Difference between revisions of "1991 AIME Problems"
| Line 1: | Line 1: | ||
== Problem 1 == | == Problem 1 == | ||
| + | Find <math>x^2+y^2_{}</math> if <math>x_{}^{}</math> and <math>y_{}^{}</math> are positive integers such that | ||
| + | <center><math>xy_{}^{}+x+y = 71</math></center> | ||
| + | <center><math>x^2y+xy^2 = 880^{}_{}.</math></center> | ||
[[1991 AIME Problems/Problem 1|Solution]] | [[1991 AIME Problems/Problem 1|Solution]] | ||
== Problem 2 == | == Problem 2 == | ||
| + | Rectangle <math>ABCD_{}^{}</math> has sides <math>\overline {AB}</math> of length 4 and <math>\overline {CB}</math> of length 3. Divide <math>\overline {AB}</math> into 168 congruent segments with points <math>A_{}^{}=P_0, P_1, \ldots, P_{168}=B</math>, and divide <math>\overline {CB}</math> into 168 congruent segments with points <math>C_{}^{}=Q_0, Q_1, \ldots, Q_{168}=B</math>. For <math>1_{}^{} \le k \le 167</math>, draw the segments <math>\overline {P_kQ_k}</math>. Repeat this construction on the sides <math>\overline {AD}</math> and <math>\overline {CD}</math>, and then draw the diagonal <math>\overline {AC}</math>. Find the sum of the lengths of the 335 parallel segments drawn. | ||
[[1991 AIME Problems/Problem 2|Solution]] | [[1991 AIME Problems/Problem 2|Solution]] | ||
== Problem 3 == | == Problem 3 == | ||
| + | Expanding <math>(1+0.2)^{1000}_{}</math> by the binomial theorem and doing no further manipulation gives | ||
| + | <center><math>{1000 \choose 0}(0.2)^0+{1000 \choose 1}(0.2)^1+{1000 \choose 2}(0.2)^2+\cdots+{1000 \choose 1000}(0.2)^{1000}</math></center> | ||
| + | <center><math>= A_0 + A_1 + A_2 + \cdots + A_{1000},</math></center> | ||
| + | where <math>A_k = {1000 \choose k}(0.2)^k</math> for <math>k = 0,1,2,\ldots,1000</math>. For which <math>k_{}^{}</math> is <math>A_k^{}</math> the largest? | ||
[[1991 AIME Problems/Problem 3|Solution]] | [[1991 AIME Problems/Problem 3|Solution]] | ||
== Problem 4 == | == Problem 4 == | ||
| + | How many real numbers <math>x^{}_{}</math> satisfy the equation <math>\frac{1}{5}\log_2 x = \sin (5\pi x)</math>? | ||
[[1991 AIME Problems/Problem 4|Solution]] | [[1991 AIME Problems/Problem 4|Solution]] | ||
== Problem 5 == | == Problem 5 == | ||
| + | Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will be <math>20_{}^{}!</math> the resulting product? | ||
[[1991 AIME Problems/Problem 5|Solution]] | [[1991 AIME Problems/Problem 5|Solution]] | ||
== Problem 6 == | == Problem 6 == | ||
| + | Suppose <math>r^{}_{}</math> is a real number for which | ||
| + | <center><math> | ||
| + | \left\lfloor r + \frac{19}{100} \right\rfloor + \left\lfloor r + \frac{20}{100} \right\rfloor + \left\lfloor r + \frac{21}{100} \right\rfloor + \cdots + \left\lfloor r + \frac{91}{100} \right\rfloor = 546. | ||
| + | </math></center> | ||
| + | Find <math>\lfloor 100r \rfloor</math>. (For real <math>x^{}_{}</math>, <math>\lfloor x \rfloor</math> is the greatest integer less than or equal to <math>x^{}_{}</math>.) | ||
[[1991 AIME Problems/Problem 6|Solution]] | [[1991 AIME Problems/Problem 6|Solution]] | ||
== Problem 7 == | == Problem 7 == | ||
| − | + | Find <math>A^2_{}</math>, where <math>A^{}_{}</math> is the sum of the absolute values of all roots of the following equation: | |
| + | <center><math>x = \sqrt{19} + \frac{91}{{\displaystyle \sqrt{19}+\frac{91}{{\displaystyle \sqrt{19}+\frac{91}{{\displaystyle \sqrt{19}+\frac{91}{{\displaystyle \sqrt{19}+\frac{91}{x}}}}}}}}} | ||
| + | </math></center> | ||
[[1991 AIME Problems/Problem 7|Solution]] | [[1991 AIME Problems/Problem 7|Solution]] | ||
== Problem 8 == | == Problem 8 == | ||
| + | For how many real numbers <math>a^{}_{}</math> does the quadratic equation <math>x^2 + ax^{}_{} + 6a=0</math> have only integer roots for <math>x^{}_{}</math>? | ||
