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{{AIME Problems|year=2021|n=I}} | {{AIME Problems|year=2021|n=I}} | ||
+ | ==Problem 1== | ||
+ | Zou and Chou are practicing their <math>100</math>-meter sprints by running <math>6</math> races against each other. Zou wins the first race, and after that, the probability that one of them wins a race is <math>\frac23</math> if they won the previous race but only <math>\frac13</math> if they lost the previous race. The probability that Zou will win exactly <math>5</math> of the <math>6</math> races is <math>\frac mn</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | ||
+ | |||
+ | [[2021 AIME I Problems/Problem 1|Solution]] | ||
+ | |||
+ | ==Problem 2== | ||
+ | In the diagram below, <math>ABCD</math> is a rectangle with side lengths <math>AB=3</math> and <math>BC=11</math>, and <math>AECF</math> is a rectangle with side lengths <math>AF=7</math> and <math>FC=9,</math> as shown. The area of the shaded region common to the interiors of both rectangles is <math>\frac mn</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | ||
+ | |||
+ | <asy> | ||
+ | pair A, B, C, D, E, F; | ||
+ | A = (0,3); | ||
+ | B=(0,0); | ||
+ | C=(11,0); | ||
+ | D=(11,3); | ||
+ | E=foot(C, A, (9/4,0)); | ||
+ | F=foot(A, C, (35/4,3)); | ||
+ | draw(A--B--C--D--cycle); | ||
+ | draw(A--E--C--F--cycle); | ||
+ | filldraw(A--(9/4,0)--C--(35/4,3)--cycle,gray*0.5+0.5*lightgray); | ||
+ | dot(A^^B^^C^^D^^E^^F); | ||
+ | label("$A$", A, W); | ||
+ | label("$B$", B, W); | ||
+ | label("$C$", C, (1,0)); | ||
+ | label("$D$", D, (1,0)); | ||
+ | label("$F$", F, N); | ||
+ | label("$E$", E, S); | ||
+ | </asy> | ||
+ | |||
+ | [[2021 AIME I Problems/Problem 2|Solution]] | ||
+ | |||
+ | ==Problem 3== | ||
+ | Find the number of positive integers less than <math>1000</math> that can be expressed as the difference of two integral powers of <math>2.</math> | ||
+ | |||
+ | [[2021 AIME I Problems/Problem 3|Solution]] | ||
+ | |||
+ | ==Problem 4== | ||
+ | Find the number of ways <math>66</math> identical coins can be separated into three nonempty piles so that there are fewer coins in the first pile than in the second pile and fewer coins in the second pile than in the third pile. | ||
+ | |||
+ | [[2021 AIME I Problems/Problem 4|Solution]] | ||
+ | |||
+ | ==Problem 5== | ||
+ | Call a three-term strictly increasing arithmetic sequence of integers special if the sum of the squares of the three terms equals the product of the middle term and the square of the common difference. Find the sum of the third terms of all special sequences. | ||
+ | |||
+ | [[2021 AIME I Problems/Problem 5|Solution]] | ||
+ | |||
==Problem 6== | ==Problem 6== | ||
Segments <math>\overline{AB}, \overline{AC},</math> and <math>\overline{AD}</math> are edges of a cube and <math>\overline{AG}</math> is a diagonal through the center of the cube. Point <math>P</math> satisfies <math>PB=60\sqrt{10}, PC=60\sqrt{5}, PD=120\sqrt{2},</math> and <math>PG=36\sqrt{7}</math>. What is <math>PA</math>? | Segments <math>\overline{AB}, \overline{AC},</math> and <math>\overline{AD}</math> are edges of a cube and <math>\overline{AG}</math> is a diagonal through the center of the cube. Point <math>P</math> satisfies <math>PB=60\sqrt{10}, PC=60\sqrt{5}, PD=120\sqrt{2},</math> and <math>PG=36\sqrt{7}</math>. What is <math>PA</math>? | ||
− | |||
[[2021 AIME I Problems/Problem 6|Solution]] | [[2021 AIME I Problems/Problem 6|Solution]] | ||
Line 29: | Line 73: | ||
==Problem 11== | ==Problem 11== | ||
− | Let <math>ABCD</math> be a cyclic quadrilateral with <math>AB=4,BC=5,CD=6,</math> and <math>DA=7</math> | + | Let <math>ABCD</math> be a cyclic quadrilateral with <math>AB=4,BC=5,CD=6,</math> and <math>DA=7.</math> Let <math>A_1</math> and <math>C_1</math> be the feet of the perpendiculars from <math>A</math> and <math>C,</math> respectively, to line <math>BD,</math> and let <math>B_1</math> and <math>D_1</math> be the feet of the perpendiculars from <math>B</math> and <math>D,</math> respectively, to line <math>AC.</math> The perimeter of <math>A_1B_1C_1D_1</math> is <math>\frac mn,</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n.</math> |
[[2021 AIME I Problems/Problem 11|Solution]] | [[2021 AIME I Problems/Problem 11|Solution]] | ||
==Problem 12== | ==Problem 12== | ||
