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Difference between revisions of "2021 JMPSC Invitationals Problems"

(Created page with "#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 c...")
 
 
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==Problem 1==
 
==Problem 1==
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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==
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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==
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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==
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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>,
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<cmath>x_{n+1}+y_{n+1} = 2x_n + 2y_n,</cmath>
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<cmath>x_{n+1}-y_{n+1}=3x_n-3y_n.</cmath>
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Find <math>x_5</math>.
  
 
[[2021 JMPSC Invitationals Problems/Problem 4|Solution]]
 
[[2021 JMPSC Invitationals Problems/Problem 4|Solution]]
  
 
==Problem 5==
 
==Problem 5==
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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>?
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<center>
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[[File:Invites5.png|400px]]
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</center>
  
 
[[2021 JMPSC Invitationals Problems/Problem 5|Solution]]
 
[[2021 JMPSC Invitationals Problems/Problem 5|Solution]]
  
 
==Problem 6==
 
==Problem 6==
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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==
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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==
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Let <math>x</math> and <math>y</math> be real numbers that satisfy
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<cmath>(x+y)^2(20x+21y) = 12</cmath>
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<cmath>(x+y)(20x+21y)^2 = 18.</cmath>
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Find <math>21x+20y</math>.
  
 
[[2021 JMPSC Invitationals Problems/Problem 8|Solution]]
 
[[2021 JMPSC Invitationals Problems/Problem 8|Solution]]
  
 
==Problem 9==
 
==Problem 9==
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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==
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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==
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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==
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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==
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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==
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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==
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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]]
 +
  
  
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[[2021 JMPSC Invitationals Problems/Problem 15|Solution]]
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==See also==
 +
#[[2021 JMPSC Sprint Problems|2021 JMPSC Sprint Problems]]
 +
#[[2021 JMPSC Accuracy Problems|2021 JMPSC Accuracy Problems]]
 +
#[[2021 JMPSC Invitationals Answer Key|2021 JMPSC Invitationals Answer Key]]
 +
#[[JMPSC Problems and Solutions|All JMPSC Problems and Solutions]]
 +
{{JMPSC Notice}}

Latest revision as of 17:39, 11 July 2021

  1. This is a fifteen question free-response test. Each question has exactly one integer answer.
  2. You have 80 minutes to complete the test.
  3. You will receive 15 points for each correct answer, and 0 points for each problem left unanswered or incorrect.
  4. Figures are not necessarily drawn to scale.
  5. 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.

Problem 1

The equation $ax^2 + 5x = 4,$ where $a$ is some constant, has $x = 1$ as a solution. What is the other solution?

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?

Solution

Problem 3

There are exactly $5$ even positive integers less than or equal to $100$ that are divisible by $x$. What is the sum of all possible positive integer values of $x$?

Solution

Problem 4

Let $(x_n)_{n\geq 0}$ and $(y_n)_{n\geq 0}$ be sequences of real numbers such that $x_0 = 3$, $y_0 = 1$, and, for all positive integers $n$, \[x_{n+1}+y_{n+1} = 2x_n + 2y_n,\] \[x_{n+1}-y_{n+1}=3x_n-3y_n.\] Find $x_5$.

Solution

Problem 5

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

Invites5.png

Solution

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 $\frac{m}{n}$. If $m$ and $n$ are relatively prime positive integers, find $m+n$.

Solution

Problem 7

In a $3 \times 3$ 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 $45^\circ$ angle with the sides of the grid. Note that there are more than two diagonals.)

Solution

Problem 8

Let $x$ and $y$ be real numbers that satisfy \[(x+y)^2(20x+21y) = 12\] \[(x+y)(20x+21y)^2 = 18.\] Find $21x+20y$.

Solution

Problem 9

In $\triangle ABC$, let $D$ be on $\overline{AB}$ such that $AD=DC$. If $\angle ADC=2\angle ABC$, $AD=13$, and $BC=10$, find $AC.$

Solution

Problem 10

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

Solution

Problem 11

For some $n$, the arithmetic progression \[4,9,14,\ldots,n\] has exactly $36$ perfect squares. Find the maximum possible value of $n.$

Solution

Problem 12

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

Solution

Problem 13

Let $p$ be a prime and $n$ be an odd integer (not necessarily positive) such that \[\dfrac{p^{n+p+2021}}{(p+n)^2}\] is an integer. Find the sum of all distinct possible values of $p \cdot n$.

Solution

Problem 14

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

Solution

Problem 15

Abhishek is choosing positive integer factors of $2021 \times 2^{2021}$ 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.

Solution


See also

  1. 2021 JMPSC Sprint Problems
  2. 2021 JMPSC Accuracy Problems
  3. 2021 JMPSC Invitationals Answer Key
  4. All JMPSC Problems and Solutions

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