ _ 6 | _ 2 _ | 9 _ _
3 _ _ | _ y _ | _ _ 1
2 _ _ | 8 4 3 | _ _ 7

$\textbf{(A) }7\qquad \textbf{(B) }8\qquad \textbf{(C) }9\qquad \textbf{(D) }10\qquad \textbf{(E) }11\qquad \textbf{(F) }12\qquad \textbf{(G) }13\qquad$ $\textbf{(H) }14\qquad \textbf{(I) }15\qquad \textbf{(J) }16\qquad \textbf{(K) }17\qquad \textbf{(L) }18\qquad \textbf{(M) }19\qquad$ $\textbf{(N) }20\qquad \textbf{(O) }21\qquad \textbf{(P) }22\qquad \textbf{(Q) }23\qquad \textbf{(R) }24\qquad \textbf{(S) }25$

Solution

Problem 20

Sudoku is a puzzle game that has one and only one solution for each puzzle. Digits from 1 to 9 must go into each space on the $9 \times 9$ grid such that every row, column, and $3 \times 3$ square contains one and only one of each digit.

Find the sum of $w + x + y + z$ by solving the Sudoku puzzle below.

_ _ _ | _ 4 _ | _ z _
1 _ 6 | _ _ _ | 7 _ 3
5 _ _ | 9 _ _ | _ _ 2
---------------------
_ 8 3 | w 2 _ | 5 _ _
2 _ _ | 5 _ 9 | _ _ 7
_ _ 7 | _ 8 _ | 9 2 _
---------------------
3 _ _ | _ _ 1 | _ _ 6
8 _ 9 | x _ _ | 3 _ 5
_ y _ | _ 3 _ | _ _ _

$\textbf{(A) }2\qquad \textbf{(B) }4\qquad \textbf{(C) }6\qquad \textbf{(D) }8\qquad \textbf{(E) }9\qquad \textbf{(F) }10\qquad \textbf{(G) }11\qquad$ $\textbf{(H) }12\qquad \textbf{(I) }13\qquad \textbf{(J) }14\qquad \textbf{(K) }15\qquad \textbf{(L) }18\qquad \textbf{(M) }19\qquad$ $\textbf{(N) }20\qquad \textbf{(O) }23\qquad \textbf{(P) }24\qquad \textbf{(Q) }25\qquad \textbf{(R) }26\qquad \textbf{(S) }28\qquad \textbf{(T) }30\qquad$

Solution

Short Answer Section

Problem 21

What is the last (rightmost) digit of $3^{2006}$?

Solution

Problem 22

Triangle $ABC$ has sidelengths $AB=75$, $BC=100$, and $CA=125$. Point $D$ is the foot of the altitude from $B$, and $E$ lies on segment $BC$ such that $DE\perp BC$. Find the area of the triangle $BDE$.

[asy] import olympiad; size(170); defaultpen(linewidth(0.7)+fontsize(11pt)); pair A = origin, B = (9,12), C = (25,0), D = foot(B,A,C), E = foot(D,B,C); draw(A--B--C--A^^B--D--E); label("$A$",A,SW); label("$B$",B,N); label("$C$",C,SE); label("$D$",D,S); label("$E$",E,NE); [/asy]

Solution

Problem 23

Jack and Jill are playing a chance game. They take turns alternately rolling a fair six sided die labeled with the integers 1 through 6 as usual (fair meaning the numbers appear with equal probability.) Jack wins if a prime number appears when he rolls, while Jill wins if when she rolls a number greater than 1 appears. The game terminates as soon as one of them has won. If Jack rolls first in a game, then the probability of that Jill wins the game can be expressed as $\tfrac mn$ where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.

Solution

Problem 24

Points $D$ and $E$ are chosen on side $BC$ of triangle $ABC$ such that $E$ is between $B$ and $D$ and $BE=1$, $ED=DC=3$. If $\angle BAD=\angle EAC=90^\circ$, the area of $ABC$ can be expressed as $\tfrac{p\sqrt q}r$, where $p$ and $r$ are relatively prime positive integers and $q$ is a positive integer not divisible by the square of any prime. Compute $p+q+r$.

