Problem 5 Six boxes contain apples, bananas, and cucumbers. The number of apples in each […]
Problem 1 A palindrome is a positive integer number that remains the same when its […]
Problem 1 Let us call a natural number interesting if any of its two consecutive […]
Wilson’s Theorem states that $p$ is prime if and only if $$(p-1)! \equiv -1 \pmod{19}$$ […]
Problem 4 The six-pointed star is regular with all the interior angles of the $12$ […]
Problem 1 Find all triples $(a,b,c)$ of positive integers such that: $$a + b + […]
Let some quantity $Q(x)$ change its value by $\pm 1$, starting at an integer number […]
Problem 1 How many sets of integers greater than $1$, with two or more elements, […]
Problem 1 A subset of $10$ elements of the set $\{1,2,3,…,90\}$ is called Boricua if […]
Problem 1 Let $O$ be the center of the circumcircle of an acute triangle $ABC$. […]
Problem 1 Positive integers $a$, $b$, $n$ satisfy the equality $$ \frac{a}{b} = \frac{a^2+n^2}{b^2+n^2} $$ […]
Problem 1 Let $O$ be the center of the circumcircle of an acute triangle $ABC$. […]
Problem 1 Positive integers $a$, $b$, $n$ satisfy the equality $$ \frac{a}{b} = \frac{a^2+n^2}{b^2+n^2} $$ […]
Complementary Counting consists in counting the total number of elements (universal set) and subtracting the […]
Diophantine equations are equations that are solved in integer numbers. We can solve some diophantine […]
Diophantine equations are equations that are solved in integer numbers. We can solve some diophantine […]
Problem 1 Let $N=2021^2k+2021$, where $k$ is a positive integer. Alfredo calculates the sum of […]
Problem 4 How many numbers $\overline{abcd}$ with different digits satisfy the following property: if we […]
Problem 1 Ana and Beto are playing a game. Ana writes a whole number on […]
Titu’s Lemma states that for all positive real numbers $x_1$, $x_2$, … , $x_n$ and […]
Diophantine equations are equations that are solved in integer numbers. We can solve some diophantine […]
Bernoulli’s Inequality states that for real numbers $x \geq -1$, $r \geq 0$ it holds […]
Problem 1 Let $ABC$ be an acute scalene triangle such that $AB <AC$. The midpoints […]
In order to find the derivative $f'(x)$ of a particular function $f(x)$ we need to […]
Day 2 Problem 1 Consider a triangle $ABC$ with $BC>AC$. The circle with […]
Day 1 Problem 1 A four-digit positive integer is called $virtual$ if it […]
Let $x_1$, $x_2$ and $x_3$ be the roots of the polynomial $$ ax^3 + bx^2 […]
AMC 8 Preparation Book presents the most popular methods and techniques that are used to […]
Let $x_1$ and $x_2$ be the solutions of the quadratic equation $$ ax^2+bx+c$$ Vieta’s Formulas […]
An invariant is a quantity or a property of an object that is unchanged under-considered […]
Euler’s Theorem states that for a positive integer $n$ and an integer $a$ relatively prime […]
Bezout’s Identity states that for any natural numbers $a$ and $b$, there exist integers $x$ […]
Mathematical Induction is used to prove that the statement of the problem $P(n)$ is true […]
Mathematical Induction is used to prove that the statement of the problem $P(n)$ is true […]
Mathematical Induction is used to prove that the statement of the problem is true for […]
Mathematical Induction is used to prove that the statement of the problem is true for […]
Problem 1 We have $10,000$ identical equilateral triangles. Consider the largest regular hexagon that can […]
Inequality of Arithmetic and Geometric Means (AM-GM) states that for all positive real numbers $x_1$, […]
Inequality of Arithmetic and Geometric Means (AM-GM) states that for all positive real numbers $x_1$, […]
Inequality of Arithmetic and Geometric Means (AM-GM) states that for the positive real numbers $x_1$, […]
