2007 iTest Problems

Multiple Choice Section

Problem 1

A twin prime pair is a set of two primes $(p, q)$ such that $q$ is $2$ greater than $p$ . What is the arithmetic mean of the two primes in the smallest twin prime pair?

$\mathrm{(A)}\, 4$

Solution

Problem 2

Find $a + b$ if $a$ and $b$ satisfy $3a + 7b = 1977$ and $5a + b = 2007$ .

$\mathrm{(A)}\, 488\quad\mathrm{(B)}\, 498$

Solution

Problem 3

An abundant number is a natural number that's proper divisors sum is greater than the number. Which one of the following natural numbers is an abundant number?

$\mathrm{(A)}\, 14\quad\mathrm{(B)}\, 28\quad\mathrm{(C)}\, 56$

Solution

Problem 4

Star flips a quarter four times. Find the probability that the quarter lands heads exactly twice.

$\mathrm{(A)}\,\frac{1}{8}\quad\mathrm{(B)}\,\frac{3}{16}\quad\mathrm{(C)}\,\frac{3}{8}\quad\mathrm{(D)}\,\frac{1}{2}$

Solution

Problem 5

Compute the sum of all twenty-one terms of the geometric series $1 + 2 + 4 + 8 + \ldots + 1048576$ .

$\mathrm{(A)}\,2097149\quad\mathrm{(B)}\,2097151\quad\mathrm{(C)}\,2097153\quad\mathrm{(D)}\,2097157\quad\mathrm{(E)}\,2097161$

Solution

Problem 6

Find the units digit of the sum

$\sum_{i=1}^{100}(i!)^{2}$

$\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,3\quad\mathrm{(D)}\,5\quad\mathrm{(E)}\,7\quad\mathrm{(F)}\,9$

Solution

Problem 7

An equilateral triangle with side length $1$ has the same area as a square with side length $s$ . Find $s$ .

$\mathrm{(A)}\,\frac{\sqrt[4]{3}}{2}\quad\mathrm{(B)}\,\frac{\sqrt[4]{3}}{\sqrt{2}}\quad\mathrm{(C)}\,1\quad\mathrm{(D)}\,\frac{3}{4}\quad\mathrm{(E)}\,\frac{4}{3}\quad\mathrm{(F)}\,\sqrt{3}\quad\mathrm{(G)}\,\frac{\sqrt6}{2}$

Solution

Problem 8

Joe is right at the middle of a train tunnel and he realizes that a train is coming. The train travels at a speed of 50 miles per hour, and Joe can run at a speed of 10 miles per hour. Joe hears the train whistle when the train is a half mile from the point where it will enter the tunnel. At that point in time, Joe can run toward the train and just exit the tunnel as the train meets him. Instead, Joe runs away from the train when he hears the whistle. How many seconds does he have to spare (before the train is upon him) when he gets to the tunnel entrance?

$\mathrm{(A)}\,7.2\quad\mathrm{(B)}\,14.4\quad\mathrm{(C)}\,36\quad\mathrm{(D)}\,10\quad\mathrm{(E)}\,12\quad\mathrm{(F)}\,2.4\quad\mathrm{(G)}\,25.2\quad\mathrm{(H)}\,123456789$

Solution

Problem 9

Suppose that $m$ and $n$ are positive integers such that $m < n$ , the geometric mean of $m$ and $n$ is greater than $2007$ , and the arithmetic mean of $m$ and $n$ is less than $2007$ . How many pairs $(m, n)$ satisfy these conditions?

$\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,2\quad\mathrm{(D)}\,3\quad\mathrm{(E)}\,4\quad\mathrm{(F)}\,5\quad\mathrm{(G)}\,6\quad\mathrm{(H)}\,7\quad\mathrm{(I)}\,2007$

Solution

Problem 10

My grandparents are Arthur, Bertha, Christoph, and Dolores. My oldest grandparent is only $4$ years older than my youngest grandparent. Each grandfather is two years older than his wife. If Bertha is younger than Dolores, what is the difference between Bertha’s age and the mean of my grandparents’ ages?

$\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,2\quad\mathrm{(D)}\,3\quad\mathrm{(E)}\,4\quad\mathrm{(F)}\,5\quad\mathrm{(G)}\,6\quad\mathrm{(H)}\,7\quad\mathrm{(I)}\,8\quad\mathrm{(J)}\,2007$

Solution

Problem 11

Consider the "tower of power" $2^{2^{2^{\cdot^{\cdot^\cdot^{2}}}}}$ (Error compiling LaTeX. Unknown error_msg), where there are 2007 twos including the base. What is the last (units digit) of this number?

