# Problems

2 Easy Problems

Problem 1: Solve for $latex x$:  $latex \pi^{1-x}=e^x$

Problem 2: Find the domain of $latex f(x)=\sqrt{4x-x^2}$

Simultaneous Equations/Lines

Find the Fahrenheit temperature, $latex x$, that is equal to $latex x$ degrees Celsius.

Graph Theory/Logic

Cominatorics and Inclusion/Exclusion:

Take 2 prime numbers, for example, 3 and 11 and one "large" number, for example 15000. How many numbers between 1 and 15000, inclusive, are divisible by 3 or 11? By adjusting how large the number is you can prevent students from trying to count their way to an answer. Moreover, changing the 3 numbers will allow you to create many similar problems. Calculating how many numbers are divisible by one of 3 primes can check for a good grasp of Inclusion/Exclusion.

Natural logs

If you use a calculator you'll find that $latex \ln(3^{\ln(7)})$ is the same as $latex \ln(7^{\ln(3)})$. But are they approximately the same or exactly the same? For what values of $latex a$ and $latex b$ is $latex \ln(a^{\ln(b)})$ the same as $latex \ln(b^{\ln(a)})$. Prove your assertion.

de Moivre's Formula

Use induction to show, for all positive integers n, $latex (\cos(\theta)+i \sin(\theta))^n = \cos(n\theta)+i\sin(n\theta)$

Pythagorean Theorem and Distance in 3D

Find the distance from point F to point D.

Combinatorics

How many squares does the grid contain? How many rectangles?

Pythagorean Theorem and simultaneous equations:

Dropping an altitude from the right angle of a triangle with sides of lengths 3, 4, and 5 breaks the hypotenuse into 2 pieces. Find the length of each piece of each piece as well as the height of the triangle.

Here's the LaTeX code for the diagram: GeomProb1 (.tex)  GeomProb (cropped PDF)

Induction and Fibonacci numbers:

Let $latex F_n$ denote the $latex \mbox{n}^{\mbox{th}}$ Fibonacci number. Prove by induction that $latex F_{n}=\dfrac {1} {\sqrt {5}}\left[ \left( \dfrac {1+\sqrt {5}} {2}\right) ^{n}-\left( \dfrac {1-\sqrt {5}} {2}\right)^n \right]$

Combinatorics: Combinations and calculators

Suppose a calculator displays 10 digits and it allows exponents from E-99 up to E99; that is, numbers are of the form:

$latex \pm 1 \times 10^{-99}$ to $latex \pm 9.999999999 \times10^{99}$

How many different numbers can the calculator represent? (Just make sure you don't forget 0.)

Combinatorics: Combinations and graph theory.

How many triangles are contained in the graph $latex K_5 -e$? It might help to count the triangles in  $latex K_5$ and throw out the "triangles" containing the missing edge.

Combinatorics: The Pigeonhole Principle

The vertices of a square defined by points (0,0), (0,1), (1,1), (1,0) are 4 points such that any two are at least 1 unit away. Prove that it is impossible  to find 5 points in the square such that any two are at least .71 units apart.

Sequences: Sequences, geometry, inverse trig functions

Calculate the hypotenuse of each right triangle from smallest to largest. If the diagram were to be continued find a formula for the $latex n^{th}$ hpotenuse. Calculate $latex \theta_i$ for each triangle. If the diagram were to be continued what's a formula for the $latex n^{th}$ angle?

Geometry: geometry, trigonometry

Find the area of the blue region enclosed by the 3 circles.

Geometry: geometry, trigonometry

Find the area of the blue region enclosed by the 3 circles. Geom2 (tex) Geom2 (PDF)

Exponents: Exponents, divisibility rules

How many zeros are at the end of $latex (22)^{78}(15)^{91}(75)^{67}(12)^{127}$?

Special Triangles, proofs: Using your knowledge of 30-60-90 right triangles, prove that the area of an equilateral triangle with side of length $latex s$ is given by $latex A=\frac{\sqrt{3}}{4}s^2$. Use this formula to find the area of a regular hexagon with sides of length $latex s$.

Trigonometry

Find the length of the belt in terms of $latex \cot(\theta)$

Calculus: Let $latex f(x)=x^2\sin(1/x^2)$ if $latex x \neq 0$ and $latex f(0)=0$. Find the derivative of $f(x)$ at 0.

Combinatorics: Determine all the divisors of N. For example, determine all the divisors of 1225.

Trigonometry: Prove: $latex \frac{\pi}{4}=4\arctan\left(\frac{1}{5}\right)-\arctan\left(\frac{1}{239}\right)$

Combinatorics: Find the number of triangles whose vertices use the points below.

Complex numbers: (Leibniz) Show that $latex \sqrt{6}=\sqrt{1+\sqrt{-3}}+\sqrt{1-\sqrt{-3}}$

Functions: Find a function whose domain is the positive real numbers and whose range is the integers.

Graph Theory/Logic: For the 8 squares below (corners aren't included) squares are adjacent up/down/left/right/diagonally. Fill in each square with a number from 1 through 8 (one time each) so that adjacent squares don't contain consecutive integers. Problem from here.Graph theory/Logic:

There are $latex n \geq 2$ people are at a party. Prove that there are two people who know the same number of people. Assumption: Two people either both know each other or they don't. That is, it's impossible for A to know B but for B not to know A. Also assume that a people don't "know themselves".