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Difference between revisions of "2021 JMPSC Accuracy Problems"
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==Problem 2==
 
==Problem 2==
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Three distinct even positive integers are chosen between <math>1</math> and <math>100,</math> inclusive. What is the largest possible average of these three integers?
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[[2021 JMPSC Accuracy Problems/Problem 2|Solution]]
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==Problem 3==
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In a regular octagon, the sum of any three consecutive sides is <math>90.</math> A square is constructed using one of the sides of this octagon. What is the area of the square?
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<center>
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[[File:Octagonsquare.jpg|250px]]
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</center>
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[[2021 JMPSC Accuracy Problems/Problem 3|Solution]]
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==Problem 4==
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If <math>\frac{x+2}{6}</math> is its own reciprocal, find the product of all possible values of <math>x.</math>
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[[2021 JMPSC Accuracy Problems/Problem 4|Solution]]
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==Problem 5==
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Let <math>n!=n \cdot (n-1) \cdot (n-2) \cdots 2 \cdot 1</math> for all positive integers <math>n</math>. Find the value of <math>x</math> that satisfies <cmath>\frac{5!x}{2022!}=\frac{20}{2021!}.</cmath>
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[[2021 JMPSC Accuracy Problems/Problem 5|Solution]]
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==Problem 6==
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In quadrilateral <math>ABCD</math>, diagonal <math>\overline{AC}</math> bisects both <math>\angle BAD</math> and <math>\angle BCD</math>. If <math>AB=15</math> and <math>BC=13</math>, find the perimeter of <math>ABCD</math>.
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[[2021 JMPSC Accuracy Problems/Problem 6|Solution]]
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==Problem 7==
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If <math>A</math>, <math>B</math>, and <math>C</math> each represent a single digit and they satisfy the equation <cmath>\begin{array}{cccc}& A & B & C \\ \times & &  &3 \\ \hline  & 7 & 9 & C\end{array},</cmath> find <math>3A+2B+C</math>.
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[[2021 JMPSC Accuracy Problems/Problem 7|Solution]]
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==Problem 8==
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How many triangles are bounded by segments in the figure and contain the red triangle? (Do not include the red triangle in your total.)
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<center>
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[[File:Sprint2 .png|300px]]
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</center>
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[[2021 JMPSC Accuracy Problems/Problem 8|Solution]]
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==Problem 9==
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If <math>x_1,x_2,\ldots,x_{10}</math> is a strictly increasing sequence of positive integers that satisfies <cmath>\frac{1}{2}<\frac{2}{x_1}<\frac{3}{x_2}< \cdots < \frac{11}{x_{10}},</cmath> find <math>x_1+x_2+\cdots+x_{10}</math>.
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[[2021 JMPSC Accuracy Problems/Problem 9|Solution]]
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==Problem 10==
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In a certain school, each class has an equal number of students. If the number of classes was to increase by <math>1</math>, then each class would have <math>20</math> students. If the number of classes was to decrease by <math>1</math>, then each class would have <math>30</math> students. How many students are in each class?
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[[2021 JMPSC Accuracy Problems/Problem 10|Solution]]
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==Problem 11==
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If <math>a : b : c : d=1 : 2 : 3 : 4</math> and <math>a</math>, <math>b</math>, <math>c</math>, and <math>d</math> are divisors of <math>252</math>, what is the maximum value of <math>a</math>?
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[[2021 JMPSC Accuracy Problems/Problem 11|Solution]]
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==Problem 12==
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A rectangle with base <math>1</math> and height <math>2</math> is inscribed in an equilateral triangle. Another rectangle with height <math>1</math> is also inscribed in the triangle. The base of the second rectangle can be written as a fully simplified fraction <math>\frac{a+b\sqrt{3}}{c}</math> such that <math>gcd(a,b,c)=1.</math> Find <math>a+b+c</math>.
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<center>
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[[File:Sprint13.jpg|400px]]
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</center>
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[[2021 JMPSC Accuracy Problems/Problem 12|Solution]]
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==Problem 13==
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Let <math>x</math> and <math>y</math> be nonnegative integers such that <math>(x+y)^2+(xy)^2=25.</math> Find the sum of all possible values of <math>x.</math>
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[[2021 JMPSC Accuracy Problems/Problem 13|Solution]]
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==Problem 14==
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What is the leftmost digit of the product <cmath>\underbrace{161616 \cdots 16}_{100 \text{ digits }} \times \underbrace{252525 \cdots 25}_{100 \text{ digits }}?</cmath>
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[[2021 JMPSC Accuracy Problems/Problem 14|Solution]]
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==Problem 15==
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For all positive integers <math>n,</math> define the function <math>f(n)</math> to output <math>4\underbrace{777 \cdots 7}_{n\ \text{sevens}}5.</math> For example, <math>f(1)=475</math>, <math>f(2)=4775</math>, and <math>f(3)=47775.</math> Find the last three digits of <cmath>\frac{f(1)+f(2)+ \cdots + f(100)}{25}.</cmath>
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[[2021 JMPSC Accuracy Problems/Problem 15|Solution]]

