# Difference between revisions of "2021 JMPSC Invitationals Problems"

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==Problem 1== | ==Problem 1== | ||

− | + | The equation <math>ax^2 + 5x = 4,</math> where <math>a</math> is some constant, has <math>x = 1</math> as a solution. What is the other solution? | |

[[2021 JMPSC Invitationals Problems/Problem 1|Solution]] | [[2021 JMPSC Invitationals Problems/Problem 1|Solution]] | ||

==Problem 2== | ==Problem 2== | ||

− | + | Two quadrilaterals are drawn on the plane such that they share no sides. What is the maximum possible number of intersections of the boundaries of the two quadrilaterals? | |

[[2021 JMPSC Invitationals Problems/Problem 2|Solution]] | [[2021 JMPSC Invitationals Problems/Problem 2|Solution]] | ||

==Problem 3== | ==Problem 3== | ||

− | + | There are exactly <math>5</math> even positive integers less than or equal to <math>100</math> that are divisible by <math>x</math>. What is the sum of all possible positive integer values of <math>x</math>? | |

[[2021 JMPSC Invitationals Problems/Problem 3|Solution]] | [[2021 JMPSC Invitationals Problems/Problem 3|Solution]] | ||

==Problem 4== | ==Problem 4== | ||

− | + | Let <math>(x_n)_{n\geq 0}</math> and <math>(y_n)_{n\geq 0}</math> be sequences of real numbers such that <math>x_0 = 3</math>, <math>y_0 = 1</math>, and, for all positive integers <math>n</math>, | |

+ | <cmath>x_{n+1}+y_{n+1} = 2x_n + 2y_n,</cmath> | ||

+ | <cmath>x_{n+1}-y_{n+1}=3x_n-3y_n.</cmath> | ||

+ | Find <math>x_5</math>. | ||

[[2021 JMPSC Invitationals Problems/Problem 4|Solution]] | [[2021 JMPSC Invitationals Problems/Problem 4|Solution]] | ||

==Problem 5== | ==Problem 5== | ||

− | + | An <math>n</math>-pointed fork is a figure that consists of two parts: a handle that weighs <math>12</math> ounces and <math>n</math> "skewers" that each weigh a nonzero integer weight (in ounces). Suppose <math>n</math> is a positive integer such that there exists a fork with weight <math>n^2.</math> What is the sum of all possible values of <math>n</math>? | |

+ | <center> | ||

+ | [[File:Invites5.png|400px]] | ||

+ | </center> | ||

[[2021 JMPSC Invitationals Problems/Problem 5|Solution]] | [[2021 JMPSC Invitationals Problems/Problem 5|Solution]] | ||

==Problem 6== | ==Problem 6== | ||

− | + | Five friends decide to meet together for a party. However, they did not plan the party well, and at noon, every friend leaves their own house and travels to one of the other four friends' houses, chosen uniformly at random. The probability that every friend sees another friend in the house they chose can be expressed in the form <math>\frac{m}{n}</math>. If <math>m</math> and <math>n</math> are relatively prime positive integers, find <math>m+n</math>. | |

[[2021 JMPSC Invitationals Problems/Problem 6|Solution]] | [[2021 JMPSC Invitationals Problems/Problem 6|Solution]] | ||

==Problem 7== | ==Problem 7== | ||

− | + | In a <math>3 \times 3</math> grid with nine square cells, how many ways can Jacob shade in some nonzero number of cells such that each row, column, and diagonal contains at most one shaded cell? (A diagonal is a set of squares such that their centers lie on a line that makes a <math>45^\circ</math> angle with the sides of the grid. Note that there are more than two diagonals.) | |

[[2021 JMPSC Invitationals Problems/Problem 7|Solution]] | [[2021 JMPSC Invitationals Problems/Problem 7|Solution]] | ||

==Problem 8== | ==Problem 8== | ||

− | + | Let <math>x</math> and <math>y</math> be real numbers that satisfy | |

+ | <cmath>(x+y)^2(20x+21y) = 12</cmath> | ||

+ | <cmath>(x+y)(20x+21y)^2 = 18.</cmath> | ||

+ | Find <math>21x+20y</math>. | ||

[[2021 JMPSC Invitationals Problems/Problem 8|Solution]] | [[2021 JMPSC Invitationals Problems/Problem 8|Solution]] | ||

==Problem 9== | ==Problem 9== | ||

− | + | In <math>\triangle ABC</math>, let <math>D</math> be on <math>\overline{AB}</math> such that <math>AD=DC</math>. If <math>\angle ADC=2\angle ABC</math>, <math>AD=13</math>, and <math>BC=10</math>, find <math>AC.</math> | |

[[2021 JMPSC Invitationals Problems/Problem 9|Solution]] | [[2021 JMPSC Invitationals Problems/Problem 9|Solution]] | ||

==Problem 10== | ==Problem 10== | ||

− | + | A point <math>P</math> is chosen in isosceles trapezoid <math>ABCD</math> with <math>AB=4</math>, <math>BC=20</math>, <math>CD=28</math>, and <math>DA=20</math>. If the sum of the areas of <math>PBC</math> and <math>PDA</math> is <math>144</math>, then the area of <math>PAB</math> can be written as <math>\frac{m}{n},</math> where <math>m</math> and <math>n</math> are relatively prime. Find <math>m+n.</math> | |

[[2021 JMPSC Invitationals Problems/Problem 10|Solution]] | [[2021 JMPSC Invitationals Problems/Problem 10|Solution]] | ||

