Math Is Fun Forum

1.Positive integers m and n satisfy

Find

2.Find the positive integer n such that

3.A trapezoid has side lengths

and

in some order. Find its area

4.A rectangle with integer side lengths has the property that its area minus 5 times its perimeter equals

Find the minimum possible perimeter of this rectangle.

5.For real numbers a, b, and c, the roots of the polynomial

form an
arithmetic progression. Find

A bullet with a mass of 20 grams is horizontally fired at a body with a mass of 100 grams, suspended by a string with a length of 6 meters, . The bullet comes to rest in the body. Find the vertical distance that the combined system of the body and the bullet moves after the collision

Olivia cut a rectangle into exactly nine squares. On inspection, he observed that the area of one square was 64 cm²
, the areas of two other squares were 16 cm²
, and the rest of them were 4 cm²
each.
What was the perimeter of the original rectangl

The opposite sides of a convex hexagon

are parallel, the distances separating the parallel pairs of sides are equal,
and the angles at vertices A and D are right angles. Prove that the diagonals BE
and CF enclose an angle of

Find, without calculus, the largest possible value of

(sin 5x + cos 3x)/(sin 4x + cos 4x)

Prove that if two right-angled triangles

have the same perimeter and the same area,
then they are congruent.

find x

In an acute-angled triangle ABC, a tangent is drawn to the inscribed circle, parallel to side BC. The tangent intersects side AC at point D. F is the orthogonal projection of point D onto the side BC. Show that AB=AD+BF

hard