$\text{(A) }0\qquad \text{(B) }1\qquad \text{(C) }2\qquad \text{(D) }3\qquad \text{(E) }4\qquad \text{(F) }5\qquad \text{(G) }6\qquad \text{(H) }7\qquad \text{(I) }8\qquad \text{(J) }9\qquad \text{(K) }2007\qquad$

Solution

Problem 12

My Frisbee group often calls "best of five" to finish our games when it's getting dark, since we don't keep score. The game ends after one of the two teams scores three points (total, not necessarily consecutive). If every possible sequence of scores is equally likely, what is the expected score of the losing team.

$\text{(A) }\frac{2}{3}\qquad \text{(B) }1\qquad \text{(C) }\frac{3}{2}\qquad \text{(D) }\frac{8}{5}\qquad \text{(E) }\frac{5}{8}\qquad \text{(F) }2\qquad\\ \\ \text{(G) }0\qquad \text{(H) }\frac{5}{2}\qquad \text{(I) }\frac{2}{5}\qquad \text{(J) }\frac{3}{4}\qquad \text{(K) }\frac{4}{3}\qquad \text{(L) }2007\qquad$

Solution

Problem 13

What is the smallest positive integer $k$ such that the number ${{2k}\choose k}$ ends in two zeros?

$\text{(A) } 3 \quad \text{(B) } 4 \quad \text{(C) } 5 \quad \text{(D) } 6 \quad \text{(E) } 7 \quad \text{(F) } 8 \quad \text{(G) } 9 \quad \text{(H) } 10 \quad \text{(I) } 11 \quad \text{(J) } 12 \quad \text{(K) } 13 \quad \text{(L) } 14 \quad \text{(M) } 2007\quad$

Solution

Problem 14

Let $\phi(n)$ be the number of positive integers $k< n$ which are relatively prime to $n$ . For how many distinct values of $n$ is $\phi(n)=12$ ?

$\text{(A) } 0 \quad \text{(B) } 1 \quad \text{(C) } 2 \quad \text{(D) } 3 \quad \text{(E) } 4 \quad \text{(F) } 5 \quad \text{(G) } 6 \quad \text{(H) } 7 \quad \text{(I) } 8 \quad \text{(J) } 9 \quad \text{(K) } 10 \quad \text{(L) } 11 \quad \text{(M) } 12\quad \text{(N) } 13\quad$

Solution

Problem 15

Form a pentagon by taking a square of side length 1 and an equilateral triangle of side length 1 and placing the triangle so that one of its sides coincides with a side of the square. Then "circumscribe" a circle around the pentagon, passing through three of its vertices, so that the circle passes through exactly one of the vertices of the equilateral triangle, and through exactly two vertices of the square. What is the radius of the circle?

$\text{(A) }\qquad \text{(B) }\qquad \text{(C) }\qquad \text{(D) }\qquad \text{(E) }\qquad \text{(F) }\qquad \text{(G) }\qquad \text{(H) }\qquad \$$ \text{(I) }\qquad \text{(J) }\qquad \text{(K) }\qquad \text{(L) }\qquad \text{(M) }\qquad$

Problem 16

How many lattice points lie within or on the border of the circle in the $xy$ -plane defined by the equation $x^2+y^2=100$

$\text{(A) }1\qquad \text{(B) }2\qquad \text{(C) }4\qquad \text{(D) }5\qquad \text{(E) }41\qquad \text{(F) }42\qquad \text{(G) }69\qquad \text{(H) }76\qquad \text{(I) }130\qquad \\ \text{(J) }133\qquad \text{(K) }233\qquad \text{(L) }311\qquad \text{(M) }317\qquad \text{(N) }420\qquad \text{(O) }520\qquad \text{(P) }2007$

Problem 17

Problem 18

Problem 19

Problem 20

Find the largest integer $n$ such that $2007^{1024}-1$ is divisible by $2^n$

$\text{(A) } 1\qquad \text{(B) } 2\qquad \text{(C) } 3\qquad \text{(D) } 4\qquad \text{(E) } 5\qquad \text{(F) } 6\qquad \text{(G) } 7\qquad \text{(H) } 8\qquad \$ (Error compiling LaTeX. Unknown error_msg) $\text{(I) } 9\qquad \text{(J) } 10\qquad \text{(K) } 11\qquad \text{(L) } 12\qquad \text{(M) } 13\qquad \text{(N) } 14\qquad \text{(O) } 15\qquad \text{(P) } 16\qquad \$ (Error compiling LaTeX. Unknown error_msg) $\text{(Q) } 55\qquad \text{(R) } 63\qquad \text{(S) } 64\qquad \text{(T) } 2007\qquad$

Problem 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 \$ (Error compiling LaTeX. Unknown error_msg)?

