Great problems handpicked from the book, divided into categories.
Topic: Angle Chasing
3.45 Let \(ABCD\) be a square, and let \(E\), \(F\) be points such that \(DA = DE = DF = DC\), and \(\angle ADE=\angle EDF=\angle FDC\). Prove that \(\triangle BEF\) is equilateral. Lorem ipsum set dolor Lorem Elsass ipsum Oberschaeffolsheim Heineken Lorem ipsum set dolor Lorem Elsass ipsum Oberschaeffolsheim Heineken Lorem ipsum set dolor
3.45 Let \(ABCD\) be a square, and let \(E\), \(F\) be points such that \(DA = DE = DF = DC\), and \(\angle ADE=\angle EDF=\angle FDC\). Prove that \(\triangle BEF\) is equilateral. Lorem ipsum set dolor Lorem Elsass ipsum Oberschaeffolsheim Heineken Lorem ipsum set dolor Lorem Elsass ipsum Oberschaeffolsheim Heineken Lorem ipsum set dolor
Number Theory
Topic: LCM & GCF
Prove that \((a,b)\) must be less than or equal to \(a\),\(b\), and \(a-b\), where \(a>b\) Topic: LCM & GCF
Prove \((m,n)[m,n]=mn\), where \((m,n)\) is the GCF of \(m,n\), and \([m,n]\) is the LCM of \(m,n\) Topic: LCM & GCF
Prove \((l,m,n)[lm,ln,mn]=lmn\) Topic: LCM & GCF
Prove \((lm,mn,nl)[l,m,n]=lmn\) Topic: LCM & GCF
Prove \([(l,m),(m,n),(n,l)] =lmn\) Topic: Number Bases
For each of \(n=84\) and \(n=88\), find the smallest integer multiple of \(n\) whose base 10 representation consists entirely of \(6's\), and \(7's\). Topic: Number Bases
When the number \(n\) is written in base \(b\) its representation is the two-digit number AB where \(A=b-2\) and \(B=2\). What is the representation of \(n\) in base \((b-1)\) Topic: Prime Proof
Denote \({ p }_{ k }\) the \({ k }^{ th }\) prime number. Show that \({ p }_{ 1 }{ p }_{ 2 }...{ p }_{ n }+1\) cannot be the perfect square of an integer. Topic: Square Proof
Prove that it is impossible for three consecutive squares to sum to another perfect square.
Prove that \((a,b)\) must be less than or equal to \(a\),\(b\), and \(a-b\), where \(a>b\) Topic: LCM & GCF
Prove \((m,n)[m,n]=mn\), where \((m,n)\) is the GCF of \(m,n\), and \([m,n]\) is the LCM of \(m,n\) Topic: LCM & GCF
Prove \((l,m,n)[lm,ln,mn]=lmn\) Topic: LCM & GCF
Prove \((lm,mn,nl)[l,m,n]=lmn\) Topic: LCM & GCF
Prove \([(l,m),(m,n),(n,l)] =lmn\) Topic: Number Bases
For each of \(n=84\) and \(n=88\), find the smallest integer multiple of \(n\) whose base 10 representation consists entirely of \(6's\), and \(7's\). Topic: Number Bases
When the number \(n\) is written in base \(b\) its representation is the two-digit number AB where \(A=b-2\) and \(B=2\). What is the representation of \(n\) in base \((b-1)\) Topic: Prime Proof
Denote \({ p }_{ k }\) the \({ k }^{ th }\) prime number. Show that \({ p }_{ 1 }{ p }_{ 2 }...{ p }_{ n }+1\) cannot be the perfect square of an integer. Topic: Square Proof
Prove that it is impossible for three consecutive squares to sum to another perfect square.
