1.0 Algebra
Commutativity of Set Union

For any two sets \( A \) and \( B \), prove that \( A \cup B = B \cup A \).
1.5 Algebra
Set Difference and Union Identity

For any two sets \( A \) and \( B \), prove that \( (A \setminus B) \cup B = A \cup B \).
1.5 Algebra Inequalities
Quadratic inequality

Solve the inequality \(x^2 - 4x + 3 \le 0\).
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.
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 Algebra
Quadratic equation with parameter

Solve the equation \(x^2 + (k-1)x + k = 0\) in terms of \(k\). (solve for \(x\))
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.
3.0 Algebra
Lambert W Function Transcendental Equation

Find the real solution for the equation
$$ xe^x = 3 $$
Express your answer using the Lambert W function.
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.
2.5 Algebra Trigonometry
Finite Sum of \(\sin(kx)\)

Find a closed-form expression for the sum \( \sum_{k=1}^{N} \sin(kx) \).
3.0 Geometry
Ceva's Theorem Application

Given a triangle with concurrent cevians, prove that the product of ratios of divided sides equals one.
1.0 Algebra Number Theory
Product of Two Even Integers

Prove that the product of any two even integers is an even integer.
2.5 Algebra Number Theory
Pell-type Diophantine Equation

Find all integer solutions \((x,y)\) to the equation \(x^2 - 7y^2 = 1\).
3.0 Algebra
Iterated Functional Equation

Find all functions \(f: \mathbb{R} \to \mathbb{R}\) such that \(f(x^2 - y^2) = (x-y)(f(x)+f(y))\) for all \(x, y \in \mathbb{R}\).
1.5 Geometry
Sum of Interior Angles of a Quadrilateral

Prove that the sum of the measures of the interior angles of any convex quadrilateral is \( 360^\circ \).
3.0 Geometry
Menelaus' Theorem

Prove that for points lying on the sides of a triangle and collinear, the product of the ratios of the segments equals one.
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 Number Theory Combinatorics
Remainder of a Factorial Sum

Find the remainder when \(1! + 2! + 3! + \dots + 100!\) is divided by \(12\).
2.5 Geometry
Incenter and Incircle

Prove that the incenter of a triangle is equidistant from all sides and lies at the intersection of the angle bisectors.
1.5 Number Theory
Prove the Formula for the Sum of Squares

Use mathematical induction to prove the formula for the sum of the first n squares: $$1^2 + 2^2 + \dots + n^2 = \frac{n(n+1)(2n+1)}{6}$$.
1.5 Number Theory
Prove the Formula for Triangular Numbers

Use mathematical induction to prove the formula for the nth triangular number: $$T_n = \frac{n(n+1)}{2}$$.
2.0 Algebra
System of Linear Equations

Solve the following system of linear equations:
$$ \\begin{cases}
3x + 2y - z = 10 \\
x - 3y + 2z = -4 \\
2x + y + 3z = 7
\\end{cases} $$
2.5 Trigonometry
Sine Sum Formula

Prove that $$\sin(A + B) = \sin A \cos B + \cos A \sin B$$.
2.0 Calculus
Indefinite Integral of \(e^x \cos x\)

Calculate the indefinite integral \( \int e^x \cos x \, dx \).
1.5 Algebra Number Theory
Integer Solutions for Reciprocal Sum

Find all positive integer solutions \((x,y)\) to the equation
$$ \frac{1}{x} + \frac{1}{y} = \frac{1}{3} $$
2.0 Geometry
Midline Theorem

Prove that the segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length.
1.5 Algebra Number Theory
Modular Inverse Calculation (Prime Modulus)

Find the smallest positive integer \(x\) such that \(17x \equiv 1 \pmod{101}\).
1.0 Combinatorics
Letter Arrangements

How many distinct 4-letter arrangements can be made using the letters M, A, T, H if each letter can be used only once?
3.0 Algebra Calculus Trigonometry
Sum of Telescoping Series with Arctangent

Find the sum of the series:$$ \sum_{n=1}^{N} \arctan\left(\frac{1}{n^2+n+1}\right) $$
2.0 Algebra Number Theory
Sum of Rational and Irrational

Prove that the sum of a rational number and an irrational number is always an irrational number.
2.0 Algebra Number Theory
Evaluating Large Powers Modulo a Prime

Calculate the remainder when \(5^{123456}\) is divided by \(17\).
2.0 Algebra
Radical Nested Equation

Solve for \( x \) in the infinite nested radical: \n $$ \sqrt{x + \sqrt{x + \sqrt{x + \dots}}} = 5 $$
1.5 Geometry
Quadrilateral inside a Circle

O is the centre of the circle. Lengths of AB and BC are both 10cm. The area of the quadrilateral OABC is 120cm^2. Calculate the radius of the circle
2.0 Geometry
Cyclic Quadrilateral Angles

Prove that in a cyclic quadrilateral, the sum of opposite angles is 180 degrees.
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?
1.0 Geometry
Equality of Vertical Angles

Prove that when two straight lines intersect, the vertical angles formed are equal in measure.
2.0 Geometry
Median Ratio

In triangle ABC, prove that the centroid divides each median in a 2:1 ratio.
1.0 Combinatorics
Movie Seating Arrangement

7 friends are going to watch a movie and sit in a single row of 7 seats.
In how many different orders can they sit?
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.
2.0 Calculus
Indefinite Integral of \(\frac{1}{\sin x \cos x}\)

Evaluate the indefinite integral \( \int \frac{1}{\sin x \cos x} \, dx \).
2.0 Algebra Number Theory
Linear Diophantine Equation