[[1991 AIME Problems/Problem 8|Solution]] | [[1991 AIME Problems/Problem 8|Solution]] | ||
== Problem 9 == | == Problem 9 == | ||
| + | Suppose that <math>\sec x+\tan x=\frac{22}7</math> and that <math>\csc x+\cot x=\frac mn,</math> where <math>\frac mn</math> is in lowest terms. Find <math>m+n^{}_{}.</math> | ||
[[1991 AIME Problems/Problem 9|Solution]] | [[1991 AIME Problems/Problem 9|Solution]] | ||
== Problem 10 == | == Problem 10 == | ||
| + | Two three-letter strings, <math>aaa^{}_{}</math> and <math>bbb^{}_{}</math>, are transmitted electronically. Each string is sent letter by letter. Due to faulty equipment, each of the six letters has a 1/3 chance of being received incorrectly, as an <math>a^{}_{}</math> when it should have been a <math>b^{}_{}</math>, or as a <math>b^{}_{}</math> when it should be an <math>a^{}_{}</math>. However, whether a given letter is received correctly or incorrectly is independent of the reception of any other letter. Let <math>S_a^{}</math> be the three-letter string received when <math>aaa^{}_{}</math> is transmitted and let <math>S_b^{}</math> be the three-letter string received when <math>bbb^{}_{}</math> is transmitted. Let <math>\displaystyle p</math> be the probability that <math>S_a^{}</math> comes before <math>S_b^{}</math> in alphabetical order. When <math>\displaystyle p</math> is written as a fraction in lowest terms, what is its numerator? | ||
[[1991 AIME Problems/Problem 10|Solution]] | [[1991 AIME Problems/Problem 10|Solution]] | ||
== Problem 11 == | == Problem 11 == | ||
| + | Twelve congruent disks are placed on a circle <math>C^{}_{}</math> of radius 1 in such a way that the twelve disks cover <math>C^{}_{}</math>, no two of the disks overlap, and so that each of the twelve disks is tangent to its two neighbors. The resulting arrangement of disks is shown in the figure below. The sum of the areas of the twelve disks can be written in the from <math>\pi(a-b\sqrt{c})</math>, where <math>a,b,c^{}_{}</math> are positive integers and <math>c^{}_{}</math> is not divisible by the square of any prime. Find <math>a+b+c^{}_{}</math>. | ||
| + | |||
| + | [[Image:AIME_1991_Problem_11.gif]] | ||
[[1991 AIME Problems/Problem 11|Solution]] | [[1991 AIME Problems/Problem 11|Solution]] | ||
== Problem 12 == | == Problem 12 == | ||
| + | Rhombus <math>PQRS^{}_{}</math> is inscribed in rectangle <math>ABCD^{}_{}</math> so that vertices <math>P^{}_{}</math>, <math>Q^{}_{}</math>, <math>R^{}_{}</math>, and <math>S^{}_{}</math> are interior points on sides <math>\overline{AB}</math>, <math>\overline{BC}</math>, <math>\overline{CD}</math>, and <math>\overline{DA}</math>, respectively. It is given that <math>PB^{}_{}=15</math>, <math>BQ^{}_{}=20</math>, <math>PR^{}_{}=30</math>, and <math>QS^{}_{}=40</math>. Let <math>m/n^{}_{}</math>, in lowest terms, denote the perimeter of <math>ABCD^{}_{}</math>. Find <math>m+n^{}_{}</math>. | ||
[[1991 AIME Problems/Problem 12|Solution]] | [[1991 AIME Problems/Problem 12|Solution]] | ||
== Problem 13 == | == Problem 13 == | ||
| + | A drawer contains a mixture of red socks and blue socks, at most 1991 in all. It so happens that, when two socks are selected randomly without replacement, there is a probability of exactly <math>\displaystyle \frac{1}{2}</math> that both are red or both are blue. What is the largest possible number of red socks in the drawer that is consistent with this data? | ||
[[1991 AIME Problems/Problem 13|Solution]] | [[1991 AIME Problems/Problem 13|Solution]] | ||
== Problem 14 == | == Problem 14 == | ||
| + | A hexagon is inscribed in a circle. Five of the sides have length 81 and the sixth, denoted by <math>\overline{AB}</math>, has length 31. Find the sum of the lengths of the three diagonals that can be drawn from <math>A_{}^{}</math>. | ||