Algebra
Topic: Rates
Two bicyclists are seven-eights of the way through a mile-long tunnel when a train approaches the closer end at \(40\) mph. The riders take off at the same speed in opposite directions and each escapes the tunnel as the train passes them. How fast did they ride? Topic: Rates
Two men men starting at a point on a circular \(1\)-mile race track walk in opposite directions with uniform speeds and meet in \(6\) minutes, but if they walk in the same direction, it requires one hour for the faster walker to gain a lap. What is the rate of the slower walker? Topic: Rates
Two dogs, each traveling \(10\frac { ft }{ sec }\), run toward each other from \(500\) feet apart. As they run, a flea flies from the nose of one dog to the nose of the other at \(25\frac { ft }{ sec } \). The flea flies between the dogs in this manner until it is crushed when the dogs collide. How far did the flea fly? Topic: Proportions
Given that \(x\) is directly proportional to \(y\) and to \(z\) and is inversely proportional to \(w\), and that \(x=4\) when \((w,y,z)=(6,8,5)\), what is \(x\) when \((w,y,z)=(4,10,9)\)? Topic: Proportions
It is four o' clock now. How many minutes will pass before the minute and hour hands of a clock are coincedent (at the same exact place)? Topic: Proportions
A metric calendar has \(1\) metric year equivalent to our calendar year of \(365\) days. A metric year is divided into \(10\) equal metric months; a metric month is divided into \(10\) equal metric weeks; a metric week is divided into \(10\) equal metric days. To the nearest day of our calendar, how many days are there in \(4\) metric months, \(5\) metric weeks and \(8\) metric days? Topic: Proportions
A woman has part of \($4500\) invested at \(4%\) and the rest at \(6%\). If her annual return on each investment is the same, then what is the average rate of interest which she realizes on the \($4500\)? Topic: Proportions
Tennis coaches High Lob and Low Smash decided to share with their assistants (Love and Vantage) the money they earned from tennis lessons. They agreed on the following ratios: \(\text{ Lob}:\text{ Love}=17:12\), \(\text{ Love}:\text{ Smash}=3:4\), and \(\text{ Smash}:\text{ Vantage} = 32:15\). If their earnings totaled \($3,150\), how much did Love receive? Topic: Nested Radicals of Irrationals
Find \(\sqrt { 34-24\sqrt { 2 } } \) Topic: Nested Radicals of Imaginariess
Find \(\sqrt { 5-12i } \)
Two bicyclists are seven-eights of the way through a mile-long tunnel when a train approaches the closer end at \(40\) mph. The riders take off at the same speed in opposite directions and each escapes the tunnel as the train passes them. How fast did they ride? Topic: Rates
Two men men starting at a point on a circular \(1\)-mile race track walk in opposite directions with uniform speeds and meet in \(6\) minutes, but if they walk in the same direction, it requires one hour for the faster walker to gain a lap. What is the rate of the slower walker? Topic: Rates
Two dogs, each traveling \(10\frac { ft }{ sec }\), run toward each other from \(500\) feet apart. As they run, a flea flies from the nose of one dog to the nose of the other at \(25\frac { ft }{ sec } \). The flea flies between the dogs in this manner until it is crushed when the dogs collide. How far did the flea fly? Topic: Proportions
Given that \(x\) is directly proportional to \(y\) and to \(z\) and is inversely proportional to \(w\), and that \(x=4\) when \((w,y,z)=(6,8,5)\), what is \(x\) when \((w,y,z)=(4,10,9)\)? Topic: Proportions
It is four o' clock now. How many minutes will pass before the minute and hour hands of a clock are coincedent (at the same exact place)? Topic: Proportions
A metric calendar has \(1\) metric year equivalent to our calendar year of \(365\) days. A metric year is divided into \(10\) equal metric months; a metric month is divided into \(10\) equal metric weeks; a metric week is divided into \(10\) equal metric days. To the nearest day of our calendar, how many days are there in \(4\) metric months, \(5\) metric weeks and \(8\) metric days? Topic: Proportions
A woman has part of \($4500\) invested at \(4%\) and the rest at \(6%\). If her annual return on each investment is the same, then what is the average rate of interest which she realizes on the \($4500\)? Topic: Proportions
Tennis coaches High Lob and Low Smash decided to share with their assistants (Love and Vantage) the money they earned from tennis lessons. They agreed on the following ratios: \(\text{ Lob}:\text{ Love}=17:12\), \(\text{ Love}:\text{ Smash}=3:4\), and \(\text{ Smash}:\text{ Vantage} = 32:15\). If their earnings totaled \($3,150\), how much did Love receive? Topic: Nested Radicals of Irrationals
Find \(\sqrt { 34-24\sqrt { 2 } } \) Topic: Nested Radicals of Imaginariess
Find \(\sqrt { 5-12i } \)