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
Apr 30, 2023
Problem 1 A palindrome is a positive integer number that remains the same when its...
Wilson’s Theorem
May 9, 2022
Wilson’s Theorem states that $p$ is prime if and only if $$(p-1)! \equiv -1 \pmod{19}$$...
Discrete Continuity
Apr 25, 2022
Let some quantity $Q(x)$ change its value by $\pm 1$, starting at an integer number...
Polish Mathematical Olympiad, 2021
Nov 20, 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
Oct 8, 2021
Complementary Counting consists in counting the total number of elements (universal set) and subtracting the...
How to Solve Diophantine Equations: Modulus
Oct 8, 2021
Diophantine equations are equations that are solved in integer numbers. We can solve some diophantine...
Diophantine Equations: Factorization
Oct 8, 2021
Diophantine equations are equations that are solved in integer numbers. We can solve some diophantine...
Team Selection Test for Iberoamerican Math Olympiad 2021
Sep 20, 2021
Problem 1 Let $N=2021^2k+2021$, where $k$ is a positive integer. Alfredo calculates the sum of...
Titu’s Lemma
Sep 14, 2021
Titu’s Lemma states that for all positive real numbers $x_1$, $x_2$, … , $x_n$ and...
How to Solve Diophantine Equations: Quadratic Formula
Sep 14, 2021
Diophantine equations are equations that are solved in integer numbers. We can solve some diophantine...
Bernoulli’s Inequality
Sep 6, 2021
Bernoulli’s Inequality states that for real numbers $x \geq -1$, $r \geq 0$ it holds...
How to Find the Derivative
Aug 30, 2021
In order to find the derivative $f'(x)$ of a particular function $f(x)$ we need to...
Vieta’s Formulas for Cubic Polynomial
Aug 24, 2021
Let $x_1$, $x_2$ and $x_3$ be the roots of the polynomial $$ ax^3 + bx^2...
Vieta’s Formulas
Aug 16, 2021
Let $x_1$ and $x_2$ be the solutions of the quadratic equation $$ ax^2+bx+c$$ Vieta’s Formulas...
Invariant
Aug 16, 2021
An invariant is a quantity or a property of an object that is unchanged under-considered...
Euler’s Theorem
Jul 14, 2021
Euler’s Theorem states that for a positive integer $n$ and an integer $a$ relatively prime...
Bezout’s Identity
Jun 15, 2021
Bezout’s Identity states that for any natural numbers $a$ and $b$, there exist integers $x$...
Mathematical Induction: Inequalities
Jun 15, 2021
Mathematical Induction is used to prove that the statement of the problem $P(n)$ is true...
Mathematical Induction: Combinatorics
Jun 15, 2021
Mathematical Induction is used to prove that the statement of the problem $P(n)$ is true...
Mathematical Induction: Algebra
Jun 15, 2021
Mathematical Induction is used to prove that the statement of the problem is true for...
Mathematical Induction: Number Theory
Jun 15, 2021
Mathematical Induction is used to prove that the statement of the problem is true for...
AM-GM Inequality for n Variables
Jun 14, 2021
Inequality of Arithmetic and Geometric Means (AM-GM) states that for all positive real numbers $x_1$,...
AM-GM Inequality for Three Variables
Jun 14, 2021
Inequality of Arithmetic and Geometric Means (AM-GM) states that for all positive real numbers $x_1$,...
AM-GM Inequality for Two Variables
Jun 14, 2021
Inequality of Arithmetic and Geometric Means (AM-GM) states that for the positive real numbers $x_1$,...
Team Selection Process for IMO in Ukraine
Jun 14, 2021
The team selection process for IMO in Ukraine consists of 4 rounds of olympiad exams...
Fermat’s Little Theorem
Jun 14, 2021
Fermat’s Little Theorem states that for a prime $p$ and an integer $a$ not divisible...
Pigeonhole Principle
Jun 14, 2021
Let $m,n \in \mathbb{N}$ and $n>m$. The Pigeonhole Principle states that if $n$ items are...
Monovariant
Jun 12, 2021
Monovariant is a quantity related to the object that is monotonous under-considered operations. Problem (Tournament...
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