1234567

123456ui
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 Calculus
Derivative of \(f(x) = \ln(\tan x)\)

Find the derivative of the function \(f(x) = \ln(\tan x)\) with respect to \(x\).
2.0 Number Theory Combinatorics
Remainder of a Factorial Sum

Find the remainder when \(1! + 2! + 3! + \dots + 100!\) is divided by \(12\).
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 Calculus
Limit of \(\frac{\cos x}{\pi/2 - x}\) as \(x \to \pi/2\)

Calculate the limit \( \lim_{x \to \pi/2} \frac{\cos x}{\pi/2 - x} \).
1.5 Calculus
Indefinite Integral of \(\tan(2x)\)

Evaluate the indefinite integral \( \int \tan(2x) \, dx \).
2.0 Calculus
Limit of \(\frac{\sin x - x}{x^3}\) as \(x \to 0\)

Evaluate the limit:$$ \lim_{x \to 0} \frac{\sin x - x}{x^3} $$
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 Algebra Inequalities
AM-GM Inequality Application

Prove that for any positive real numbers \(a, b, c\),
$$ (a+b+c) \left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right) \geq 9 $$
2.0 Algebra
Polynomial Roots and Vieta's Formulas

Let \(P(x) = x^4 + ax^3 + bx^2 + cx + d\) be a polynomial with real coefficients. If \(P(1) = 0\), \(P(2) = 0\), and \(P(3) = 0\), and the sum of the roots of \(P(x)\) is 10, find the value of \(d\).
1.5 Calculus
Derivative of \(h(x) = \arcsin(\sqrt{x})\)

Find the derivative of the function \(h(x) = \arcsin(\sqrt{x})\) with respect to \(x\).
2.0 Number Theory Combinatorics
Prove the Formula for the Sum of Fibonacci Numbers by Induction

Let \( F_k \) be the \( k \)-th Fibonacci number defined by \( F_1 = 1 \), \( F_2 = 1 \), and \( F_k = F_{k-1} + F_{k-2} \) for \( k \geq 3 \).
Prove by mathematical induction that for all positive integers \( n \):
$$ \sum_{i=1}^n F_i = F_{n+2} - 1 $$
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?
1.0 Algebra Number Theory
Product of Two Even Integers

Prove that the product of any two even integers is an even integer.
1.5 Algebra
Set Difference and Union Identity

For any two sets \( A \) and \( B \), prove that \( (A \setminus B) \cup B = A \cup B \).
1.0 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 Algebra Number Theory
Modular Inverse Calculation (Prime Modulus)

Find the smallest positive integer \(x\) such that \(17x \equiv 1 \pmod{101}\).
2.0 Number Theory Combinatorics
Postage Stamp Problem

Prove by strong induction that every integer amount of postage greater than or equal to 12 cents can be formed using only 4-cent and 5-cent stamps.
1.0 Combinatorics
Card Hand Combinations

From a standard deck of 52 playing cards, how many different ways can you select exactly 2 aces?
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.
1.5 Algebra Inequalities
Absolute Value Product Property

For any real numbers \( a \) and \( b \), prove that \( |a \cdot b| = |a| \cdot |b| \).
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 Geometry
Exterior Angle Theorem for Triangles

Given a triangle \( \triangle ABC \), with \( D \) a point on the line containing \( BC \) such that \( C \) is between \( B \) and \( D \). Prove that the measure of the exterior angle \( \angle ACD \) is equal to the sum of the measures of the two opposite interior angles, \( \angle BAC + \angle ABC \).
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.
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?
1.5 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.
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.
1.0 Algebra
Quadratic equation with parameter

Solve the equation \(x^2 + (k-1)x + k = 0\) in terms of \(k\). (solve for \(x\))
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 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.
3.5 Algebra Number Theory
Higher Order Diophantine System

Determine all integer solutions to the system: \n $$ x + y = u + v \\ x^2 + y^2 = u^2 + v^2 $$
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 \(p=2\).
2.5 Number Theory Probability
Prime-sum probability

Three fair dice are rolled. What is the probability that their sum is a prime number?
1.0 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 Combinatorics
Group Photo Lineup

6 friends are lining up side-by-side for a group photo.
In how many different orders can they arrange themselves?
2.5 Number Theory
Prime Number Divisibility Condition

Find all prime numbers \(p\) such that \(p\) divides \(2^{p-1} + 1\).
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 Number Theory
Divisibility test

