Difference between revisions of "Mock AIME 4 2005-2006/Problems"
Line 90: | Line 90: | ||
http://www.mathlinks.ro/Forum/latexrender/pictures/2ea0ce047615395691113a82d6c190b3.gif | http://www.mathlinks.ro/Forum/latexrender/pictures/2ea0ce047615395691113a82d6c190b3.gif | ||
− | + | == Problem 1 == | |
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
1. A 5-digit number is leet if and only if the sum of the first 2 digits, the sum of the last 2 digits and the middle digit are equal. How many 5-digit leet numbers exist? | 1. A 5-digit number is leet if and only if the sum of the first 2 digits, the sum of the last 2 digits and the middle digit are equal. How many 5-digit leet numbers exist? | ||
− | + | [[Mock AIME 5 2005-2006/Problem 14|Solution]] | |
− | + | == Problem 2 == | |
2. Qin Shi Huang wants to count the number of warriors he has to invade China. He puts his warriors into lines with the most people such that they have even length. The people left over are the remainder. He makes 2 lines, with a remainder of 1, 3 lines with a remainder of 2, 4 lines with a remainder of 3, 5 lines with a remainder of 4, and 6 lines with a remainder of 5. Find the minimum number of warriors that he has. | 2. Qin Shi Huang wants to count the number of warriors he has to invade China. He puts his warriors into lines with the most people such that they have even length. The people left over are the remainder. He makes 2 lines, with a remainder of 1, 3 lines with a remainder of 2, 4 lines with a remainder of 3, 5 lines with a remainder of 4, and 6 lines with a remainder of 5. Find the minimum number of warriors that he has. | ||
− | + | [[Mock AIME 5 2005-2006/Problem 14|Solution]] | |
− | + | == Problem 3 == | |
3. <math>T_1</math> is a regular tetrahedron. Tetrahedron <math>T_2</math> is formed by connecting the centers of the faces of <math>T_1</math>. Generally, a new tetrahedron <math>T_{n+1}</math> is formed by connecting the centers of the faces of <math>T_n</math>. <math>V_n</math> is the volume of tetrahedron <math>T_n</math>. <math>\frac{V_{2006}}{V_1}=\frac{m}{n}</math> where <math>m</math> and <math>n</math> are coprime positive integers, find the remainder when <math>m+n</math> is divided by <math>1000</math>. | 3. <math>T_1</math> is a regular tetrahedron. Tetrahedron <math>T_2</math> is formed by connecting the centers of the faces of <math>T_1</math>. Generally, a new tetrahedron <math>T_{n+1}</math> is formed by connecting the centers of the faces of <math>T_n</math>. <math>V_n</math> is the volume of tetrahedron <math>T_n</math>. <math>\frac{V_{2006}}{V_1}=\frac{m}{n}</math> where <math>m</math> and <math>n</math> are coprime positive integers, find the remainder when <math>m+n</math> is divided by <math>1000</math>. | ||
− | + | [[Mock AIME 5 2005-2006/Problem 14|Solution]] | |
− | + | == Problem 4 == | |
4. Let <math>P(x)=\sum_{i=1}^{20}(-x)^{20-i}(x+i)^i</math>. Let <math>K</math> be the product of the roots. How many digits are does <math>\lfloor K \rfloor</math> have where <math>\lfloor x \rfloor</math> denotes the greatest integer less than or equal to <math>x</math>? | 4. Let <math>P(x)=\sum_{i=1}^{20}(-x)^{20-i}(x+i)^i</math>. Let <math>K</math> be the product of the roots. How many digits are does <math>\lfloor K \rfloor</math> have where <math>\lfloor x \rfloor</math> denotes the greatest integer less than or equal to <math>x</math>? | ||
− | + | [[Mock AIME 5 2005-2006/Problem 14|Solution]] | |
− | + | == Problem 5 == | |
