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## Algebra, Germany MO 2019, Problem 6

Understand the problem Suppose that real numbers  and  satisfy the following equations: Show that  must be equal to  or . Note: It is not required to show the existence of such numbers . Source of the problem Germany MO 2019, Problem 6 Topic Algebra, Simultaneous...

## Geometry, Israel MO 2019, Problem 3

Understand the problem Six congruent isosceles triangles have been put together as described in the picture below. Prove that points M, F, C lie on one line. Source of the problem Israel MO 2019 Problem 3 Topic Geometry Difficulty Level 6/10 Suggested Book Challenges...

## Combinatorics, Israel MO 2014, Problem 4

Understand the problem The three-digit number 999 has a special property: It is divisible by 27, and its digit sum is also divisible by 27. The four-digit number 5778 also has this property, as it is divisible by 27 and its digit sum is also divisible by 27. How many...

## Number Theory – Germany MO 2019, Problem 4

Understand the problem Show that for each non-negative integer  there are unique non-negative integers  and  such that we have Source of the problem Germany MO, 2019, Problem 4 Topic Number Theory Difficulty Level 4/10 Suggested Book Challenges and Thrills of Pre...

## Polynomial, Vietnam MO 2014 Problem 2

Understand the problem Given the polynomial  where  is a positive integer. Prove that  can’t be written as a product of  non-constant polynomials with integer coefficients. Source of the problem Vietnam MO 2014 Problem 2 Topic Polynomial, Algebra Difficulty...

## Number Theory – Italy MO 2019, Problem 2

Understand the problem Let  be prime numbers Prove that if  is a perfect square then  is not a perfect square for any positive integer  Source of the problem Italy MO 2019, Problem 2 Topic Number Theory Difficulty Level 5/10 Suggested Book Challenges and Thrills of...