[asy] import olympiad; size(200); defaultpen(linewidth(0.7)+fontsize(11pt)); pair D = origin, E = (3,0), C = (-3,0), B = (4,0); path circ1 = arc(D,3,0,180), circ2 = arc(B/2,2,0,180); pair A = intersectionpoint(circ1, circ2); draw(E--A--C--B--A--D); label("$A$",A,N); label("$B$",B,SE); label("$C$",C,SW); label("$D$",D,S); label("$E$",E,S); [/asy]

Solution

Problem 25

The expression \[\dfrac{(1+2+\cdots + 10)(1^3+2^3+\cdots + 10^3)}{(1^2+2^2+\cdots + 10^2)^2}\] reduces to $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution

Problem 26

A rectangle has area $A$ and perimeter $P$. The largest possible value of $\tfrac A{P^2}$ can be expressed as $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.

Solution

Problem 27

Line $\ell$ passes through $A$ and into the interior of the equilateral triangle $ABC$. $D$ and $E$ are the orthogonal projections of $B$ and $C$ onto $\ell$ respectively. If $DE=1$ and $2BD=CE$, then the area of $ABC$ can be expressed as $m\sqrt n$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Determine $m+n$.

[asy] import olympiad; size(250); defaultpen(linewidth(0.7)+fontsize(11pt)); real r = 31, t = -10; pair A = origin, B = dir(r-60), C = dir(r); pair X = -0.8 * dir(t), Y = 2 * dir(t); pair D = foot(B,X,Y), E = foot(C,X,Y); draw(A--B--C--A^^X--Y^^B--D^^C--E); label("$A$",A,S); label("$B$",B,S); label("$C$",C,N); label("$D$",D,dir(B--D)); label("$E$",E,dir(C--E)); [/asy]

Solution

Problem 28

The largest prime factor of $999999999999$ is greater than $2006$. Determine the remainder obtained when this prime factor is divided by $2006$.

Solution

Problem 29

The altitudes in triangle $ABC$ have lengths 10, 12, and 15. The area of $ABC$ can be expressed as $\tfrac{m\sqrt n}p$, where $m$ and $p$ are relatively prime positive integers and $n$ is a positive integer not divisible by the square of any prime. Find $m + n + p$.

[asy] import olympiad; size(200); defaultpen(linewidth(0.7)+fontsize(11pt)); pair A = (9,8.5), B = origin, C = (15,0); draw(A--B--C--cycle); pair D = foot(A,B,C), E = foot(B,C,A), F = foot(C,A,B); draw(A--D^^B--E^^C--F); label("$A$",A,N); label("$B$",B,SW); label("$C$",C,SE); [/asy]

Solution

Problem 30

Triangle $ABC$ is equilateral. Points $D$ and $E$ are the midpoints of segments $BC$ and $AC$ respectively. $F$ is the point on segment $AB$ such that $2BF=AF$. Let $P$ denote the intersection of $AD$ and $EF$, The value of $EP/PF$ can be expressed as $m/n$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

[asy] import olympiad; size(150); defaultpen(linewidth(0.7) + fontsize(11pt)); pair A = origin, B = (1,0), C = dir(60), D = (B+C)/2, E = C/2, F = 2*B/3, P = intersectionpoint(A--D,E--F); draw(A--B--C--A--D^^E--F); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,N); label("$D$",D,NE); label("$E$",E,NW); label("$F$",F,S); label("$P$",P,N); [/asy]

Solution

Problem 31

The value of the infinite series \[\sum_{n=2}^\infty\dfrac{n^4+n^3+n^2-n+1}{n^6-1}\] can be expressed as $\tfrac pq$ where $p$ and $q$ are relatively prime positive numbers. Compute $p+q$.

Solution

Problem 32

Triangle $ABC$ is scalene. Points $P$ and $Q$ are on segment $BC$ with $P$ between $B$ and $Q$ such that $BP=21$, $PQ=35$, and $QC=100$. If $AP$ and $AQ$ trisect $\angle A$, then $\tfrac{AB}{AC}$ can be written uniquely as $\tfrac{p\sqrt q}r$, where $p$ and $r$ are relatively prime positive integers and $q$ is a positive integer not divisible by the square of any prime. Determine $p+q+r$.

Solution

Problem 33

Six students sit in a group and chat during a complicated mathematical lecture. The professor, annoyed by the chatter, splits the group into two or more smaller groups. However, the smaller groups with at least two members continue to produce chatter, so the professor again chooses one noisy group and splits it into smaller groups. This process continues until the professor achieves the silence he needs to teach Algebraic Combinatorics. Suppose the procedure can be carried out in $N$ ways, where the order of group breaking matters (if A and B are disjoint groups, then breaking up group A and then B is considered different form breaking up group B and then A even if the resulting partitions are identical) and where a group of students is treated as an unordered set of people. Compute the remainder obtained when $N$ is divided by $2006$.