5. A parabola <math>P: y=x^2</math> is rotated <math>135</math> degrees clockwise about the origin to <math>P'</math>. This image is translated upward <math>\frac{8+\sqrt{2}}{2}</math> to <math>P''</math>. Point <math>A: (0,0)</math>, <math>B: (256,0)</math>, and <math>C</math> is in Quadrant I, on <math>P''</math>. If the area of <math>\triangle ABC</math> is at a maximum, it is <math>a\sqrt{b}+c</math> where <math>a</math>, <math>b</math> and <math>c</math> are integers and <math>b</math> is square free, find <math>a+b+c</math>. | 5. A parabola <math>P: y=x^2</math> is rotated <math>135</math> degrees clockwise about the origin to <math>P'</math>. This image is translated upward <math>\frac{8+\sqrt{2}}{2}</math> to <math>P''</math>. Point <math>A: (0,0)</math>, <math>B: (256,0)</math>, and <math>C</math> is in Quadrant I, on <math>P''</math>. If the area of <math>\triangle ABC</math> is at a maximum, it is <math>a\sqrt{b}+c</math> where <math>a</math>, <math>b</math> and <math>c</math> are integers and <math>b</math> is square free, find <math>a+b+c</math>. | ||
− | + | [[Mock AIME 5 2005-2006/Problem 14|Solution]] | |
− | + | == Problem 6 == | |
6. Define a sequence <math>a_0=2006</math> and <math>a_{n+1}=(n+1)^{a_n}</math> for all positive integers <math>n</math>. Find the remainder when <math>a_{2007}</math> is divided by <math>1000</math>. | 6. Define a sequence <math>a_0=2006</math> and <math>a_{n+1}=(n+1)^{a_n}</math> for all positive integers <math>n</math>. Find the remainder when <math>a_{2007}</math> is divided by <math>1000</math>. | ||
− | + | [[Mock AIME 5 2005-2006/Problem 14|Solution]] | |
− | + | == Problem 7 == | |
− | |||
7. <math>f(x)</math> is a function that satisfies <math>3f(x)=2x+1-f(\frac{1}{1-x})</math> for all defined <math>x</math>. Suppose that the sum of the zeros of <math>f(x)=\frac{m}{n}</math> where <math>m</math> and <math>n</math> are coprime positive integers, find <math>m^2+n^2</math>. | 7. <math>f(x)</math> is a function that satisfies <math>3f(x)=2x+1-f(\frac{1}{1-x})</math> for all defined <math>x</math>. Suppose that the sum of the zeros of <math>f(x)=\frac{m}{n}</math> where <math>m</math> and <math>n</math> are coprime positive integers, find <math>m^2+n^2</math>. | ||
− | + | [[Mock AIME 5 2005-2006/Problem 14|Solution]] | |
− | + | == Problem 8 == | |
− | |||
8. <math>R</math> is a solution to <math>x+\frac{1}{x}=\frac{ \sin210^{\circ} }{\sin285^{\circ} }</math>. Suppose that <math>\frac{1}{R^{2006}}+R^{2006}=A</math> find <math>\lfloor A^{10} \rfloor</math> where <math>\lfloor x \rfloor</math> is the greatest integer less than or equal to <math>x</math>. | 8. <math>R</math> is a solution to <math>x+\frac{1}{x}=\frac{ \sin210^{\circ} }{\sin285^{\circ} }</math>. Suppose that <math>\frac{1}{R^{2006}}+R^{2006}=A</math> find <math>\lfloor A^{10} \rfloor</math> where <math>\lfloor x \rfloor</math> is the greatest integer less than or equal to <math>x</math>. | ||
− | + | [[Mock AIME 5 2005-2006/Problem 14|Solution]] | |
+ | == Problem 9 == | ||
9. Zeus, Athena, and Posideon arrive at Mount Olympus at a random time between 12:00 pm and 12:00 am, and stay for 3 hours. All three hours does not need to fall within 12 pm to 12 am. If any of the 2 gods see each other during 12 pm to 12 am, it will be a good day. The probability of it being a good day is <math>\frac{m}{n}</math> where <math>m</math> and <math>n</math> are coprime positive integers, find <math>m+n</math>. | 9. Zeus, Athena, and Posideon arrive at Mount Olympus at a random time between 12:00 pm and 12:00 am, and stay for 3 hours. All three hours does not need to fall within 12 pm to 12 am. If any of the 2 gods see each other during 12 pm to 12 am, it will be a good day. The probability of it being a good day is <math>\frac{m}{n}</math> where <math>m</math> and <math>n</math> are coprime positive integers, find <math>m+n</math>. | ||
