Puerto Rico Team Selection Test, 2023. Day 2
Problem 5 Six boxes contain apples, bananas, and cucumbers. The number of apples in each...
Puerto Rico Team Selection Test, 2023. Day 1
Problem 1 A palindrome is a positive integer number that remains the same when its...
Wilson’s Theorem
Wilson’s Theorem states that $p$ is prime if and only if $$(p-1)! \equiv -1 \pmod{19}$$...
Discrete Continuity
Let some quantity $Q(x)$ change its value by $\pm 1$, starting at an integer number...
Polish Mathematical Olympiad, 2021
Problem 1 Positive integers $a$, $b$, $n$ satisfy the equality $$ \frac{a}{b} = \frac{a^2+n^2}{b^2+n^2} $$...
Complementary Counting
Complementary Counting consists in counting the total number of elements (universal set) and subtracting the...
How to Solve Diophantine Equations: Modulus
Diophantine equations are equations that are solved in integer numbers. We can solve some diophantine...
Diophantine Equations: Factorization
Diophantine equations are equations that are solved in integer numbers. We can solve some diophantine...
Team Selection Test for Iberoamerican Math Olympiad 2021
Problem 1 Let $N=2021^2k+2021$, where $k$ is a positive integer. Alfredo calculates the sum of...
Titu’s Lemma
Titu’s Lemma states that for all positive real numbers $x_1$, $x_2$, … , $x_n$ and...
How to Solve Diophantine Equations: Quadratic Formula
Diophantine equations are equations that are solved in integer numbers. We can solve some diophantine...
Bernoulli’s Inequality
Bernoulli’s Inequality states that for real numbers $x \geq -1$, $r \geq 0$ it holds...
How to Find the Derivative
In order to find the derivative $f'(x)$ of a particular function $f(x)$ we need to...
Vieta’s Formulas for Cubic Polynomial
Let $x_1$, $x_2$ and $x_3$ be the roots of the polynomial $$ ax^3 + bx^2...
Vieta’s Formulas
Let $x_1$ and $x_2$ be the solutions of the quadratic equation $$ ax^2+bx+c$$ Vieta’s Formulas...
Invariant
An invariant is a quantity or a property of an object that is unchanged under-considered...
Euler’s Theorem
Euler’s Theorem states that for a positive integer $n$ and an integer $a$ relatively prime...
Bezout’s Identity
Bezout’s Identity states that for any natural numbers $a$ and $b$, there exist integers $x$...
Mathematical Induction: Inequalities
Mathematical Induction is used to prove that the statement of the problem $P(n)$ is true...
Mathematical Induction: Combinatorics
Mathematical Induction is used to prove that the statement of the problem $P(n)$ is true...
Mathematical Induction: Algebra
Mathematical Induction is used to prove that the statement of the problem is true for...
Mathematical Induction: Number Theory
Mathematical Induction is used to prove that the statement of the problem is true for...
AM-GM Inequality for n Variables
Inequality of Arithmetic and Geometric Means (AM-GM) states that for all positive real numbers $x_1$,...
AM-GM Inequality for Three Variables
Inequality of Arithmetic and Geometric Means (AM-GM) states that for all positive real numbers $x_1$,...
AM-GM Inequality for Two Variables
Inequality of Arithmetic and Geometric Means (AM-GM) states that for the positive real numbers $x_1$,...
Team Selection Process for IMO in Ukraine
The team selection process for IMO in Ukraine consists of 4 rounds of olympiad exams...
Fermat’s Little Theorem
Fermat’s Little Theorem states that for a prime $p$ and an integer $a$ not divisible...
Pigeonhole Principle
Let $m,n \in \mathbb{N}$ and $n>m$. The Pigeonhole Principle states that if $n$ items are...
Monovariant
Monovariant is a quantity related to the object that is monotonous under-considered operations. Problem (Tournament...
