MathDB

p1. For each unit square on a

5 \times 9

board write the number

1

0

. Then calculate the sum of all the numbers in each column and also in each row so are obtained

14

numbers. Suppose H is a set containing those numbers. Determine the maximum number of members of

H

.
p2. The natural number

k> 2

is said to be beautiful if for every natural number

n \ge 4

with

5n + 1

a perfect square number, can be found real numbers

a_1, a_2,..., a_k

such that

n+1= a_1^2+ a_2^2+...+a_k^2.

Find the smallest beautiful number
[url=https://artofproblemsolving.com/community/c6h2370986p19378672]p3. Given triangle

ABC

, the three altitudes intersect at point

H

. Determine all points

X

on the side

BC

so that the symmetric of

H

wrt point

X

lies on the circumcircle of triangle

ABC

.
[url=https://artofproblemsolving.com/community/c6h2685865p23301414]p4. Let

a

b

, and

c

be real numbers whose absolute value is not greater than

1

. Prove that

\sqrt{|a-b|}+\sqrt{|b-c|}+\sqrt{|c-a|}\le 2+\sqrt2

p5. On a

2017\times n

chessboard , Ani and Banu play a game. First player choose a square and then color it red. Next player selects a square from an area that has not been colored red and then color it in red. The selected square can be any size but must accurately cover a number of square units on a chessboard. Then the two players take turns doing the same thing. One player is said to have won, if the next player can no longer continue the game. If Ani gets the first turn, determine all values of

n\ge 2017

until Ani has a strategy to win the game.

Problem Statement