Is This Your Card?

A magician randomly draws one card from a standard card deck. The probability that the card drawn is a heart or a diamond can be written as , where  and  are positive, coprime integers. What is the value of ?

Details and assumptions

A standard deck of cards consists of 52 cards, formed by 13 ranks (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q and K) of 4 suits (clubs, diamonds, hearts and spades).

Point of Discontinuity

f is a function defined on the positive reals such that . For what value of  is  undefined?

The Curve of "Love"

If we talk about love, we know that love is one of abstract things in this world. Like Mathematics, love just can feel with mind, heart, and sense. We know the symbol of love, isn't right? With mathematics, we can draw the symbol of love in Cartesian Coordinate. Its so amazing right? Mathematics like a magic, all things that we think can't happen, but in mathematics, it can be happen.

This is the equation of "Love" :

x⁶ + 3x⁴ y² - 3x⁴ + 3x² y⁴ - x² y³ - 6x² y² + 3x² + y⁶ - 3y⁴ + 3y² = 1

And if we draw the equation in Cartesian Coordinate (I use GeoGebra) we get:

Believe it !!! It so amazing, with Mathematics we can do anything, so please loving mathematics !!!

Slope through two points

Find the slope of a line that passes through the points (3,−26)and (8,19).

X-intercept of a line

What is the x-intercept of the line y=2x−96?

The y-intercept

What is the y-intercept of the line with slope m=1 that passes through the point (19, 33)?

Minimario's diagonal

Let ABCD be a square with side length 2 and let X be a point outside the square such that XA=XB=2√. The length of the longest diagonal in pentagon AXBCD can be expressed as x√. What is the value of x?

Maximizing Rectangle Area

A rectangle has perimeter equal to 28. What is the maximum possible area of this rectangle?

Area of Inscribed Isosceles Triangle

ABC is an isosceles triangle with AB=BC. The circumcircle of ABC has radius 8. Given that AC is a diameter of the circumcircle, what is the area of triangle ABC?

Details and assumptions

The circumcircle of a triangle ABC passes through A,B,C.

Angles in a Acute Triangle

In acute triangle △ABC, ∠CAB=34∘ and E is the foot of the perpendicular from B to AC. A point H on line segment BE is chosen, and X is the foot of the perpendicular from H to AB. What is the measure (in degrees) of ∠EHX?