− | + | [[Mock AIME 5 2005-2006/Problem 14|Solution]] | |
− | + | == Problem 10 == | |
− | |||
10. Define <math>S= \tan^2{1^{\circ}}+\tan^2{3^{\circ}}+\tan^2{5^{\circ}}+...+\tan^2{87^{\circ}}+\tan^2{89^{\circ}}</math>. Find the remainder when <math>S</math> is divided by <math>1000</math>. | 10. Define <math>S= \tan^2{1^{\circ}}+\tan^2{3^{\circ}}+\tan^2{5^{\circ}}+...+\tan^2{87^{\circ}}+\tan^2{89^{\circ}}</math>. Find the remainder when <math>S</math> is divided by <math>1000</math>. | ||
− | + | [[Mock AIME 5 2005-2006/Problem 14|Solution]] | |
− | + | == Problem 11 == | |
11. <math>\triangle ABC</math> is isosceles with <math>\angle C= 90^{\circ}</math>. A point <math>P</math> lies inside the triangle such that <math>AP=33</math>, <math>CP=28\sqrt{2}</math>, and <math>BP=65</math>. Let <math>A</math> be the area of <math>\triangle ABC</math>. Find the remainder when <math>2A</math> is divided by <math>1000</math>. | 11. <math>\triangle ABC</math> is isosceles with <math>\angle C= 90^{\circ}</math>. A point <math>P</math> lies inside the triangle such that <math>AP=33</math>, <math>CP=28\sqrt{2}</math>, and <math>BP=65</math>. Let <math>A</math> be the area of <math>\triangle ABC</math>. Find the remainder when <math>2A</math> is divided by <math>1000</math>. | ||
− | + | [[Mock AIME 5 2005-2006/Problem 14|Solution]] | |
− | + | == Problem 12 == | |
− | |||
12. There exists a line <math>L</math> with points <math>D</math>,<math>E</math>,<math>F</math> with <math>E</math> in between <math>D</math> and <math>F</math>. Point <math>A</math>, not on the line is such that <math>\overline{AF}=6</math>, <math>\overline{AD}=\frac{36}{7}</math>, <math>\overline{AE}=\frac{12}{\sqrt{7}}</math> with <math>\angle AEF > 90</math>. Construct <math>E'</math> on ray <math>AE</math> such that <math>(\overline{AE})(\overline{AE'})=36</math> and <math>\overline{FE'}=3</math>. Point <math>G</math> is on ray <math>AD</math> such that <math>\overline{AG}=7</math>. If <math>2*(\overline{E'G})=a+\sqrt{b}</math> where <math>a</math> and <math>b</math> are integers, then find <math>a+b</math>. | 12. There exists a line <math>L</math> with points <math>D</math>,<math>E</math>,<math>F</math> with <math>E</math> in between <math>D</math> and <math>F</math>. Point <math>A</math>, not on the line is such that <math>\overline{AF}=6</math>, <math>\overline{AD}=\frac{36}{7}</math>, <math>\overline{AE}=\frac{12}{\sqrt{7}}</math> with <math>\angle AEF > 90</math>. Construct <math>E'</math> on ray <math>AE</math> such that <math>(\overline{AE})(\overline{AE'})=36</math> and <math>\overline{FE'}=3</math>. Point <math>G</math> is on ray <math>AD</math> such that <math>\overline{AG}=7</math>. If <math>2*(\overline{E'G})=a+\sqrt{b}</math> where <math>a</math> and <math>b</math> are integers, then find <math>a+b</math>. | ||
− | + | [[Mock AIME 5 2005-2006/Problem 14|Solution]] | |
− | + | == Problem 13 == | |
− | |||
