1.5 Algebra Inequalities
AM-GM Inequality Application

Prove that for any positive real numbers \(a, b, c\),
$$ (a+b+c) \left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right) \geq 9 $$
1.0 Combinatorics
Fast Food Meal Deals

A fast-food restaurant offers a meal deal where customers can choose 1 main course from 3 options, 1 side dish from 4 options, and 1 drink from 2 options.
How many different meal deals can be chosen?
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
1.0 Algebra Number Theory
Difference of Squares of Consecutive Integers

Prove that the difference between the squares of any two consecutive integers is always an odd integer.
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?
1.5 Algebra Number Theory
Closure of Rationals Under Addition

Prove that the sum of any two rational numbers is always a rational number.
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 Combinatorics
Selecting a Basketball Team

A basketball coach needs to select a starting lineup of 5 players from a squad of 12 players.
How many different starting lineups are possible, assuming the order of players in the lineup does not matter?
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?
1.0 Geometry
Triangle with Given Perimeter

Given a triangle with perimeter 30, prove that the sum of the altitudes is less than 20.
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 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.
1.5 Geometry
Triangle area using coordinates

Find the area of the triangle with vertices \((1,2)\), \((4,5)\), and \((6,1)\)
2.0 Algebra Number Theory
Linear Diophantine Equation

Find all integer solutions \((x,y)\) to the equation \(17x + 23y = 100\).
3.0 Algebra Number Theory
Linear Diophantine Equation in Three Variables

Find all integer solutions \( (x, y, z) \) that satisfy the equation: \n $$ 15x + 21y + 35z = 1 $$
1.5 Algebra Number Theory
Prove the Formula for the Sum of an Arithmetic Series by Induction

Use mathematical induction to prove the formula for the sum of the first \( n \) terms of an arithmetic series, which is given by
$$ S_n = \sum_{k=1}^n (a + (k-1)d) = \frac{n}{2}(2a + (n-1)d) $$
for all positive integers \( n \ge 1 \), where \( a \) is the first term and \( d \) is the common difference.
1.0 Combinatorics
Pizza Topping Choices

A pizza restaurant offers 8 different toppings. A customer wants to choose exactly 3 different toppings for their pizza.
How many different combinations of toppings are possible?
2.0 Calculus
Indefinite Integral of \(e^x \cos x\)

Calculate the indefinite integral \( \int e^x \cos x \, dx \).
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 Geometry
Median Ratio

In triangle ABC, prove that the centroid divides each median in a 2:1 ratio.
1.5 Algebra
Intersection with a Superset

For any two sets \( A \) and \( B \), prove that if \( A \subseteq B \), then \( A \cap B = A \).
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 $$\frac{BD}{DC} = \frac{AB}{AC}$$.
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1.5 Algebra Number Theory
Prove the Formula for the Sum of Powers of Two by Induction

Use the principle of mathematical induction to prove that for any non-negative integer \( n \), the sum of the first \( n+1 \) powers of two is given by the formula:
$$ \sum_{k=0}^{n} 2^k = 2^{n+1} - 1 $$
1.5 Algebra Number Theory
Divisibility by 7 via Induction

Prove by induction that for every non-negative integer \( n \), the number \( 3^{2n+1} + 2^{n+2} \) is divisible by 7.
3.0 Algebra
Symmetric System of Equations

Solve the following system of equations for real numbers \(x, y, z\):
$$ x+y+z = 6 $$
$$ x^2+y^2+z^2 = 14 $$
$$ x^3+y^3+z^3 = 36 $$
1.0 Calculus
Definite Integral of \(\frac{1}{\sqrt{1-x^2}}\) from 0 to 1

Compute the definite integral \( \int_0^1 \frac{1}{\sqrt{1-x^2}} \, dx \).
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 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 Calculus
Limit of \(\frac{\sin x - x}{x^3}\) as \(x \to 0\)

Evaluate the limit:$$ \lim_{x \to 0} \frac{\sin x - x}{x^3} $$
2.0 Geometry
Cyclic Quadrilateral Angle Property

Prove that in a cyclic quadrilateral, the sum of the opposite angles is 180 degrees.
1.0 Algebra
Polynomial factorization

Factor the polynomial \(x^3 - 3x^2 + 3x - 1\).
2.5 Number Theory Probability
Prime-sum probability

Three fair dice are rolled. What is the probability that their sum is a prime number?
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.
2.5 Algebra Number Theory
Solving a Modular Power Congruence

Find the smallest positive integer \(x\) such that \(x^{10} \equiv 3 \pmod{11}\).
3.5 Algebra Number Theory
Integer Solutions for Cubic System

Find all integer solutions \((x,y,z)\) to the system:
$$ x+y+z = 3 $$
$$ x^3+y^3+z^3 = 3 $$
1.5 Algebra
Associativity of Set Intersection

For any three sets \( A, B, C \), prove that \( (A \cap B) \cap C = A \cap (B \cap C) \).
1.0 Number Theory
Geometric progression partial sum

Compute the sum of the first 16 terms of the geometric sequence \(2, 6, 18, \dots\).
1.5 Combinatorics
School Committee Formation

A school club has 15 members, consisting of 8 boys and 7 girls. They need to form a committee of 4 members.

