1.0 Combinatorics
Lottery Ticket Selections

In a local lottery, players choose 4 distinct numbers from the integers 1 to 20. The order in which the numbers are chosen does not matter.
How many different lottery tickets are possible?
1.0 Calculus
Derivative of \(f(x) = \ln(\tan x)\)

Find the derivative of the function \(f(x) = \ln(\tan x)\) with respect to \(x\).
1.5 Calculus
Definite Integral of \(\sin^2 x\)

Compute the definite integral \( \int_0^{\pi/2} \sin^2 x \, dx \).
1.5 Algebra Number Theory
Parity of Square Roots

Prove that if \( x \) is an integer such that \( x^2 \) is an even integer, then \( x \) must also be an even integer.
1.5 Geometry
Angle Bisector in a Triangle

In triangle ABC, the internal angle bisector of angle A meets side BC at point D. Prove that $$\frac{BD}{DC} = \frac{AB}{AC}$$.
1.5 Geometry
Triangle area using coordinates

Find the area of the triangle with vertices \((1,2)\), \((4,5)\), and \((6,1)\)
1.5 Calculus
Indefinite Integral of \(\tan(2x)\)

Evaluate the indefinite integral \( \int \tan(2x) \, dx \).
2.0 Calculus
Indefinite Integral of \(e^x \cos x\)

Calculate the indefinite integral \( \int e^x \cos x \, dx \).
1.5 Algebra Inequalities
AM-GM Inequality Application

Prove that for any positive real numbers \(a, b, c\),
$$ (a+b+c) \left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right) \geq 9 $$
1.5 Number Theory
Prove the Formula for Triangular Numbers

Use mathematical induction to prove the formula for the nth triangular number: $$T_n = \frac{n(n+1)}{2}$$.
1.5 Combinatorics
Arranging Books on a Shelf

Jessica has 7 distinct books she wants to arrange on a bookshelf.

a) In how many different ways can she arrange all 7 books?

b) If 3 of the books are history books and must be kept together, in how many ways can she arrange the books?
1.5 Algebra Number Theory
Prove the Sum of \(k \cdot k!\) Formula by Induction

Prove by mathematical induction that for all positive integers \(n\), the following formula holds:ç$$ \sum_{k=1}^{n} k \cdot k! = (n+1)! - 1 $$
3.5 Algebra
Logarithmic Lambert W Form

Solve for \( x \) in the equation: \n $$ 2^x = x^2 $$ \n identify all real roots using the appropriate branches of the W function.
3.5 Algebra
Cyclic System of Equations

Find all real solutions to the system: \n $$ x_1^2 = x_2 + 2, \quad x_2^2 = x_3 + 2, \quad x_3^2 = x_1 + 2 $$
1.0 Algebra Number Theory
Product of Two Even Integers

Prove that the product of any two even integers is an even integer.
3.0 Algebra
Quartic Equation with Special Symmetry

Solve the equation: \n $$ (x-1)^4 + (x-5)^4 = 82 $$
1.0 Algebra
Quadratic equation with parameter

Solve the equation \(x^2 + (k-1)x + k = 0\) in terms of \(k\). (solve for \(x\))
1.5 Geometry
Pythagorean Theorem

Prove that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
2.0 Geometry
Inscribed Angle Theorem

In a circle, an inscribed angle subtending an arc is equal to half the measure of the corresponding central angle.
1.5 Calculus
Derivative of \(h(x) = \arcsin(\sqrt{x})\)

Find the derivative of the function \(h(x) = \arcsin(\sqrt{x})\) with respect to \(x\).
2.5 Algebra Number Theory
Sum of Powers Modulo a Prime

Let \(p\) be a prime number greater than 3. Evaluate the sum \(S = \sum_{k=1}^{p-1} k^{p-2} \pmod p\).
1.5 Geometry
Triangle Inequality

Prove that in any triangle, the sum of the lengths of any two sides is greater than the length of the third side.
2.0 Algebra Calculus
Sum of Maclaurin Series for Cosine

Find the sum of the infinite series \( \sum_{n=0}^{\infty} \frac{(-1)^n \pi^{2n}}{(2n)!} \).
4.0 Algebra Number Theory
Exponential Diophantine Equation

Find all pairs of positive integers \( (n, k) \) such that: \n $$ n! + 1 = k^2 $$
2.0 Number Theory
Prove a Divisibility Rule by Induction

Use the principle of mathematical induction to prove that for every positive integer \(n \geq 1\), the expression
$$
3^{2n+2} - 8n - 9
$$
is divisible by 64.
3.0 Algebra
Polynomial Functional Equation

Find all polynomials \( P(x) \) such that \( P(x^2) = (P(x))^2 \) for all real \( x \).
1.0 Geometry
Equality of Vertical Angles

Prove that when two straight lines intersect, the vertical angles formed are equal in measure.
3.5 Geometry
Simson Line

Prove that for a point on the circumcircle of a triangle, the feet of the perpendiculars to the three sides are collinear, forming the Simson line.
2.5 Algebra
Solving an Equation with Logarithms