13. <math>\triangle VA_0A_1</math> is isosceles with base <math>\overline{{A_1A_0}}</math>. Construct <math>A_2</math> on segment <math>\overline{{A_0V}}</math> such that <math>\overline{A_0A_1}=\overline{A_1A_2}=b</math>. Construct <math>A_3</math> on <math>\overline{A_1V}</math> such that <math>b=\overline{A_2A_3}</math>. Contiue this pattern: construct <math>\overline{A_{2n}A_{2n+1}}=b</math> with <math>A_{2n+1}</math> on segment <math>\overline{VA_1}</math> and <math>\overline{A_{2n+1}A_{2n+2}}=b</math> with <math>A_{2n+2}</math> on segment <math>\overline{VA_0}</math>. The points <math>A_n</math> do not coincide and <math>\angle VA_1A_0=90-\frac{1}{2006}</math>. Suppose <math>A_k</math> is the last point you can construct on the perimeter of the triangle. Find the remainder when <math>k</math> is divided by <math>1000</math>. | 13. <math>\triangle VA_0A_1</math> is isosceles with base <math>\overline{{A_1A_0}}</math>. Construct <math>A_2</math> on segment <math>\overline{{A_0V}}</math> such that <math>\overline{A_0A_1}=\overline{A_1A_2}=b</math>. Construct <math>A_3</math> on <math>\overline{A_1V}</math> such that <math>b=\overline{A_2A_3}</math>. Contiue this pattern: construct <math>\overline{A_{2n}A_{2n+1}}=b</math> with <math>A_{2n+1}</math> on segment <math>\overline{VA_1}</math> and <math>\overline{A_{2n+1}A_{2n+2}}=b</math> with <math>A_{2n+2}</math> on segment <math>\overline{VA_0}</math>. The points <math>A_n</math> do not coincide and <math>\angle VA_1A_0=90-\frac{1}{2006}</math>. Suppose <math>A_k</math> is the last point you can construct on the perimeter of the triangle. Find the remainder when <math>k</math> is divided by <math>1000</math>. | ||
− | + | [[Mock AIME 5 2005-2006/Problem 14|Solution]] | |
− | + | == Problem 14 == | |
14. <math>P</math> is the probability that if you flip a fair coin, <math>20</math> heads will occur before <math>19</math> tails. If <math>P=\frac{m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive integers, find the remainder when <math>m+n</math> is divided by <math>1000</math>. | 14. <math>P</math> is the probability that if you flip a fair coin, <math>20</math> heads will occur before <math>19</math> tails. If <math>P=\frac{m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive integers, find the remainder when <math>m+n</math> is divided by <math>1000</math>. | ||
− | + | [[Mock AIME 5 2005-2006/Problem 14|Solution]] | |
− | + | == Problem 15 == | |
15. A regular 61-gon with verticies <math>A_1</math>, <math>A_2</math>, <math>A_3</math>,...<math>A_{61}</math> is inscribed in a circle with a radius of <math>r</math>. Suppose <math>(\overline{A_1A_2})(\overline{A_1A_3})(\overline{A_1A_4})...(\overline{A_1A_{61}})=r</math>. If <math>r^{2006}=\frac{p}{q}</math> where <math>p</math> and <math>q</math> are coprime positive integers, find the remainder when <math>p+q</math> is divided by <math>1000</math>. | 15. A regular 61-gon with verticies <math>A_1</math>, <math>A_2</math>, <math>A_3</math>,...<math>A_{61}</math> is inscribed in a circle with a radius of <math>r</math>. Suppose <math>(\overline{A_1A_2})(\overline{A_1A_3})(\overline{A_1A_4})...(\overline{A_1A_{61}})=r</math>. If <math>r^{2006}=\frac{p}{q}</math> where <math>p</math> and <math>q</math> are coprime positive integers, find the remainder when <math>p+q</math> is divided by <math>1000</math>. | ||
− | + | [[Mock AIME 5 2005-2006/Problem 14|Solution]] | |
− | [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=70988 Here is the page on AoPS] | + | == See also == |
+ | * [[Mock AIME 4 2005-2006]] | ||
+ | * [[Mock AIME]] | ||
+ | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=70988 Here is the page on AoPS] |
Revision as of 04:44, 26 February 2007
Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 See also
- 17 Problem 1
- 18 Problem 2
- 19 Problem 3
- 20 Problem 4
- 21 Problem 5
- 22 Problem 6
- 23 Problem 7
- 24 Problem 8
- 25 Problem 9
- 26 Problem 10
- 27 Problem 11
- 28 Problem 12
- 29 Problem 13
- 30 Problem 14
- 31 Problem 15
- 32 See also
Problem 1
Suppose is a positive integer. Let
be the sum of the distinct positive prime divisors of
less than
(e.g.
and
). Evaluate the remainder when
is divided by
.
Problem 2
A circle of radius
is internally tangent to a larger circle
of radius
such that the center of
lies on
. A diameter
of
is drawn tangent to
. A second line
is drawn from
tangent to
. Let the line tangent to
at
intersect
at
. Find the area of
.
Problem 3
A number
, where
denotes the
th digit in the base-
representation of
for
, is a positive integer with distinct nonzero digits such that
if
is even and
if
is odd for
(and
). Let
be the number of four-digit hailstone numbers and
be the number of three-digit hailstone numbers. Find
.
Problem 4
Let and
be integers such that
and
. Given that
and
, find the number of ordered pairs
such that
. (
is the greatest integer less than or equal to
and
is the least integer greater than or equal to
).
Problem 5
Find the largest prime divisor of .
Problem 6
,
, and
are polynomials defined by:
Find the number of distinct complex roots of .
Problem 7
A coin of radius is flipped onto an
square grid divided into
equal squares. Circles are inscribed in
of these
squares. Let
be the probability that, given that the coin lands completely within one of the smaller squares, it also lands completely within one of the circles. Let
be the probability that, when flipped onto the grid, the coin lands completely within one of the smaller squares. Let
smallest value of
such that
. Find the value of
.
Problem 8
Let be a polyhedron with
faces, all of which are equilateral triangles, squares, or regular pentagons with equal side length. Given there is at least one of each type of face and there are twice as many pentagons as triangles, what is the sum of all the possible number of vertices
can have?
Problem 9
nondistinguishable residents are moving into
distinct houses in Conformistville, with at least one resident per house. In how many ways can the residents be assigned to these houses such that there is at least one house with
residents?
Problem 10
Find the smallest positive integer such that
is divisible by all the primes between
and
.
Problem 11
Let be a subset of consecutive elements of
where
is a positive integer. Define
, where
if
has an odd number of divisors and
if
has an even number of divisors. For how many
does there exist an
such that
and
? (
denotes the cardinality of the set
, or the number of elements in
)
Problem 12
Let be a triangle with
,
, and
. Let
be the foot of the altitude from
to
and
be the point on
between
and
such that
. Extend
to meet the circumcircle of
at
. If the area of triangle
is
, where
and
are relatively prime positive integers, find
.
Problem 13
Let be the set of positive integers with only odd digits satisfying the following condition: any
with
digits must be divisible by
. Let
be the sum of the
smallest elements of
. Find the remainder upon dividing
by
.
Problem 14
Let be a triangle such that
,
, and
. Let
be the orthocenter of
(intersection of the altitudes). Let
be the midpoint of
,
be the midpoint of
, and
be the midpoint of
. Points
,
, and
are constructed on
,
, and
, respectively, such that
is the midpoint of
,
is the midpoint of
, and
is the midpoint of
. Find
.