a) How many different committees can be formed?

b) How many different committees can be formed if it must consist of 2 boys and 2 girls?
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 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$$ \sum_{k=1}^{n} k \cdot k! = (n+1)! - 1 $$
2.5 Number Theory
Fermat's Little Theorem Application

Find the remainder when \\( 2^{100} \\) is divided by 13.
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?
1.5 Geometry
Triangle area

A triangle has side lengths 5, 12, and 13. Compute its area.
1.5 Geometry
Exterior Angle Theorem

Prove that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two opposite interior angles.
1.5 Number Theory
Divisibility test

Determine whether \(2^{10} + 3^{5}\) is divisible by 7.
1.5 Geometry
Perpendicular Bisectors and Circumcenter

Given triangle ABC, the perpendicular bisectors of its sides intersect at point O. Prove that O is equidistant from A, B, and C.
2.5 Algebra
Functional Equation on Reals

Find all functions \(f: \mathbb{R} \to \mathbb{R}\) such that \(f(x+y) = f(x) + f(y)\) for all \(x,y \in \mathbb{R}\), and \(f(xy) = f(x)f(y)\) for all \(x,y \in \mathbb{R}\).
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 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?
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 \( n \ge 0 \).
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 Algebra Number Theory
Multiplicative Order Modulo a Prime

Find the smallest positive integer \(n\) such that \(3^n \equiv 1 \pmod{43}\).
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1.5 Algebra Inequalities
Quadratic inequality

Solve the inequality \(x^2 - 4x + 3 \le 0\).
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 Geometry
Median Length in Triangle

In triangle ABC, prove that the median from vertex A is shorter than the sum of half the other two sides.
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).
1.5 Calculus
Limit of \(\frac{\cos x}{\pi/2 - x}\) as \(x \to \pi/2\)

Calculate the limit \( \lim_{x \to \pi/2} \frac{\cos x}{\pi/2 - x} \).
2.0 Number Theory
Remainder of a Large Power

Find the remainder when \(3^{2023}\) is divided by \(17\).
1.5 Algebra Number Theory
Modular Inverse Calculation (Prime Modulus)

Find the smallest positive integer \(x\) such that \(17x \equiv 1 \pmod{101}\).
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 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.
1.5 Algebra
Union with a Superset

For any two sets \( A \) and \( B \), prove that if \( A \subseteq B \), then \( A \cup B = B \).
2.0 Calculus
Indefinite Integral of \(\cos^3 x\)

Evaluate the indefinite integral \( \int \cos^3 x \, dx \).
2.5 Algebra Number Theory
Divisibility of a Specific Number by 7

Prove by induction that for any natural number \( n \), the number \( N_n = 11 \cdot 10^{2n} + 10^{n+1} + 1 \) is divisible by 7.
1.5 Calculus
Definite Integral of \(\sin^2 x\)

Compute the definite integral \( \int_0^{\pi/2} \sin^2 x \, dx \).
2.5 Algebra
Nesbitt's Inequality

Let \(a, b, c\) be positive real numbers. Prove that:$$\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b} \ge \frac{3}{2}$$
3.0 Algebra Calculus Trigonometry
Sum of Telescoping Series with Arctangent

Find the sum of the series:$$ \sum_{n=1}^{N} \arctan\left(\frac{1}{n^2+n+1}\right) $$
1.5 Calculus
Indefinite Integral of \(\frac{\cos x}{1 + \sin^2 x}\)

Find the indefinite integral \( \int \frac{\cos x}{1 + \sin^2 x} \, dx \).
1.0 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 Calculus
Sum of Maclaurin Series for Cosine

Find the sum of the infinite series \( \sum_{n=0}^{\infty} \frac{(-1)^n \pi^{2n}}{(2n)!} \).
3.0 Number Theory
Fermat Numbers and Relative Primality

Let \( F_n = 2^{2^n} + 1 \) be the \( n \)-th Fermat number for \( n \ge 0 \).
a) Prove by induction that for any \( n \ge 1 \), \( F_n = F_0 \cdot F_1 \cdot \dots \cdot F_{n-1} + 2 \).
b) Use the result from part (a) to show that any two distinct Fermat numbers \( F_m \) and \( F_n \) (where \( m \ne n \)) are relatively prime, i.e., \( \gcd(F_m, F_n) = 1 \).
2.0 Geometry
Law of Cosines

Prove that in any triangle, the square of a side equals the sum of the squares of the other two sides minus twice the product of those sides and the cosine of the included angle.
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 Calculus
Limit of \((\sin x)^x\) as \(x \to 0^+\)

Determine the limit \( \lim_{x \to 0^+} (\sin x)^x \).
1.5 Calculus
Indefinite Integral of \(\tan(2x)\)

Evaluate the indefinite integral \( \int \tan(2x) \, dx \).
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) \(a^3 + b^3 + c^3 + ab^2 + bc^2 + ca^2 \ge 2(a^2b + b^2c + c^2a)\)
3.0 Number Theory
Prime Divisor of Sum of Squares

Let \(p\) be a prime number. If \(p\) divides \(n^2 + 1\) for some integer \(n\), prove that \(p \equiv 1 \pmod{4}\) or \(p=2\).