In maths class Albrecht had to compute $(a+2b-3)^2$ . His result was $a^2 +4b^2-9$ . ‘This is not correct’ said his teacher, ‘try substituting positive integers for $a$ and $b$.’ Albrecht did so, but his result proved to be correct. What numbers could he substitute? a) Show a good substitution. b) Give all the pairs that Albrecht could substitute and prove that there are no more.
2020 Dürer Math Competition (First Round)
Category $E$
Initially we have a $2 \times 2$ table with at least one grain of wheat on each cell. In each step we may perform one of the following two kinds of moves: $i.$ If there is at least one grain on every cell of a row, we can take away one grain from each cell in that row. $ii.$ We can double the number of grains on each cell of an arbitrary column. a) Show that it is possible to reach the empty table using the above moves, starting from the position down below. b) Show that it is possible to reach the empty table from any starting position. c) Prove that the same is true for the $8 \times 8$ tables as well.
a) Is it possible that the sum of all the positive divisors of two different natural numbers are equal? b) Show that if the product of all the positive divisors of two natural numbers are equal, then the two numbers must be equal.
Let $ABC$ be an acute triangle with side $AB$ of length $1$. Say we reflect the points $A$ and $B$ across the midpoints of $BC$ and $AC$, respectively to obtain the points $A’$ and $B’$ . Assume that the orthocenters of triangles $ ABC$, $A’BC$ and $B’AC$ form an equilateral triangle. a) Prove that triangle $ABC$ is isosceles. b) What is the length of the altitude of $ABC$ through $C$?
We call a table of size $n \times n$ self-describing if each cell of the table contains the total number of even numbers in its row and column other than itself. How many self-describing tables of size a) $3 \times 3$ exist? b) $4 \times 4$ exist? c) $5 \times 5$ exist? Two tables are different if they differ in at least one cell.
Category $E^+$
a) Is it possible that the sum of all the positive divisors of two different natural numbers are equal? b) Is it possible that the product of all the positive divisors of two different natural numbers are equal?
How many ways can you fill a table of size $n\times n$ with integers such that each cell contains the total number of even numbers in its row and column other than itself? Two tables are different if they differ in at least one cell.
At least how many non-zero real numbers do we have to select such that every one of them can be written as a sum of $2019$ other selected numbers and a) the selected numbers are not necessarily different? b) the selected numbers are pairwise different?
Suppose that you are given the foot of the altitude from vertex $A$ of a scalene triangle $ABC$, the midpoint of the arc with endpoints $B$ and $C$, not containing $A$ of the circumscribed circle of $ABC$, and also a third point $P$. Construct the triangle from these three points if $P$ is the a) orthocenter b) centroid c) incenter of the triangle.
Let $p$ be prime and $ k > 1$ be a divisor of $p-1$. Show that if a polynomial of degree $k$ with integer coefficients attains every possible value modulo $ p$ that is $(0,1,\dots, p-1)$ at integer inputs then its leading coefficient must be divisible by $p$. NoteNote: the leading coefficient of a polynomial of degree d is the coefficient of the $x_d$ term.