Determine whether \(2^{10} + 3^{5}\) is divisible by 7.
1.0 Calculus
Definite Integral of \(\frac{1}{\sqrt{1-x^2}}\) from 0 to 1

Compute the definite integral \( \int_0^1 \frac{1}{\sqrt{1-x^2}} \, dx \).
1.0 Geometry
Angle Sum in a Triangle

Prove that the sum of the interior angles of any triangle is equal to $$180^\circ$$.
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.
3.5 Geometry
Nine-Point Circle

Prove that the midpoints of the sides, the feet of the altitudes, and the midpoints between vertices and orthocenter of a triangle lie on a circle called the nine-point circle.
1.0 Algebra Inequalities
Non-negativity of a Real Square

For any real number \( x \), prove that \( x^2 \geq 0 \).
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 Calculus
Indefinite Integral of \(\frac{1}{\sin x \cos x}\)

Evaluate the indefinite integral \( \int \frac{1}{\sin x \cos x} \, dx \).
2.5 Algebra Number Theory
Pell-type Diophantine Equation

Find all integer solutions \((x,y)\) to the equation \(x^2 - 7y^2 = 1\).
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 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$$ \sum_{k=1}^{n} k \cdot k! = (n+1)! - 1 $$
4.0 Algebra Number Theory
Exponential Diophantine Equation

Find all pairs of positive integers \( (n, k) \) such that: \n $$ n! + 1 = k^2 $$
1.5 Geometry
Triangle Area with Coordinates

Find the area of the triangle with vertices (1,2), (4,5), and (6,1).
1.0 Combinatorics
City Tour Itinerary

A tourist wants to visit 3 specific cities out of a list of 7 possible cities. The order in which the tourist visits these cities matters.
How many different travel itineraries are possible?
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.
2.0 Algebra Number Theory
Linear Diophantine Equation

Find all integer solutions \((x,y)\) to the equation \(17x + 23y = 100\).
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
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.
1.5 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 \( \int \cos^3 x \, dx \).
3.5 Algebra Number Theory
Integer Solutions for Cubic System

Find all integer solutions \((x,y,z)\) to the system:
$$ x+y+z = 3 $$
$$ x^3+y^3+z^3 = 3 $$
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
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:$$ \sum_{n=1}^{N} \arctan\left(\frac{1}{n^2+n+1}\right) $$
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: $$1^2 + 2^2 + \dots + n^2 = \frac{n(n+1)(2n+1)}{6}$$.
5.0
wefrg

daEFSz
1.0 Combinatorics
Custom Computer Configurations

A computer manufacturer allows customers to customize their orders. There are 3 CPU options, 2 RAM options, and 4 storage options.
How many different custom computer configurations are possible?
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 \( n \), \( \gcd(F_n, F_{n+1}) = 1 \).
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 Combinatorics
Permutations with Restrictions

How many distinct 5-letter words can be formed using the letters A, B, C, D, E, F, G, H if each letter can be used at most once and the word must contain the letter A?
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 Number Theory Combinatorics
Combinatorics with restriction

From a group of 17 men and 23 women, how many 3-person committees can be formed that include at least one woman?
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.
2.0 Algebra Inequalities
Integer Solutions to a System of Inequalities

Find all integer pairs \((x,y)\) that satisfy the following system of inequalities:
$$ x^2 + y^2 \leq 16 $$
$$ y \geq x+1 $$
2.5 Algebra Number Theory
Recurrence Invariant Identity

Consider the sequence defined by \( x_1=1, x_2=3 \) and \( x_n = 4x_{n-1} - x_{n-2} \) for \( n \ge 3 \).
Prove by induction that \( x_n^2 - 4x_n x_{n-1} + x_{n-1}^2 = -3 \) for all \( n \ge 2 \).
3.0 Algebra
Exponential-Polynomial Equation

Solve the equation \(e^x = x^2\) for real \(x\). Express your answer using the Lambert W function.
1.0 Number Theory
Prime Factorization

Find all prime factors of $$ 2^{64}-1$$.
1.5 Geometry
Triangle area

A triangle has side lengths 5, 12, and 13. Compute its area.
1.5 Algebra
Intersection with a Superset

For any two sets \( A \) and \( B \), prove that if \( A \subseteq B \), then \( A \cap B = A \).
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.
1.0 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 Geometry
Inscribed Angle Theorem

In a circle, an inscribed angle subtending an arc is equal to half the measure of the corresponding central angle.