Find all integer solutions \((x,y)\) to the equation \(17x + 23y = 100\).
2.0 Algebra
Uniqueness of Additive Identity

Prove that the additive identity in the set of real numbers (i.e., the number 0 such that \( a+0=a \) for any real \( a \)) is unique.
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.
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.
2.0 Geometry
Angle in a Circle

In a circle, prove that the angle subtended by a diameter is always a right angle.
1.5 Calculus
Indefinite Integral of \(\frac{\cos x}{1 + \sin^2 x}\)

Find the indefinite integral \( \int \frac{\cos x}{1 + \sin^2 x} \, dx \).
1.5 Calculus
Limit of \(\frac{1 - \cos x}{x^2}\) as \(x \to 0\)

Compute the limit \( \lim_{x \to 0} \frac{1 - \cos x}{x^2} \).
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.
3.0 Geometry
Ceva's Theorem

Prove that for concurrent lines from the vertices of a triangle to the opposite sides, the product of the ratios of the divided segments equals one.
2.5 Number Theory
Divisibility of Powers

Prove that $$7^n - 1$$ is divisible by 6 for all $$n \in \mathbb{N}$$.
2.0 Geometry
Area of a Triangle by Heron

Prove that the area of a triangle with sides a, b, c can be computed as $$\sqrt{s(s-a)(s-b)(s-c)}$$, where s is the semiperimeter.
2.0 Number Theory
Sum of squares

Compute the sum of squares of the first 1000 positive integers: $$1^2 + 2^2 + \dots + 1000^2.$$
4.0 Algebra Number Theory
Exponential Diophantine Equation

Find all pairs of positive integers \( (n, k) \) such that: \n $$ n! + 1 = k^2 $$
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.
2.0 Geometry
Area Using Heron's Formula

Prove that the area of a triangle with sides a, b, c can be computed as the square root of s(s-a)(s-b)(s-c), where s is the semiperimeter.
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).
2.0 Algebra Number Theory
Recurrence Relation: Closed Form and Parity

Let a sequence \( \{a_n\} \) be defined by \( a_1 = 1, a_2 = 8 \), and \( a_{n+2} = 8a_{n+1} - 15a_n \) for \( n \ge 1 \).
a) Prove by induction that \( a_n = 5^{n-1} + 3^{n-1} \) for all \( n \ge 1 \).
b) Using the closed form from part (a), or otherwise, prove that \( a_n \) is always an even number for all \( n \ge 1 \).
2.5 Calculus Trigonometry
Indefinite Integral of \(\sec^3 x\)

Find the indefinite integral:$$ \int \sec^3 x \, dx $$
1.5 Algebra Inequalities
Absolute Value Product Property

For any real numbers \( a \) and \( b \), prove that \( |a \cdot b| = |a| \cdot |b| \).
3.5 Algebra
System of Symmetric Rational Equations

Solve the system of equations for complex numbers \( x, y, z \): \n $$ \begin{cases} x+y+z = 3 \ x^2+y^2+z^2 = 3 \ x^3+y^3+z^3 = 3 \end{cases} $$
4.0 Algebra Inequalities
System of Inequality and Constraints

Let \( a, b, c \) be non-negative real numbers such that \( a+b+c=1 \). Find the maximum value of: \n $$ ab + bc + ca - 2abc $$
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?
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: $$1^3 + 2^3 + \dots + n^3 = \frac{n^2(n+1)^2}{4}$$.
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$$ \sum_{k=1}^{n} k \cdot k! = (n+1)! - 1 $$
1.5 Geometry
Triangle area

A triangle has side lengths 5, 12, and 13. Compute its area.
1.0 Calculus
Derivative of \(f(x) = \frac{\sin x}{x}\)

Find the derivative of the function \(f(x) = \frac{\sin x}{x}\) with respect to \(x\).
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 Algebra
Associativity of Set Intersection

For any three sets \( A, B, C \), prove that \( (A \cap B) \cap C = A \cap (B \cap C) \).
2.0 Calculus
Second Derivative of \(g(x) = \cos(x^2)\)

Determine the second derivative of the function \(g(x) = \cos(x^2)\) with respect to \(x\).
1.0 Combinatorics
Music Playlist Creation

A music streaming service allows users to create playlists. If a user has 10 favorite songs and wants to create a playlist of 4 songs, where the order of the songs matters, how many different playlists are possible?
2.5 Inequalities
Problem 1

Let \(a, b, c\) be positive real numbers. Prove that: i) \(a^2 + b^2 + c^2 \ge ab + bc + ca\) ii) \(a^3 + b^3 + c^3 + ab^2 + bc^2 + ca^2 \ge 2(a^2b + b^2c + c^2a)\)
2.0 Number Theory Combinatorics
Stars and bars distribution

In how many ways can \(n\) identical candies be distributed among \(k\) children?
1.0 Number Theory
Geometric series partial sum

Compute the sum of the series \(1 + 2 + 4 + 8 + \dots + 2^n\).
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.
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.
1.5 Calculus Trigonometry
Definite Integral of \(\cos^2 x\)

Evaluate the definite integral:$$ \int_{0}^{\pi/2} \cos^2 x \, dx $$
1.0 Calculus
Derivative of \(f(x) = x \sin x\)

Find the derivative of the function \(f(x) = x \sin x\).
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?
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 \( n \ge 0 \).
3.0 Algebra
Polynomial Functional Equation

Find all polynomials \( P(x) \) such that \( P(x^2) = (P(x))^2 \) for all real \( x \).