Difference between revisions of "2006 iTest Problems"

Latest revision as of 23:02, 3 November 2023

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

@@ Line 538: / Line 538: @@
 [[2006 iTest Problems/Problem 37|Solution]]
-==Solution==
-To begin, let's number the equations as follows
-<cmath>\begin{align}
-x^2 + xy + y^2 &= 1 \\
-y^2 + yz +z^2 &= 2 \\
-z^2 + zx +x^2 &= 3
-\end{align}</cmath>
-In order to solve for <math>y^2</math>, we begin by subtracting the first equation from the second
-equation:
-<cmath>\begin{align*}
-&=y^2 + yz +z^2 -(x^2 + xy + y^2)\\
-&= z^2 + yz - xy - x^2 \\
-&=(z-x)(x + y + z) \\
-\end{align*}</cmath>
-Similarily, subtracting the second eqaution from the  third equation:
-<cmath>\begin{align*}
-&= x^2 + xz +z^2 -(y^2 + zy + z^2)\\
-&= x^2 + xz - zy - y^2 \\
-&=(x-y)(x + y + z) \\
-\end{align*}</cmath>
-Subtracting the factored equations we see that
-<cmath>\begin{align*}
-&=(x-y)(x + y + z)-(z-x)(x + y + z)\\
-&=(x+y+z)((x-y)-(z-x))\\
-&=(x+y+z)(2x-y-z)\\
-\end{align*}</cmath>
-Therefore, we can conclude that either <math>x+y+z=0 \Rightarrow x=-z-y</math>, or that <math>2x-y-z=0 \Rightarrow x=\frac{z+y}{2}</math>.  As we are told that <math>y^2</math> has a unique value, we can use either option.
-Continuing with the second option, we can plug <math>x=\frac{z+y}{2}</math> into the first equation:
-<cmath>\begin{align*}
-&=x^2+xy+y^2\\
-&=\left(\frac{z+y}{2}\right)^2 + \left(\frac{z+y}{2}\right)\cdot y +y^2 \\
-&=\frac{z^2+2yz+y^2}{4} + \frac{yz+y^2}{2} + y^2
-\end{align*}</cmath>
-Multiplying each side of the resulting equation by four:
-<cmath>\begin{align*}
-&=4 \cdot \left(\frac{z^2+2yz+y^2}{4} + \frac{yz+y^2}{2} + y^2 \right)\\
-&=(z^2+2yz+y^2)+(2yz+2y^2)+4y^2\\
-&=z^2+4yz+7y^2
-\end{align*}</cmath>
-We can then subtract twice our original second equation, <math>y^2+yz+z^2=2</math>, from the equation from the previous step, yielding:
-<cmath>\begin{align*}
--2\cdot2&=(z^2+4yz+7y^2)-2(y^2+yz+z^2)\\
-&=5y^2+2yz-z^2\\
-\end{align*}</cmath>
-Rearranging the new equation, we see that:
-<cmath>\begin{align*}
-y^2 &=z^2-2yz\\
-y^2 &= z^2-2yz+y^2\\
-y^2 &=(z-y)^2\\
-\sqrt{6y^2}&=\sqrt{(z-y)^2}\\
-y\sqrt{6}&=z-y\\
-\end{align*}</cmath>
-We can ignore the negative root of the RHS because both y and z are positive, and because <math>z>y</math> which is a result of (3) being greater than (1).
-Continuing to manipulate the newest equation yields:
-<cmath>\begin{align*}
-z-y &=y\sqrt{6}\\
-z&=y\sqrt{6}+y\\
-z&=y(\sqrt{6}+1)
-\end{align*}</cmath>
-Plugging this back into (2)
-<cmath>\begin{align*}
-y^2 + yz +z^2 =& \quad 2\\
-y^2+y \cdot y(\sqrt{6}+1) + (y(\sqrt{6}+1))^2 =&\\
-y^2\cdot(1+(\sqrt{6}+1)+(\sqrt{6}+1)^2)=&\\
-y^2\cdot(1+(\sqrt{6}+1)+(6+2\sqrt{6}+1))=&\\
-y^2\cdot(9+3\sqrt{6})=&
-\end{align*}</cmath>
-Solving for <math>y^2</math> yields that <math>y^2= \frac{2}{9+3\sqrt{6}} \Rightarrow y^2=\frac{2}{9+3\sqrt{6}} \cdot \frac{9-3\sqrt{6}}{9-3\sqrt{6}} = \frac{18-6\sqrt{6}}{27} = \frac{6-2\sqrt{6}}{9}</math>
-Therefore, the solution is <math> 6+2+6+9 = \boxed{23}</math>
 ===Problem 38===
@@ Line 682: / Line 590: @@
 ====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]]

2006 iTest (Problems, Answer Key)
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Difference between revisions of "2006 iTest Problems"

Latest revision as of 23:02, 3 November 2023

Contents

Multiple Choice Section

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Short Answer Section

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

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

Ultimate Question

Problem 41

Problem U1

Problem U2

Problem U3

Problem 42

Problem U4

Problem U5

Problem U6

Problem U7

Problem 43

Problem U8

Problem U9

Problem U10

See Also