p1. Among the numbers 51 and 41 there are infinitely many fractional numbers. Find 999 decimal numbers between 51 and 41 so that the difference between the next fractional number with the previous fraction constant.
(i.e. If x1,x2,x3,x4,...,x999 is a fraction that meant, then x2−x1=x3−x3=...=xn−xn−1=...=x999−x998)
p2. The pattern in the image below is: "Next image obtained by adding an image of a black equilateral triangle connecting midpoints of the sides of each white triangle that is left in the previous image." The pattern is continuous to infinity.
https://cdn.artofproblemsolving.com/attachments/e/f/81a6b4d20607c7508169c00391541248b8f31e.png
It is known that the area of the triangle in Figure 1 is 1 unit area. Find the total area of the area formed by the black triangles in figure 5. Also find the total area of the area formed by the black triangles in the 20th figure.
p3. For each pair of natural numbers a and b, we define a∗b=ab+a−b. The natural number x is said to be the constituent of the natural number n if there is a natural number y that satisfies x∗y=n. For example, 2 is a constituent of 6 because there is a natural number 4 so that 2∗4=2⋅4+2−4=8+2−4=6. Find all constituent of 2005.
p4. Three people want to eat at a restaurant. To find who pays them to make a game. Each tossing one coin at a time. If the result is all heads or all tails, then they toss again. If not, then "odd person" (i.e. the person whose coin appears different from the two other's coins) who pay. Determine the number of all possible outcomes, if the game ends in tossing:
a. First.
b. Second.
c. Third.
d. Tenth.
p5. Given the equation x2+3y2=n, where x and y are integers. If n<20 what number is n, and which is the respective pair (x,y) ? Show that it is impossible to solve x2+3y2=8 in integers. algebrageometrycombinatoricsnumber theoryindonesia juniors