This document contains a problem set covering topics in algebra, number theory, proof-based problems, and other areas of mathematics. There are over 30 multi-part problems presented across seven sections dealing with concepts like congruences, quadratic residues, functions, proofs involving trigonometric functions and geometry, and number fields.
This document contains a problem set covering topics in algebra, number theory, proof-based problems, and other areas of mathematics. There are over 30 multi-part problems presented across seven sections dealing with concepts like congruences, quadratic residues, functions, proofs involving trigonometric functions and geometry, and number fields.
This document contains a problem set covering topics in algebra, number theory, proof-based problems, and other areas of mathematics. There are over 30 multi-part problems presented across seven sections dealing with concepts like congruences, quadratic residues, functions, proofs involving trigonometric functions and geometry, and number fields.
This document contains a problem set covering topics in algebra, number theory, proof-based problems, and other areas of mathematics. There are over 30 multi-part problems presented across seven sections dealing with concepts like congruences, quadratic residues, functions, proofs involving trigonometric functions and geometry, and number fields.
(a) x2 + 3x − 18 (b) x4 − 2x3 − 83x2 + 84x + 1764 if we know (x + 6) is a factor. (c) 2x − 3x
3. Graph f (x) = x2 − sin x (Note all I’m looking for is a sketch with x intercepts and y intercepts labelled, it is totally ok to just estimate the graph with a table of values as it is a difficult to function to graph) 4. Suppose f satisfies f (x + y) = f (x) + f (y) for all x and y. (a) Prove that f (x1 + x2 + ... + xn ) = f (x1 ) + f (x2 ) + ... + f (xn ) (b) Prove that there exists some number c such that f (x) = cx for all rational numbers x. 5. For what values of a, b, c, d does ax + b f (x) = cx + d such that f (f (x)) = x for all x ̸= −d/c
1 Section Two: Number Theory
Definition: Let n ∈ Z+ . Define a relation ” ≡ ”on Z by a ≡ b (mod n) iff
(if and only if) n|a − b for all a, b ∈ Z . What this is saying is that two numbers a and b are congruent modulo n if and only if n divides a − b, or in other words, the remained when n is divided by a is the same as when it is divided by b.
1. Are the following true or false?
(a) 3 ≡ 8 (mod 5).
(b) n ≡ −n (mod 7). (c) n ≡ −n (mod 2n). (d) 17 ≡ 13 (mod 7) 2. The following are properties of mod.
(a) If a ≡ b (mod n), then a + kn ≡ b (mod n), for k ∈ Z.
(b) If a ≡ b (mod n) and c ≡ d (mod n), then a + c ≡ b + d (mod n).
(c) a · b (mod c) = a (mod c) · b (mod c) . 3. Generally, an integer a is a quadratic residue modulo p if it is congruent to a perfect square mod p. (a) Show that 8 is not a quadratic residue mod 19. (b) Find all quadratic residues mod 7. p+1 (c) Prove that for any prime p, there are 2 quadratic residues mod p. (d) When is 17 a quadratic residue mod p? (e) When is 6 a quadratic residue mod p? Why is this problem harder than the last? 4. Find the roots x2 ≡ 11 (mod 2017).
5. Find values of x and y as natural numbers such that 10x2 + 7 = 18x + 11y . 6. Given a triangle ABC, with sides a, b, c such that a is opposite to A, b is opposite to B and c is opposite to C; prove that: aA + bB + cC 60 ≤ < 90 a+b+c
2 Section 3: Proof Based Problems
1. An odd function satisfies f (−x) = −f (x), while an even function satisfies
f (−x) = f (x). x+1 (a) Write x−1 as the sum of an odd and even function. (b) Show that any function can be written as the sum of an odd and even function. 2. (a) Show that Asin(x + B) can be expressed as asin x + bcos x for some a and b. Also find the value of A and B in terms of a and b. (b) Graph the function: √ f (x) = 3sin x + cos x.
3. Let us define arcsin(x) as the inverse of sin(x), that is arcsin(sin(x)) = x
and vice versa. Prove that p p arcsin(a) + arcsin(b) = arcsin(a 1 − b2 + b 1 − a2 )
For −π/2 < arcsin a + arcsin b < π/2.
4. A knot is defined as a closed, non-self-intersecting curve that is embedded in three dimensions and cannot be untangled to produce a simple loop. In other words, if i have a string that I loop around such that every time the string over itself there is an over-crossing and and under- crossing then I merge the two ends of the string together such that I cannot untangle the knot, that is a knot (feel free to look on the internet to clarify what a knot is in a mathematical sense). Prove that no matter how you cut a knot into two parts and connect the edges of the string (see diagram), if it can be wrapped around a donut (torus) then one part will be untangleable. (Notice how in the diagram the right piece is untangleable)
3 5. (a) If on a circle, 3 point on the circumference are chosen at random, what is the probability of the triangle created by connecting them including the origin in its area? (b) On an n-dimensional sphere, if n + 1 points are selected at random on it’s surface, what is the probability of the origin being in the shape formed by connecting those points in that n-space? (Note that this should come directly from part one) 6. Consider the unit circle, with B(1, 0) and A in the first quadrant. Extend OA to C such that CB is perpendicular on the x-axis. (a) Consider triangles OAB and OCB, prove that if 0 < x < π/4, then sinx < x < tanx (b) Using the previous problem, conclude that
sinx cosx < <1 x sinx and prove that as x approaches zero, x equals one. (c) Use this to find as x approaches zero the value of: 1 − cosx x √ 7. Notice that 13 = (2 − 3i)(2 + 3i) where i = −1. √ We call the complex numbers a quadratic field with discriminant of D = −1 and numbers of the form a + bD, where and and b are real numbers. In general we define a field of degree n as follows: for a degree n, a polynomial P (x) with degree n with discriminant d. We build a field as follows with addition and multiplication remaining the same. n X F = ai · di i=0
(a) Show that any prime P in the real numbers that is congruent to one modulo four will have a decomposition similar to that of 13 in the complex numbers. (b) Find a general form of prime that split under a quadratic field with discriminant d. ie find g satisfying Ps = g(a, b) where the field is defined by a + bd. (c) We call a prime P ramified in a quadratic field if P satisfies: P = u(a + bd)2 , where u is a unit of the field (namely, 1, -1, d, -d). Find all ramified P in (a) the complex numbers, (b) any quadratic field.