Solution

Problem 34

For each positive integer $n$ let $S_n$ denote the set of positive integers $k$ such that $n^k-1$ is divisible by $2006$. Define the function $P(n)$ by the rule \[P(n):=\begin{cases}\min(s)_{s\in S_n}&\text{if }S_n\neq\emptyset,\\0&\text{otherwise}.\end{cases}\] Let $d$ be the least upper bound of $\{P(1),P(2),P(3),\ldots\}$ and let $m$ be the number of integers $i$ such that $1\leq i\leq 2006$ and $P(i) = d$. Compute the value of $d+m$.

Solution

Problem 35

Compute the $\textit{number}$ of ordered quadruples $(w,x,y,z)$ of complex numbers (not necessarily nonreal) such that the following system is satisfied: \begin{align*} wxyz&=1\\ wxy^2 + wx^2z + w^2yz + xyz^2&=2\\ wx^2y + w^2y^2 + w^2xz + xy^2z + x^2z^2 + ywz^2 &= -3\\ w^2xy + x^2yz + wy^2z + wxz^2 &= -1 \end{align*}

Solution

Problem 36

Let $\alpha$ denote $\cos^{-1}(\tfrac 23)$. The recursive sequence $a_0,a_1,a_2,\ldots$ satisfies $a_0 = 1$ and, for all positive integers $n$, \[a_n = \dfrac{\cos(n\alpha) - (a_1a_{n-1} + \cdots + a_{n-1}a_1)}{2a_0}.\] Suppose that the series \[\sum_{k=0}^\infty\dfrac{a_k}{2^k}\] can be expressed uniquely as $\tfrac{p\sqrt q}r$, where $p$ and $r$ are coprime positive integers and $q$ is not divisible by the square of any prime. Find the value of $p+q+r$.

Solution

Problem 37

The positive reals $x$, $y$, $z$ satisfy the relations \begin{align*} x^2+xy+y^2&=1,\\ y^2+yz+z^2&=2,\\ z^2+zx+x^2&=3. \end{align*} The value of $y^2$ can be expressed uniquely as $\tfrac{m-n\sqrt p}q$, where $m$, $n$, $p$, $q$ are positive integers such that $p$ is not divisible by the square of any prime and no prime dividing $q$ divides both $m$ and $n$. Compute $m+n+p+q$.

Solution


Problem 38

Segment $AB$ is a diameter of circle $\Gamma_1$. Point $C$ lies in the interior of segment $AB$ such that $BC=7$, and $D$ is a point on $\Gamma_1$ such that $BD=CD=10$. Segment $AC$ is a diameter of the circle $\Gamma_2$. A third circle, $\omega$, is drawn internally tangent to $\Gamma_1$, externally tangent to $\Gamma_2$, and tangent to segment $CD$. If $\omega$ is centered on the opposite side of $CD$ as $B$, then the radius of $\omega$ can be expressed as $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.

Solution

Problem 39

$ABCDEFGHIJKL$ is a regular dodecagon. The number 1 is written at the vertex A, and 0's are written at each of the other vertices. Suddenly and simultaneously, the number at each vertex is replaced by the arithmetic mean of the two numbers appearing at the adjacent vertices. If this procedure is repeated a total of $2006$ times, then the resulting number at A can be expressed as $m/n$, where $m$ and $n$ are relatively prime positive integers. Compute the remainder obtained when $m + n$ is divided by $2006$.

Solution

Problem 40

Acute triangle $ABC$ satisfies $AB=2AC$ and $AB^4+BC^4+CA^4 = 2006\cdot 10^{10}$. Tetrahedron $DEFP$ is formed by choosing points $D$, $E$, and $F$ on the segments $BC$, $CA$, and $AB$ (respectively) and folding $A$, $B$, $C$, over $EF$, $FD$, and $DE$ (respectively) to the common point $P$. Let $R$ denote the circumradius of $DEFP$. Compute the smallest positive integer $N$ for which we can be certain that $n\geq R$. It may be helpful to use $\sqrt[4]{1239} = 5.9329109\ldots$.

Solution

Ultimate Question

In the next 2 problems, the problem after will require the answer of the current problem. TNFTPP stands for the number from the previous problem. Problem 41 requires the answer to the third problem. Problem 42 requires the answer to the seventh problem. Problem 43, however, requires the sum of the answers to all ten questions.