Problem 15
colored beads are placed on a necklace (circular ring) such that each bead is adjacent to two others. The beads are labeled
,
,
,
around the circle in order. Two beads
and
, where
and
are non-negative integers, satisfy
if and only if the color of
is the same as the color of
. Given that there exists no non-negative integer
and positive integer
such that
, where all subscripts are taken
, find the minimum number of different colors of beads on the necklace.
See also
- Mock AIME 5 2005-2006
- Mock AIME
- A .pdf version of the problems
http://www.mathlinks.ro/Forum/latexrender/pictures/2ea0ce047615395691113a82d6c190b3.gif
Problem 1
1. A 5-digit number is leet if and only if the sum of the first 2 digits, the sum of the last 2 digits and the middle digit are equal. How many 5-digit leet numbers exist? Solution
Problem 2
2. Qin Shi Huang wants to count the number of warriors he has to invade China. He puts his warriors into lines with the most people such that they have even length. The people left over are the remainder. He makes 2 lines, with a remainder of 1, 3 lines with a remainder of 2, 4 lines with a remainder of 3, 5 lines with a remainder of 4, and 6 lines with a remainder of 5. Find the minimum number of warriors that he has. Solution
Problem 3
3. is a regular tetrahedron. Tetrahedron
is formed by connecting the centers of the faces of
. Generally, a new tetrahedron
is formed by connecting the centers of the faces of
.
is the volume of tetrahedron
.
where
and
are coprime positive integers, find the remainder when
is divided by
.
Solution
Problem 4
4. Let . Let
be the product of the roots. How many digits are does
have where
denotes the greatest integer less than or equal to
?
Solution
Problem 5
5. A parabola is rotated
degrees clockwise about the origin to
. This image is translated upward
to
. Point
,
, and
is in Quadrant I, on
. If the area of
is at a maximum, it is
where
,
and
are integers and
is square free, find
.
Solution
Problem 6
6. Define a sequence and
for all positive integers
. Find the remainder when
is divided by
.
Solution
Problem 7
7. is a function that satisfies
for all defined
. Suppose that the sum of the zeros of
where
and
are coprime positive integers, find
.
Solution
Problem 8
8. is a solution to
. Suppose that
find
where
is the greatest integer less than or equal to
.
Solution
Problem 9
9. Zeus, Athena, and Posideon arrive at Mount Olympus at a random time between 12:00 pm and 12:00 am, and stay for 3 hours. All three hours does not need to fall within 12 pm to 12 am. If any of the 2 gods see each other during 12 pm to 12 am, it will be a good day. The probability of it being a good day is where
and
are coprime positive integers, find
.
Solution
Problem 10
10. Define . Find the remainder when
is divided by
.
Solution
Problem 11
11. is isosceles with
. A point
lies inside the triangle such that
,
, and
. Let
be the area of
. Find the remainder when
is divided by
.
Solution
Problem 12
12. There exists a line with points
,
,
with
in between
and
. Point
, not on the line is such that
,
,
with
. Construct
on ray
such that
and
. Point
is on ray
such that
. If
where
and
are integers, then find
.
Solution
Problem 13
13. is isosceles with base
. Construct
on segment
such that
. Construct
on
such that
. Contiue this pattern: construct
with
on segment
and
with
on segment
. The points
do not coincide and
. Suppose
is the last point you can construct on the perimeter of the triangle. Find the remainder when
is divided by
.
Solution
Problem 14
14. is the probability that if you flip a fair coin,
heads will occur before
tails. If
where
and
are relatively prime positive integers, find the remainder when
is divided by
.
Solution
Problem 15
15. A regular 61-gon with verticies ,
,
,...
is inscribed in a circle with a radius of
. Suppose
. If
where
and
are coprime positive integers, find the remainder when
is divided by
.
Solution