− | Let <math>A_1A_2A_3 | + | Let <math>A_1A_2A_3\ldots A_{12}</math> be a dodecagon (<math>12</math>-gon). Three frogs initially sit at <math>A_4,A_8,</math> and <math>A_{12}</math>. At the end of each minute, simultaneously, each of the three frogs jumps to one of the two vertices adjacent to its current position, chosen randomly and independently with both choices being equally likely. All three frogs stop jumping as soon as two frogs arrive at the same vertex at the same time. The expected number of minutes until the frogs stop jumping is <math>\frac mn</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. |
[[2021 AIME I Problems/Problem 12|Solution]] | [[2021 AIME I Problems/Problem 12|Solution]] |
Latest revision as of 14:20, 9 August 2021
2021 AIME I (Answer Key) | AoPS Contest Collections | ||
Instructions
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1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |
Contents
Problem 1
Zou and Chou are practicing their -meter sprints by running races against each other. Zou wins the first race, and after that, the probability that one of them wins a race is if they won the previous race but only if they lost the previous race. The probability that Zou will win exactly of the races is , where and are relatively prime positive integers. Find .
Problem 2
In the diagram below, is a rectangle with side lengths and , and is a rectangle with side lengths and as shown. The area of the shaded region common to the interiors of both rectangles is , where and are relatively prime positive integers. Find .
Problem 3
Find the number of positive integers less than that can be expressed as the difference of two integral powers of
Problem 4
Find the number of ways identical coins can be separated into three nonempty piles so that there are fewer coins in the first pile than in the second pile and fewer coins in the second pile than in the third pile.
Problem 5
Call a three-term strictly increasing arithmetic sequence of integers special if the sum of the squares of the three terms equals the product of the middle term and the square of the common difference. Find the sum of the third terms of all special sequences.
Problem 6
Segments and are edges of a cube and is a diagonal through the center of the cube. Point satisfies and . What is ?
Problem 7
Find the number of pairs of positive integers with such that there exists a real number satisfying
Problem 8
Find the number of integers such that the equationhas distinct real solutions.
Problem 9
Let be an isosceles trapezoid with and Suppose that the distances from to the lines and are and respectively. Let be the area of Find
Problem 10
Consider the sequence of positive rational numbers defined by and for , if for relatively prime positive integers and , then
Determine the sum of all positive integers such that the rational number can be written in the form for some positive integer .
Problem 11
Let be a cyclic quadrilateral with and Let and be the feet of the perpendiculars from and respectively, to line and let and be the feet of the perpendiculars from and respectively, to line The perimeter of is where and are relatively prime positive integers. Find
Problem 12
Let be a dodecagon (-gon). Three frogs initially sit at and . At the end of each minute, simultaneously, each of the three frogs jumps to one of the two vertices adjacent to its current position, chosen randomly and independently with both choices being equally likely. All three frogs stop jumping as soon as two frogs arrive at the same vertex at the same time. The expected number of minutes until the frogs stop jumping is , where and are relatively prime positive integers. Find .
Problem 13
Circles and with radii and , respectively, intersect at distinct points and . A third circle is externally tangent to both and . Suppose line intersects at two points and such that the measure of minor arc is . Find the distance between the centers of and .
Problem 14
For any positive integer denotes the sum of the positive integer divisors of . Let be the least positive integer such that is divisible by for all positive integers . Find the sum of the prime factors in the prime factorization of .
Problem 15
Let be the set of positive integers such that the two parabolasintersect in four distinct points, and these four points lie on a circle with radius at most . Find the sum of the least element of and the greatest element of .
See also
2021 AIME I (Problems • Answer Key • Resources) | ||
Preceded by 2020 AIME II |
Followed by 2021 AIME II | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
- American Invitational Mathematics Examination
- AIME Problems and Solutions
- Mathematics competition resources
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.