For those who want to try these problems without having to find the T-values of the previous problem, a link will be here. Also, all solutions will have the T-values substituted.

Problem 41

Problem U1

Find the real number $x$ such that \[\sqrt{x-9} + \sqrt{x-6} = \sqrt{x-1}.\]

Solution

Problem U2

Let $T=TNFTPP$. Points $A$ and $B$ lie on a circle centered at $O$ such that $\angle AOB$ is right. Points $C$ and $D$ lie on radii $OA$ and $OB$ respectively such that $AC = T-3$, $CD = 5$, and $BD = 6$. Determine the area of quadrilateral $ACDB$.

[asy] draw(circle((0,0),10)); draw((0,10)--(0,0)--(10,0)--(0,10)); draw((0,3)--(4,0)); label("O",(0,0),SW); label("C",(0,3),W); label("A",(0,10),N); label("D",(4,0),S); label("B",(10,0),E); [/asy]

Solution

Problem U3

Let $T  =  TNFTPP$. When properly sorted, $T  -  35$ math books on a shelf are arranged in alphabetical order from left to right. An eager student checked out and read all of them. Unfortunately, the student did not realize how the books were sorted, and so after finishing the student put the books back on the shelf in a random order. If all arrangements are equally likely, the probability that exactly $6$ of the books were returned to their correct (original) position can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m  +  n$.

Solution

Problem 42

Problem U4

Let $T  =  TNFTPP$. As $n$ ranges over the integers, the expression $n^4  -  898n^2  +  T  -  2160$ evaluates to just one prime number. Find this prime.

Solution

Problem U5

Let $T  =  TNFTPP$, and let $S$ be the sum of the digits of $T$. In triangle $ABC$, points $D$, $E$, and $F$ are the feet of the angle bisectors of $\angle A$, $\angle B$, $\angle C$ respectively. Let point $P$ be the intersection of segments $AD$ and $BE$, and let $p$ denote the perimeter of $ABC$. If $AP  =  3PD$, $BE  =  S  -  1$, and $CF  =  9$, then the value of $\frac{AD}{p}$ can be expressed uniquely as $\frac{\sqrt{m}}{n}$ where $m$ and $n$ are positive integers such that $m$ is not divisible by the square of any prime. Find $m  +  n$.

Solution

Problem U6

Let $T = TNFTPP$. $x$ and $y$ are nonzero real numbers such that

\[18x - 4x^2 + 2x^3 - 9y - 10xy - x^2y + Ty^2 + 2xy^2 - y^3 = 0\]

The smallest possible value of $\frac{y}{x}$ is equal to $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution

Problem U7

Let $T  =  TNFTPP$. Triangle $ABC$ has integer side lengths, including $BC  =  100T  -  4$, and a right angle, $\angle ABC$. Let $r$ and $s$ denote the inradius and semiperimeter of $ABC$ respectively. Find the perimeter of the triangle ABC which minimizes $\frac{s}{r}$.

Solution

Problem 43

Problem U8

Let $T  =  TNFTPP$, and let $S$ be the sum of the digits of $T$. Cyclic quadrilateral $ABCD$ has side lengths $AB  =  S  -  11$, $BC  =  2$, $CD  =  3$, and $DA  =  10$. Let $M$ and $N$ be the midpoints of sides $AD$ and $BC$. The diagonals $AC$ and $BD$ intersect $MN$ at $P$ and $Q$ respectively. $\frac{PQ}{MN}$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Determine $m  +  n$.

Solution

Problem U9

Let $T  =  TNFTPP$. Determine the number of 5 element subsets $S$ of $\{1,2,3, \cdots ,T  +  100\}$ such that the sum of the elements of $S$ is divisible by 5.

Solution

Problem U10

Let $T  =  TNFTPP$ and let $S$ be the sum of the digits of $T$. Point $P$ in the interior of triangle $ABC$ satisfies $AP  =  S  +  51$, $BP  =  156$, and $CP  =  169$. If the sides of ABC satisfy

\[\frac{BC}{13} = \frac{CA}{14} = \frac{AB}{15}\]

then the area of triangle $ABC$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute the remainder obtained when $m  +  n$ is divided by $2006$.

Recall that you are turning in the sum of all ten answers, NOT the answer to this problem.

Solution

See Also

2006 iTest (Problems)
Preceded by:
2005 iTest
Followed by:
2007 iTest
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 U1 U2 U3 U4 U5 U6 U7 U8 U9 U10
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