Hello, I wanna know how to use 2,6,8,8 to get result equals 1 in the All ten game?

Known ellipse (a>b>0), C and D are two points on the same side of the x axis of the ellipse. The two tangents of the ellipse made through C and D respectively intersect at point P, and intersect with the x axis at points B and A, and the line AC and BD intersect at point E. Make at point F, and the reverse extension line of line EF intersects CD at point M, and intersects the ellipse at point N. Prove that P, E, F three points are collinear:

IMAGE

Let be a regular heptagon. Lines and intersect at , and lines and intersect at . Show that , , and are collinear.

A triangular corner with side lengths is cut from equilateral triangle of side length . The perimeter of the remaining quadrilateral is

IMAGE

How many primes less than have as the ones digit? (Assume the usual base ten representation)

Let be a point in parallelogram . Prove that

Source: Kazakhstan 2018

Let Prove that

An Observation on Perfect Square Trinomials — “Square Result Theorem”

Hi everyone,

I recently noticed something interesting while working with perfect square trinomials, and I wanted to share it to see if it’s a known result or just a neat observation.

My Theorem Statement (Informal):

If a real number is substituted into a perfect square trinomial, the result will always be a non-negative perfect square.

Example:

Let’s take the trinomial:

x^2 + 4x + 4 = (x + 2)^2

Try substituting different values for :1,2,3

And so on. This happens with any perfect square trinomial like:

In every case, the result of substituting a value for the variable gives a non-negative square, often of an integer or rational number.

My Question:

Is this theorem already known or named? If yes, what is it typically called? If not, is this worth developing into a more formal concept?

I’d love feedback from anyone more experienced in algebra or number theory!

Thanks in advance.
– HamiltonGafer

Call an ordered triple of positive integers if two terms of the triple multiply to the third. Find the number of permutations of the first positive integers to itself such that if


is a multiplicative triple, then is a multiplicative triple as well.

As shown in the figure below, MAIL is a cyclic quadrilateral circumscribed by a circle with diameter MI=25. In addition, \frac{AI}{IL}=\frac{6}{5}, AM=7, and MI is perpendicular to MS. Find the length of SL.
Given figure: https://imgur.com/a/ABS11ev
Answer
Solution: https://imgur.com/a/FcUnELt
Source: 2024 Mathirang Mathibay Orals/Tier 3-3