Lots of Fun Problems to Prove!!!

Many from "2000 feladat az elemi matematika korebol" by Sandor Roka

Number Theory Examples.

369. [2] Prove the following divisiblity statements:

•  9| 1033+8
• 6 | 1010+14
• 72 | 1020+ 8

373. [2] Prove that the following numbers are all composite.

• 106-57
• 10100-7
• 420-1
• 1000...001    (with 1991   0's)
• 111...1111     (with 1989 1's)
• 111...1111      (with 1990 1's)
• 111...121...111  (with the same number of 1's before as after the 2)
• 1!+2!+3!+...+100!
374. [3] Prove that 347,777,743 is a composite number.

375. [3] Prove that 49+610+320 is a composite number.

379. [4] Prove that 989*1001*1007 + 320 is a composite number.

382. [4] Prove that for any positive integer n, 19*8n + 17 is a composite number.

(There are typo's in the first two-- they should be > n and > sqrt(n) ) I'll Change it asap.)

Geometric Proofs

In each of the following problems prove that the area of the blue region is equal to the area of the red.

1284. [2]  Connect any point in the interior of a parallelagram with the four corners.

1285. [2] Connect the midpoints of two opposite sides of a convex quadrilateral to the corners as shown.

1288. [2]  In a Hexagon, with opposite sides parallel and of equal length, draw a triangle connecting alternating vertices.

1290. [2]  Divide a  regular star as shown.