[[1991 AIME Problems/Problem 14|Solution]] | [[1991 AIME Problems/Problem 14|Solution]] | ||
== Problem 15 == | == Problem 15 == | ||
| + | For positive integer <math>n_{}^{}</math>, define <math>S_n^{}</math> to be the minimum value of the sum | ||
| + | <center><math>\sum_{k=1}^n \sqrt{(2k-1)^2+a_k^2},</math></center> | ||
| + | where <math>a_1,a_2,\ldots,a_n^{}</math> are positive real numbers whose sum is 17. There is a unique positive integer <math>n^{}_{}</math> for which <math>S_n^{}</math> is also an integer. Find this <math>n^{}_{}</math>. | ||
[[1991 AIME Problems/Problem 15|Solution]] | [[1991 AIME Problems/Problem 15|Solution]] | ||
Revision as of 02:45, 2 March 2007
Contents
Problem 1
Find 
if 
and 
are positive integers such that
Problem 2
Rectangle 
has sides 
of length 4 and 
of length 3. Divide into 168 congruent segments with points 
, and divide into 168 congruent segments with points 
. For 
, draw the segments 
. Repeat this construction on the sides 
and 
, and then draw the diagonal 
. Find the sum of the lengths of the 335 parallel segments drawn.
Problem 3
Expanding 
by the binomial theorem and doing no further manipulation gives
where 
for 
. For which 
is 
the largest?
Problem 4
How many real numbers 
satisfy the equation 
?
Problem 5
Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will be 
the resulting product?
Problem 6
Suppose 
is a real number for which
Find 
. (For real , 
is the greatest integer less than or equal to .)
Problem 7
Find 
, where 
is the sum of the absolute values of all roots of the following equation:
Problem 8
For how many real numbers 
does the quadratic equation 
have only integer roots for ?
Problem 9
Suppose that 
and that 
where 
is in lowest terms. Find 
Problem 10
Two three-letter strings, 
and 
, are transmitted electronically. Each string is sent letter by letter. Due to faulty equipment, each of the six letters has a 1/3 chance of being received incorrectly, as an when it should have been a 
, or as a when it should be an . However, whether a given letter is received correctly or incorrectly is independent of the reception of any other letter. Let 
be the three-letter string received when is transmitted and let 
be the three-letter string received when is transmitted. Let 
be the probability that comes before in alphabetical order. When is written as a fraction in lowest terms, what is its numerator?
Problem 11
Twelve congruent disks are placed on a circle 
of radius 1 in such a way that the twelve disks cover , no two of the disks overlap, and so that each of the twelve disks is tangent to its two neighbors. The resulting arrangement of disks is shown in the figure below. The sum of the areas of the twelve disks can be written in the from 
, where 
are positive integers and 
is not divisible by the square of any prime. Find 
.
Problem 12
Rhombus 
is inscribed in rectangle 
so that vertices 
, 
, 
, and 
are interior points on sides 
, 
, 
, and 
, respectively. It is given that 
, 
, 
, and 
. Let 
, in lowest terms, denote the perimeter of . Find 
.
Problem 13
A drawer contains a mixture of red socks and blue socks, at most 1991 in all. It so happens that, when two socks are selected randomly without replacement, there is a probability of exactly 
that both are red or both are blue. What is the largest possible number of red socks in the drawer that is consistent with this data?
Problem 14
A hexagon is inscribed in a circle. Five of the sides have length 81 and the sixth, denoted by , has length 31. Find the sum of the lengths of the three diagonals that can be drawn from 
.
Problem 15
For positive integer 
, define 
to be the minimum value of the sum
where 
are positive real numbers whose sum is 17. There is a unique positive integer 
for which is also an integer. Find this .