Revision as of 20:48, 10 July 2021

  1. This is a fifteen question free-response test. Each question has exactly one integer answer.
  2. You have 60 minutes to complete the test.
  3. You will receive 4 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.
  6. When you finish the exam, please stay in the Zoom meeting for further instructions.

Problem 1

Find the sum of all positive multiples of $3$ that are factors of $27.$

Solution

Problem 2

Three distinct even positive integers are chosen between $1$ and $100,$ inclusive. What is the largest possible average of these three integers?

Solution

Problem 3

In a regular octagon, the sum of any three consecutive sides is $90.$ A square is constructed using one of the sides of this octagon. What is the area of the square?

Octagonsquare.jpg

Solution

Problem 4

If $\frac{x+2}{6}$ is its own reciprocal, find the product of all possible values of $x.$

Solution

Problem 5

Let $n!=n \cdot (n-1) \cdot (n-2) \cdots 2 \cdot 1$ for all positive integers $n$. Find the value of $x$ that satisfies \[\frac{5!x}{2022!}=\frac{20}{2021!}.\]

Solution

Problem 6

In quadrilateral $ABCD$, diagonal $\overline{AC}$ bisects both $\angle BAD$ and $\angle BCD$. If $AB=15$ and $BC=13$, find the perimeter of $ABCD$.

Solution

Problem 7

If $A$, $B$, and $C$ each represent a single digit and they satisfy the equation \[\begin{array}{cccc}& A & B & C \\ \times & & &3 \\ \hline & 7 & 9 & C\end{array},\] find $3A+2B+C$.

Solution

Problem 8

How many triangles are bounded by segments in the figure and contain the red triangle? (Do not include the red triangle in your total.)

Sprint2 .png

Solution

Problem 9

If $x_1,x_2,\ldots,x_{10}$ is a strictly increasing sequence of positive integers that satisfies \[\frac{1}{2}<\frac{2}{x_1}<\frac{3}{x_2}< \cdots < \frac{11}{x_{10}},\] find $x_1+x_2+\cdots+x_{10}$.

Solution

Problem 10

In a certain school, each class has an equal number of students. If the number of classes was to increase by $1$, then each class would have $20$ students. If the number of classes was to decrease by $1$, then each class would have $30$ students. How many students are in each class?

Solution

Problem 11

If $a : b : c : d=1 : 2 : 3 : 4$ and $a$, $b$, $c$, and $d$ are divisors of $252$, what is the maximum value of $a$?

Solution

Problem 12

A rectangle with base $1$ and height $2$ is inscribed in an equilateral triangle. Another rectangle with height $1$ is also inscribed in the triangle. The base of the second rectangle can be written as a fully simplified fraction $\frac{a+b\sqrt{3}}{c}$ such that $gcd(a,b,c)=1.$ Find $a+b+c$.

Sprint13.jpg

Solution

Problem 13

Let $x$ and $y$ be nonnegative integers such that $(x+y)^2+(xy)^2=25.$ Find the sum of all possible values of $x.$

Solution

Problem 14

What is the leftmost digit of the product \[\underbrace{161616 \cdots 16}_{100 \text{ digits }} \times \underbrace{252525 \cdots 25}_{100 \text{ digits }}?\]

Solution

Problem 15

For all positive integers $n,$ define the function $f(n)$ to output $4\underbrace{777 \cdots 7}_{n\ \text{sevens}}5.$ For example, $f(1)=475$, $f(2)=4775$, and $f(3)=47775.$ Find the last three digits of \[\frac{f(1)+f(2)+ \cdots + f(100)}{25}.\]

Solution

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