MathNet 2021 — Part 01

Question 1

What is the value of $\frac{(2112-2021)^2}{169}$?

Question 2

What is the value of $1234 + 2341 + 3412 + 4123$?

10,000
10,010
10,110
11,000
11,110

Question 3

The expression $\frac{2021}{2020} - \frac{2020}{2021}$ is equal to the fraction $\frac{p}{q}$ in which $p$ and $q$ are positive integers whose greatest common divisor is 1. What is $p$?

1
9
2020
2021
4041

Question 4

Let $n = 8^{2022}$. Which of the following is equal to $\frac{n}{4}$?

$4^{1010}$
$2^{2022}$
$8^{2018}$
$4^{3031}$
$4^{3032}$

Question 5

The least positive integer with exactly $2021$ distinct positive divisors can be written in the form $m \cdot 6^k$, where $m$ and $k$ are integers and $6$ is not a divisor of $m$. What is $m + k$?

Question 6

A two-digit positive integer is said to be *cuddly* if it is equal to the sum of its nonzero tens digit and the square of its units digit. How many two-digit positive integers are cuddly?

Question 7

When a certain unfair die is rolled, an even number is 3 times as likely to appear as an odd number. The die is rolled twice. What is the probability that the sum of the numbers rolled is even?

$\frac{3}{8}$
$\frac{4}{9}$
$\frac{5}{9}$
$\frac{9}{16}$
$\frac{5}{8}$

Forty slips of paper numbered $1$ to $40$ are placed in a hat. Alice and Bob each draw one number from the hat without replacement, keeping their numbers hidden from each other. Alice says, “I can’t tell who has the larger number.” Then Bob says, “I know who has the larger number.” Alice says, “You do? Is your number prime?” Bob replies, “Yes.” Alice says, “In that case, if I multiply your number by $100$ and add my number, the result is a perfect square.” What is the sum of the two numbers drawn from the hat?

$27$
$37$
$47$
$57$
$67$

Question 9

Emily sees a ship traveling at a constant speed along a straight section of a river. She walks parallel to the riverbank at a uniform rate faster than the ship. She counts 210 equal steps walking from the back of the ship to the front. Walking in the opposite direction, she counts 42 steps of the same size from the front of the ship to the back. In terms of Emily's equal steps, what is the length of the ship?

Question 10

A regular hexagon of side length $1$ is inscribed in a circle. Each minor arc of the circle determined by a side of the hexagon is reflected over that side. What is the area of the region bounded by these $6$ reflected arcs?

$\frac{5\sqrt{3}}{2} - \pi$
$3\sqrt{3} - \pi$
$4\sqrt{3} - \frac{3\pi}{2}$
$\pi - \frac{\sqrt{3}}{2}$
$\frac{\pi + \sqrt{3}}{2}$

Question 11

Which of the following conditions is sufficient to guarantee that integers $x$, $y$, and $z$ satisfy the equation
$$
x(x - y) + y(y - z) + z(z - x) = 1?
$$

$x \gt y$ and $y = z$
$x = y - 1$ and $y = z - 1$
$x = z + 1$ and $y = x + 1$
$x = z$ and $y - 1 = x$
$x + y + z = 1$

Question 12

A square with side length $3$ is inscribed in an isosceles triangle with one side of the square along the base of the triangle. A square with side length $2$ has two vertices on the other square and the other two on sides of the triangle, as shown. What is the area of the triangle?
Question illustration

$19\frac{1}{4}$
$20\frac{1}{4}$
$21\frac{3}{4}$
$22\frac{1}{2}$
$23\frac{3}{4}$

Question 13

Una rolls 6 standard 6-sided dice simultaneously and calculates the product of the 6 numbers obtained. What is the probability that the product is divisible by 4?

$\frac{3}{4}$
$\frac{57}{64}$
$\frac{59}{64}$
$\frac{187}{192}$
$\frac{63}{64}$

Question 14

How many ordered pairs $(x, y)$ of real numbers satisfy the following system of equations?
$$
\begin{aligned}
x^2 + 3y &= 9 \\
(|x| + |y| - 4)^2 &= 1
\end{aligned}
$$

Question 15

In square $ABCD$, points $P$ and $Q$ lie on $\overline{AD}$ and $\overline{AB}$, respectively. Segments $\overline{BP}$ and $\overline{CQ}$ intersect at right angles at $R$, with $BR = 6$ and $PR = 7$. What is the area of the square?

Question illustration

Question 16

Isosceles triangle $ABC$ has $AB = AC = 3\sqrt{6}$, and a circle with radius $5\sqrt{2}$ is tangent to line $AB$ at $B$ and to line $AC$ at $C$. What is the area of the circle that passes through vertices $A$, $B$, and $C$?

$24\pi$
$25\pi$
$26\pi$
$27\pi$
$28\pi$

Question 17

Five balls are arranged around a circle. Chris chooses two adjacent balls at random and interchanges them. Then Silva does the same, with her choice of adjacent balls to interchange being independent of Chris's. What is the expected number of balls that occupy their original positions after these two successive transpositions?

Question 18

Distinct lines $\ell$ and $m$ lie in the $xy$-plane. They intersect at the origin. Point $P(-1, 4)$ is reflected about line $\ell$ to point $P'$, and then $P'$ is reflected about line $m$ to point $P''$. The equation of line $\ell$ is $5x - y = 0$, and the coordinates of $P''$ are (4, 1). What is the equation of line $m$?

$5x + 2y = 0$
$3x + 2y = 0$
$x - 3y = 0$
$2x - 3y = 0$
$5x - 3y = 0$

Question 19

An architect is building a structure that will place vertical pillars at the vertices of regular hexagon $ABCDEF$, which is lying horizontally on the ground. The six pillars will hold up a flat solar panel that will not be parallel to the ground. The heights of the pillars at $A$, $B$, and $C$ are $12$, $9$, and $10$ meters, respectively. What is the height, in meters, of the pillar at $E$?

$9$
$6\sqrt{3}$
$8\sqrt{3}$
$17$
$12\sqrt{3}$

Question 20

Three identical square sheets of paper each with side length $6$ are stacked on top of each other. The middle sheet is rotated clockwise $30^\circ$ about its center and the top sheet is rotated clockwise $60^\circ$ about its center, resulting in the 24-sided polygon shown in the figure below. The area of this polygon can be expressed in the form $a - b\sqrt{c}$, where $a$, $b$, and $c$ are positive integers, and $c$ is not divisible by the square of any prime. What is $a + b + c$?

Question illustration