Find all real solutions to the equation \(x \ln x = 1\). Express your answer using the Lambert W function.
2.0 Calculus
Limit of \(\frac{\sin x - x}{x^3}\) as \(x \to 0\)

Evaluate the limit:$$ \lim_{x \to 0} \frac{\sin x - x}{x^3} $$
2.5 Algebra
Equation with x^x

Find all real solutions to the equation \(x^x = 2\) for \(x > 0\). Express your answer using the Lambert W function.
1.5 Algebra Inequalities
Absolute Value Product Property

For any real numbers \( a \) and \( b \), prove that \( |a \cdot b| = |a| \cdot |b| \).
1.0 Combinatorics
Grocery Shopping Combinations

A shopper needs to buy 1 type of fruit from 5 available options, 1 type of vegetable from 4 available options, and 1 type of drink from 6 available options.
How many different combinations of these three items can they choose?
3.0
Inequality identity with square roots

Let \(a, b, c\) be positive real numbers. Prove that:$$\frac{\sqrt{2a+b} + \sqrt{2b+c} + \sqrt{2c+a}}{\sqrt{a+b+c}} \le 3$$
1.5 Calculus
Indefinite Integral of \(\frac{\cos x}{1 + \sin^2 x}\)

Find the indefinite integral \( \int \frac{\cos x}{1 + \sin^2 x} \, dx \).
5.0
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1.0 Combinatorics
Restaurant Menu Selections

A restaurant offers a fixed-price menu where customers choose one appetizer, one main course, and one dessert. There are 5 choices for appetizers, 8 choices for main courses, and 4 choices for desserts. How many different three-course meal combinations are possible?
2.0 Algebra
System of Linear Equations

Solve the following system of linear equations:
$$ \\begin{cases}
3x + 2y - z = 10 \\
x - 3y + 2z = -4 \\
2x + y + 3z = 7
\\end{cases} $$
1.5 Combinatorics
Marble Selection from a Bag

A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles.
If you pick 3 marbles at random, how many ways can you pick exactly 1 red, 1 blue, and 1 green marble?
3.0 Algebra
Symmetric System of Equations

Solve the following system of equations for real numbers \(x, y, z\):
$$ x+y+z = 6 $$
$$ x^2+y^2+z^2 = 14 $$
$$ x^3+y^3+z^3 = 36 $$
1.5 Number Theory
Prove the Formula for the Sum of Cubes

Use mathematical induction to prove the formula for the sum of the first n cubes: $$1^3 + 2^3 + \dots + n^3 = \frac{n^2(n+1)^2}{4}$$.
2.5 Algebra Trigonometry
Finite Sum of \(\sin(kx)\)

Find a closed-form expression for the sum \( \sum_{k=1}^{N} \sin(kx) \).
1.0 Algebra
Polynomial factorization

Factor the polynomial \(x^3 - 3x^2 + 3x - 1\).
1.0 Probability Combinatorics
Probability of Picking Marbles

A bag contains 5 red marbles and 3 blue marbles. If two marbles are drawn at random without replacement, what is the probability that both are red?
2.0 Algebra Number Theory
Multiplicative Order Modulo a Prime

Find the smallest positive integer \(n\) such that \(3^n \equiv 1 \pmod{43}\).
3.0 Algebra Number Theory
Polynomial without Integer Roots

Let \(P(x)\) be a polynomial with integer coefficients. If there exist distinct integers \(a\) and \(b\) such that \(P(a) = 1\) and \(P(b) = 3\), prove that \(P(x)\) has no integer roots.
2.5 Trigonometry
Sine Sum Formula

Prove that $$\sin(A + B) = \sin A \cos B + \cos A \sin B$$.
2.5 Calculus
Derivative of \(y = (\sin x)^{\cos x}\)

Differentiate the function \(y = (\sin x)^{\cos x}\) with respect to \(x\).
2.5 Number Theory
Fermat's Little Theorem Application

Find the remainder when \\( 2^{100} \\) is divided by 13.
1.0 Combinatorics
Daily Outfit Combinations

John is packing for a trip. He has 4 different shirts, 3 different pairs of pants, and 2 different pairs of shoes. Assuming every shirt goes with every pair of pants and every pair of shoes, how many different outfits can he create?
2.0 Calculus
Indefinite Integral of \(\cos^3 x\)

Evaluate the indefinite integral \( \int \cos^3 x \, dx \).
1.0 Combinatorics
Meeting Schedule Permutations

There are 4 different meetings that need to be scheduled for Monday afternoon, one after another.
In how many different orders can these meetings be scheduled?
1.5 Algebra
Union with a Superset

For any two sets \( A \) and \( B \), prove that if \( A \subseteq B \), then \( A \cup B = B \).
1.0 Algebra Inequalities
Non-negativity of a Real Square

For any real number \( x \), prove that \( x^2 \geq 0 \).
1.5 Number Theory Inequalities
Prove Bernoulli's Inequality by Induction