==Problem 11== | ==Problem 11== | ||

− | + | For some <math>n</math>, the arithmetic progression <cmath>4,9,14,\ldots,n</cmath> has exactly <math>36</math> perfect squares. Find the maximum possible value of <math>n.</math> | |

[[2021 JMPSC Invitationals Problems/Problem 11|Solution]] | [[2021 JMPSC Invitationals Problems/Problem 11|Solution]] | ||

==Problem 12== | ==Problem 12== | ||

− | + | Rectangle <math>ABCD</math> is drawn such that <math>AB=7</math> and <math>BC=4</math>. <math>BDEF</math> is a square that contains vertex <math>C</math> in its interior. Find <math>CE^2+CF^2</math>. | |

[[2021 JMPSC Invitationals Problems/Problem 12|Solution]] | [[2021 JMPSC Invitationals Problems/Problem 12|Solution]] | ||

==Problem 13== | ==Problem 13== | ||

− | + | Let <math>p</math> be a prime and <math>n</math> be an odd integer (not necessarily positive) such that <cmath>\dfrac{p^{n+p+2021}}{(p+n)^2}</cmath> is an integer. Find the sum of all distinct possible values of <math>p \cdot n</math>. | |

[[2021 JMPSC Invitationals Problems/Problem 13|Solution]] | [[2021 JMPSC Invitationals Problems/Problem 13|Solution]] | ||

==Problem 14== | ==Problem 14== | ||

− | + | Let there be a <math>\triangle ACD</math> such that <math>AC=5</math>, <math>AD=12</math>, and <math>CD=13</math>, and let <math>B</math> be a point on <math>AD</math> such that <math>BD=7.</math> Let the circumcircle of <math>\triangle ABC</math> intersect hypotenuse <math>CD</math> at <math>E</math> and <math>C</math>. Let <math>AE</math> intersect <math>BC</math> at <math>F</math>. If the ratio <math>\tfrac{FC}{BF}</math> can be expressed as <math>\tfrac{m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime, find <math>m+n.</math> | |

[[2021 JMPSC Invitationals Problems/Problem 14|Solution]] | [[2021 JMPSC Invitationals Problems/Problem 14|Solution]] | ||

==Problem 15== | ==Problem 15== | ||

− | + | Abhishek is choosing positive integer factors of <math>2021 \times 2^{2021}</math> with replacement. After a minute passes, he chooses a random factor and writes it down. Abhishek repeats this process until the first time the product of all numbers written down is a perfect square. Find the expected number of minutes it takes for him to stop. | |

[[2021 JMPSC Invitationals Problems/Problem 15|Solution]] | [[2021 JMPSC Invitationals Problems/Problem 15|Solution]] |

## Revision as of 15:17, 11 July 2021

- This is a fifteen question free-response test. Each question has exactly one integer answer.
- You have 80 minutes to complete the test.
- You will receive 15 points for each correct answer, and 0 points for each problem left unanswered or incorrect.
- Figures are not necessarily drawn to scale.
- No aids are permitted other than scratch paper, graph paper, rulers, and erasers. No calculators, smartwatches, or computing devices are allowed. No problems on the test will require the use of a calculator.

## Contents

## Problem 1

The equation where is some constant, has as a solution. What is the other solution?

## Problem 2

Two quadrilaterals are drawn on the plane such that they share no sides. What is the maximum possible number of intersections of the boundaries of the two quadrilaterals?

## Problem 3

There are exactly even positive integers less than or equal to that are divisible by . What is the sum of all possible positive integer values of ?

## Problem 4

Let and be sequences of real numbers such that , , and, for all positive integers , Find .

## Problem 5

An -pointed fork is a figure that consists of two parts: a handle that weighs ounces and "skewers" that each weigh a nonzero integer weight (in ounces). Suppose is a positive integer such that there exists a fork with weight What is the sum of all possible values of ?

## Problem 6

Five friends decide to meet together for a party. However, they did not plan the party well, and at noon, every friend leaves their own house and travels to one of the other four friends' houses, chosen uniformly at random. The probability that every friend sees another friend in the house they chose can be expressed in the form . If and are relatively prime positive integers, find .

## Problem 7

In a grid with nine square cells, how many ways can Jacob shade in some nonzero number of cells such that each row, column, and diagonal contains at most one shaded cell? (A diagonal is a set of squares such that their centers lie on a line that makes a angle with the sides of the grid. Note that there are more than two diagonals.)

## Problem 8

Let and be real numbers that satisfy Find .

## Problem 9

In , let be on such that . If , , and , find

## Problem 10

A point is chosen in isosceles trapezoid with , , , and . If the sum of the areas of and is , then the area of can be written as where and are relatively prime. Find

## Problem 11

For some , the arithmetic progression has exactly perfect squares. Find the maximum possible value of

## Problem 12

Rectangle is drawn such that and . is a square that contains vertex in its interior. Find .

## Problem 13

Let be a prime and be an odd integer (not necessarily positive) such that is an integer. Find the sum of all distinct possible values of .

## Problem 14

Let there be a such that , , and , and let be a point on such that Let the circumcircle of intersect hypotenuse at and . Let intersect at . If the ratio can be expressed as where and are relatively prime, find

## Problem 15

Abhishek is choosing positive integer factors of with replacement. After a minute passes, he chooses a random factor and writes it down. Abhishek repeats this process until the first time the product of all numbers written down is a perfect square. Find the expected number of minutes it takes for him to stop.