On a big plane, two points with length are given. Prove that one can only use straightedge (which draws a straight line passing two drawn points) and compass (which draws a circle with a chosen radius equal to the distance of two drawn points and centered at a drawn points) to construct a line and two points on it with length in only steps (Namely, the total number of circles and straight lines drawn is at most .)

Given triangle inscribed in a circle with center . is any point not on (O). intersect at . Let be the Lemoine points of triangle respectively. Prove that are collinear.

Let be a prime number and . Prove that
Proposed by Miloš Milićev

In the city of Timbuktu, there is an orphanage. It shelters children from the new mysterious disease that causes children to explode. There are m children in the orphanage. To try to cure this disease, a mad scientist named Myla has come up with an innovative cure. She ties every child to every other child using medicinal ropes. Every child is connected to every other child using one of different ropes. She then performs a experiment that causes children, each connected to each other with the same type of rope, to be cured. Two experiments are said to be of the same type, if each of the ropes connecting the children has the same medicine imbued in it. She then unties them and lets them go back home.

We let be the minimum m such that Myla can perform at least experiments of the same type. Prove that:

For every there exists a and such that for all ,
Find the value of for every .

Let be a positive integer and let be a prime number. Prove that if , , are integers (not necessarily positive) satisfying the equations then .

Proposed by Angelo Di Pasquale, Australia

Aloo and Batata are playing a game. They are given a regular -gon, where is an even integer. At the start, a line joining two vertices is chosen arbitrarily and one of its endpoints is chosen as its pivot. Now, Aloo rotates the line around the pivot either clockwise or anti-clockwise until it passes through another vertex of the -gon. Then, the new vertex becomes the pivot and Batata again chooses to rotate the line clockwise or anti-clockwise
about the pivot. The player who moves the line into a position it has already been in (i.e. it passes through the same two vertices of the -gon it was in at a previous time) loses.
Find all such that Batata always has a winning strategy irrespective of the starting edge.

Proposed by Anik Sardar, Om Patil and Anudip Giri

Let be such that and . Prove that:

Proposed by Shreyas Deshpande

Let be a triangle with orthocentre . Let be the foot of altitudes of onto the opposite sides, respectively. Consider , the reflection of about line . Let line cut at distinct points , and let be the orthocenter of . Prove that points are concyclic.

Proposed by Mandar Kasulkar

Given an n x n chessboard, with a king starting at any square, the king's task is to visit each square in the board exactly once (essentially an open path); this king moves how a king in chess would.
However, we are allowed to place k numbers on the board of any value such that for each number A we placed on the board, the king must be in the position of that number A on its Ath square in its journey, with the starting square as its 1st square.
Suppose after we placed k numbers, there is one and only one way to complete the king's task (this includes placing the king in a starting square), find the minimum value of k set by n.

Should've put one of its tag as "Open problem"; I have no idea how to tackle this problem either.

Solve the quadratic equation:

It's about the Japanese Olympiad

I can't solve it no matter how much I think about it.

If there are people who are good at math

Please help me.