Prove Bernoulli's Inequality by mathematical induction:
If \(x\) is a real number such that \(x \ge -1\), then for every non-negative integer \(n\), the inequality \((1+x)^n \ge 1+nx\) holds.
2.0 Probability
Conditional Probability with Dice

Two fair six-sided dice are rolled. What is the probability that the sum of the numbers rolled is 7, given that at least one die shows a 3?
1.0 Combinatorics
License Plate Combinations

A certain state's license plates consist of 3 letters followed by 3 digits.
If repetition of letters and digits is allowed, how many different license plates are possible? Assume there are 26 possible letters and 10 possible digits (0-9).
3.0 Number Theory
Cannonball Problem

Prove that the sum of the first n squares is a perfect square only for n = 1 and n = 24. That is, show that $$1^2 + 2^2 + \dots + n^2 = m^2$$ has solutions only for these values of n.
2.5 Number Theory Probability
Prime-sum probability

Three fair dice are rolled. What is the probability that their sum is a prime number?
1.5 Algebra Number Theory
Fibonacci Numbers and GCD

Let \( F_n \) denote the \( n \)-th Fibonacci number, defined by \( F_1=1, F_2=1 \), and \( F_{k+2} = F_{k+1} + F_k \) for \( k \ge 1 \).
Prove by induction that for all positive integers \( n \), \( \gcd(F_n, F_{n+1}) = 1 \).
1.5 Algebra Number Theory
Closure of Rationals Under Addition

Prove that the sum of any two rational numbers is always a rational number.
3.0 Geometry
Excenter and Excircle

Prove that each excenter of a triangle is the intersection point of two external angle bisectors and the remaining internal bisector, and is the center of an excircle tangent to one side and the extensions of the other two.
1.5 Combinatorics
Counting Paths on a Grid

How many shortest paths are there from (0,0) to (m,n) in a grid if you can only move right or up?
2.5 Algebra Number Theory
Modular Recurrence Relation

Let a sequence be defined by \( a_0 = 1, a_1 = 1 \) and \( a_{n+2} = 3a_{n+1} + 4a_n \) for \( n \ge 0 \).
Prove by induction that \( a_n \equiv (-1)^n \pmod 5 \) for all \( n \ge 0 \).
2.0 Number Theory
Remainder of a Large Power

Find the remainder when \(3^{2023}\) is divided by \(17\).
3.0 Geometry
Euler Line

Prove that the centroid, circumcenter, and orthocenter of a non-equilateral triangle lie on a single line called the Euler line.
1.5 Combinatorics
Striped Flag Design

A flag has 3 horizontal stripes. There are 5 different colors available to choose from.
How many different flags can be designed if adjacent stripes must have different colors?
1.5 Algebra Number Theory
Sum of Cubes as a Perfect Square

Prove by induction that for any natural number \( n \), the sum of the first \( n \) cubes, \( S_n = 1^3 + 2^3 + \dots + n^3 \), is always a perfect square.
1.0 Number Theory
Geometric series partial sum

Compute the sum of the series \(1 + 2 + 4 + 8 + \dots + 2^n\).
2.0 Combinatorics
PIN Code Possibilities

A bank requires its customers to create a 4-digit Personal Identification Number (PIN) using digits from 0 to 9.

a) How many different 4-digit PINs are possible if digits can be repeated?

b) How many different 4-digit PINs are possible if digits cannot be repeated?

c) How many different 4-digit PINs are possible if the first digit cannot be 0 and digits cannot be repeated?
1.0 Algebra Number Theory
Product of an Even and an Integer

Prove that the product of an even integer and any integer is always an even integer.
1.5 Geometry
Triangle area

A triangle has side lengths 5, 12, and 13. Compute its area.
2.5 Algebra Number Theory
Pell-type Diophantine Equation

Find all integer solutions \((x,y)\) to the equation \(x^2 - 7y^2 = 1\).
1.5 Algebra Number Theory
Integer Solutions for Reciprocal Sum

Find all positive integer solutions \((x,y)\) to the equation
$$ \frac{1}{x} + \frac{1}{y} = \frac{1}{3} $$
1.5 Number Theory Combinatorics
Simple Pigeonhole Principle Application

Prove that among any group of 13 people, at least two people must have been born in the same month.
1.0 Combinatorics
Multiple Choice Quiz Answers

A quiz consists of 5 multiple-choice questions. Each question has 4 possible answers (A, B, C, D), and only one is correct.
How many different ways can a student answer all 5 questions, regardless of correctness?
2.0 Geometry
Median Intersection Point

In triangle ABC, medians intersect at point G. Prove that G divides each median in the ratio 2:1.
2.0 Geometry
Cyclic Quadrilateral Angle Property

Prove that in a cyclic quadrilateral, the sum of the opposite angles is 180 degrees.
2.0 Geometry
Law of Sines

Prove that in any triangle, the ratio of a side to the sine of its opposite angle is constant for all three sides.
1.0 Combinatorics
Digital Security Codes

A security system requires a 4-digit code. Each digit can be any integer from 0 to 9.
If digits can be repeated, how many different 4-digit codes are possible?