Introduction

#

Having retired from my local MAO chapter as the question writer, I felt like writing this solutions manual for every question I wrote for it.

Problems

#

Problem 1

#

Let \(m\) be the slope of the line passing through the points \((1,2)\) and \((A,3)\). For what value of \(A\) does \(m\) become undefined?

Problem 2

#

Express as a simplified radical: \(\sqrt1 \sqrt2 \sqrt3 \sqrt4 \sqrt5\)

Problem 3

#

Find \(x\) in the system: \(2x+3y=10, x+3y=8\)

Problem 4

#

If the diagonal of an isosceles right triangle measures \(2\sqrt2\), what is the area of the triangle?

Problem 5

#

How many real \(x\)-intercepts does the equation \(x^2+2x+1=0\) have?

Problem 6

#

What is the surface area of a unit sphere? Express your answer in terms of \(\pi\).

Problem 7

#

How many interior diagonals does a pentagon have?

Problem 8

#

Find the number of divisors of 200. Include 1 and 200 as divisors.

Problem 9

#

Find the radius of a circle if its inscribed square has area 1.

Problem 10

#

Give one solution to the equation: \(x^3+x^2+x+1=40\)

Problem 11

#

What positive number is equal to twice its reciprocal? Express your answer as a simplified radical.

Problem 12

#

Find \(A\): \(M^A= \sqrt[5]{\sqrt[4]{\sqrt[3]{\sqrt[2]{\sqrt[1]{M}}}}}\)

Problem 13

#

What is the surface area of a cube with volume \(\displaystyle \frac{1}{\sqrt2}\)? Express your answer in simplest radical form.

Problem 14

#

If \((a+b)^2=100\) and \(ab=20\), what is the value of \(a^2+b^2\)?

Problem 15

#

Let \(p_n\) be the nth prime, and let \(d(n)\) be the divisor count function (\(n\) and \(1\) are counted as divisors). Find the value of the sum: \(d(p_1)+d(p_2)+d(p_3)\ldots+d(p_{2024})\)

Problem 16

#

Solve for \(A+M+T\) in the following system: \(A+M=3,A+T=6,T+M=9\)

Problem 17

#

A family of lines is created by \(y=Ax+A\) for all \(A \in \mathbb{R} \setminus 0\). The intersection of all lines in this family can be expressed as \((x,y)\). Find the value of \(x^2+y^2\).

Problem 18

#

Functions \(A(p)\) and \(M(p)\) are both functions that translate a point \(p\). The function \(A(p)\) translates \(p\) right 7 units and moves it up \(2k\) units. The function \(M(p)\) translates \(p\) right \(R\) units and up \(U\) units. If \(A(M(p))\) retains the original x-coordinate of \(p\) whilst moving it up \(2k\) units, what is the value of \(R+U\)?

Problem 19

#

My friend could’ve been born in any year from 1000 to 2024. If the sum of the digits in his birth year is 26, in how many possible years could he have been born?

Problem 20

#

Find the number of possible arrangements of the letters in the word PLACEMENT. One such arrangement is TLEACMPEN.

Solutions

#

Solution 1

#

This only happens with a vertical line where both points share an \(x\) value: \(\boxed{A=1}\)

Solution 2

#

\(\sqrt1 \sqrt2 \sqrt3 \sqrt4 \sqrt5\) = \((1)(\sqrt2)(\sqrt3)(2)(\sqrt5) = \boxed{2\sqrt{30}}\)

Solution 3

#

By the Elimination Method: \((2x+3y=10) - (x+3y=8) \implies \boxed{x=2}\)

Solution 4

#

Since the diagonal of an isosceles right triangle is the lateral side multiplied by \(\sqrt2\), the lateral sides must measure \(2\) units; the area is then \(\frac{bh}{h} = \frac{2(2)}{2} = \boxed{2 \enspace \text{square units}}\)

Solution 5

#

\(x^2+2x+1 =(x+1)^2\), implying only \(\boxed{1}\) real intercept with multiplicity 2.

Solution 6

#

Surface area is equal to \(4\pi r^2\); if \(r=1\), as with a unit sphere, then the surface area is \(\boxed{4\pi}\).

Solution 7

#

Find the number of ways to choose any two vertices (5 choose 2) and then subtract the 5 sides that are simply edges: \(\binom{5}{2} - 5 = 10-5=\boxed5\)

Solution 8

#

To find the number of divisors of \(n\), find \(n\)’s prime factorization, where \(p_n\) represents the resulting exponents. Take the product of all \((p_n+1)\) Since \(200=5^2\cdot 2^3\), the number of divisors is \(\boxed{12}\)

Solution 9

#

Find the radius of a circle if its inscribed square has area 1. The diagonal must then be \(\sqrt2\), which spans the diameter. Since radius is half the diameter: \(\boxed{\frac{\sqrt2}{2}}\).

Solution 10

#

Since no specification is given for answer format, we can start searching for integer solutions: by the rational root theorem, we should only reasonably check \(\{1,3\}\). Of these, \(\boxed{3}\) satisfies the given relation.

Solution 11

#

\(n=2\cdot\frac{1}{n} \implies n^2=2 \implies n=\boxed{\sqrt2}\)

Solution 12

#

$\sqrt[5]{\sqrt[4]{\sqrt[3]{\sqrt[2]{\sqrt[1]{M}}}}} = M^{\frac{1}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5}} = M^{\frac{1}{120}} = \boxed{M^{\frac{1}{120}}}$

Solution 13

#

If the volume is \(\frac{1}{\sqrt{2}}\), then the side length is \(\sqrt[3]{\frac{1}{\sqrt{2}}}=\frac{1}{\sqrt[6]{2}}=\frac{\sqrt[6]{2^5}}{2}=\boxed{\frac{\sqrt[6]{32}}{2}}\).

Solution 14

#

\(a^2+b^2=(a+b)^2-2ab \implies a^2+b^2=100-2\cdot20=\boxed{60}\)

Solution 15

#

There are 2024 primes given, and each has 2 divisors: \(2\cdot2024=\boxed{4048}\)

Solution 16

#

Adding all equations together: \((A+M=3)+(A+T=6)+(T+M=9) \implies (2A+2M+2T=18) \implies \boxed{A+M+T=9}\)

Solution 17

#

Checking two trivial lines: the only intersection of \(y=1x+1\) and \(y=2x+2\) lies on \((-1,0)\), which implies this must be the intersection for all lines. \((-1)^2+(0)^2=\boxed{1}\).

Solution 18

#

Translation is additive so we can represent \(A(p)\) as the vector \([7,2k]\) and represent \(M(p)\) as \([R,U]\). Their composition is \([7,2k]+[R,U]=[7+R,2k+U]\). Their combination is provided to have a net effect of \([0,2k]\), implying \(R=-7\) and that \(U=0\). Then, \(R+U=\boxed{-7}\).

Solution 19

#

If the first digit were 2, there is no combination up to 2024 that has sum 26; the first digit is 1 so the rest must add to 25. Therefore, we have to count the number of ways to “subtract 2” from the maximal \(\{9,9,9\}\): \(\{8,8,9\},\{7,9,9\}.\) The triplets each have 3 permutations: \(\boxed{6}.\)

Solution 20

#

Treating each letter as distinct would yield \(9!\) arrangements. Correcting for arrangements of repeated letters: \(\frac{9!}{2!}=\boxed{181440}\)