$\text{(A) }0\qquad \text{(B) }\frac{6435 }{2^{14}}\qquad \text{(C) }\frac{6435 }{2^{13}}\qquad \text{(D) }\frac{429}{2^{12}}\qquad \text{(E) }\frac{429}{2^{11}}\qquad \text{(F) }\frac{429}{2^{10}}\qquad \text{(G) }\frac{1}{15}\qquad \text{(H) }\frac{1}{31}\qquad \$ (Error compiling LaTeX. Unknown error_msg)

$\text{(I) }\frac{1}{30}\qquad \text{(J) }\frac{1}{29}\qquad \text{(K) }\frac{6435 }{2^{15}}\qquad \text{(L) }\frac{6435 }{2^{14}}\qquad \text{(M) }\frac{6435 }{2^{13}}\qquad \text{(N) }\frac{1}{2^{7}}\qquad \text{(O) }\frac{1}{2^{14}}\qquad \text{(P) }\frac{1}{2^{15}}\qquad \$ (Error compiling LaTeX. Unknown error_msg)

$\text{(P) }\frac{2007}{2^{14}}\qquad \text{(P) }\frac{2007}{2^{15}}\qquad \text{(P) }\frac{2007}{2^{2007}}\qquad \text{(P) }\frac{1}{2007}\qquad \text{(P) }\frac{-2007}{2^{14}}\qquad$

Problem 22

Problem 23

Problem 24

Problem 25

Ted's favorite number is equal to $1\cdot{2007\choose 1}+2\cdot {2007\choose 2}+3\cdot {2007\choose 3} + \cdots + 2007\cdot {2007 \choose 2007}$

Find the remainder when Ted's favorite number is divided by 25.

$\text{(A) } 0\qquad \text{(B) } 1\qquad \text{(C) } 2\qquad \text{(D) } 3\qquad \text{(E) } 4\qquad \text{(F) } 5\qquad \text{(G) } 6\qquad \text{(H) } 7\qquad \text{(I) } 8\qquad \$ (Error compiling LaTeX. Unknown error_msg)

$\text{(J) } 9\qquad \text{(K) } 10\qquad \text{(L) } 11\qquad \text{(M) } 12\qquad \text{(N) } 13\qquad \text{(O) } 14\qquad \text{(P) } 15\qquad \text{(Q) } 16\qquad \$ (Error compiling LaTeX. Unknown error_msg)

$\text{(R) } 17\qquad \text{(S) } 18\qquad \text{(T) } 19\qquad \text{(U) } 20\qquad \text{(V) } 21\qquad \text{(A) } 22\qquad \text{(X) } 23\qquad \text{(Y) } 24$

Short Answer Section

Problem 26

Problem 27

Problem 28

Problem 29

Problem 30

Problem 31

Problem 32

Problem 33

Problem 34

Problem 35

Problem 36

Problem 37

Problem 38

Problem 39

Problem 40

Problem 41

Problem 42

Problem 43

Problem 44

Problem 45

Problem 46

Problem 47

Problem 48

Problem 49

Problem 50

Ultimate Question

Problem 51

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

Problem 52

Let $T=TNFTPP$ . Let $R$ be the region consisting of points $(x,y)$ of the Cartesian plane satisfying both $|x|-|y|\le T-500$ and $|y|\le T-500$ . Find the area of region $R$ .

Solution

Problem 53

Problem 54

Problem 55

Problem 56

Problem 57

Problem 58

Problem 59

Problem 60

Tiebreaker Questions

Problem TB1

The sum of the digits of an integer is equal to the sum of the digits of three times that integer. Prove that the integer is a multiple of 9.

Solution

Problem TB2

Factor completely over integer coefficients the polynomial $p(x)=x^8+x^5+x^4+x^3+x+1$ . Demonstrate that your factorization is complete.

Solution

Problem TB3

4014 boys and 4014 girls stand in a line holding hands, such that only the two people at the ends are not holding hands with exactly two people (an ordinary line of people). One of the two people at the ends gets tired of the hand-holding fest and leaves. Then, from the remaining line, one of the two people at the ends leaves. Then another from an end, and then another, and another. This continues until exactly half of the people from the original line remain. Prove that no matter what order the original 8028 people were standing in, that it is possible that exactly 2007 of the remaining people are girls.

Solution

Problem TB4

Circle $O$ is the circumcircle of non-isosceles triangle $ABC$ . The tangent lines to circle $O$ at points $B$ and $C$ intersect at $L_a$ , and the tangents at $A$ and $C$ intersect at $L_b$ . The external angle bisectors of triangle $ABC$ at $B$ and $C$ meet at $I_a$ and the external bisectors at $A$ and $C$ intersect at $I_b$ . Prove that lines $L_aI_a$ , $L_bI_b$ , and $AB$ are concurrent.