1. There are 25 unit squares on a 5 × 5 chessboard. Every symmetric square is painted in the same color, as shown below. The colors are all different for squares having different symmetry. How many colors will be needed for n × n chessboard, where n is odd?

^{92}is divided by 92?

3. The number of divisors of every natural number from 1 to 1000 is calculated. Which natural number has the highest number of divisors?

4. Twenty cubical blocks are arranged as shown. First, 10 are arranged in a triangular pattern; then a layer of 6, arranged in a triangular pattern, is centered on the 10; then a layer of 3, arranged in a triangular pattern, is centered on the 6; and finally one block is centered on top of the third layer. The blocks in the bottom layer are numbered 1 through 10 in some order. Each block in layers 2, 3 and 4 is assigned the number which is the sum of the numbers assigned to the three blocks on which it rests. Find the smallest possible number which could be assigned to the top block.

5. Let the set A = {512, 513, 514, … 1023}. What is the probability that these numbers, when written in base 2, have more 0’s than 1’s?

6. Grihasthi Lal was married just a day ago, but due to some official work he has to leave his wife and go to country X. Grihasthi Lal promises his wife that he will be back before the day is over. He leaves for country X on the plane at 9:00am and arrives in country X on the same day at 1:00pm, local time. Two hour later, his plane leaves country X for his own country, and he finally arrives back in his country at 9:00pm, local time. The flight takes the same time in each direction. What time is it in country X when Grihasthi Lal reaches back at 9:00pm?

7. Raghav, the drunkard, is marooned on an island, left with a full one-liter bottle of ‘Old Monk’ rum. To make the supply of rum last longer, he drinks 10ml of rum on the first day, and fills the bottle with water. The next day, he drinks 20ml of the contents of the bottle and again fills the bottle with water. The next day he drinks 30ml and repeats the same process, and so on, till the bottle is empty. How much water did Raghav drink?

8. Let N be a number such that N is divisible by every natural number less than the cube root of N. What is highest possible value of N?

9. A number sequence has 100 elements. Any of its elements (except for the first and last element) is equal to the product of its neighbors. The product of the first 50 elements, just as the product of all the elements is 27. What is the sum of the first and the second element?

10. In the adjoining figure, a wooden cube has edges of length 3 meters. Square holes of side one meter, centered in each face, are cut through to the opposite face. The edges of the whole are parallel to the edges of the cube. The entire surface area including the inside, in square meters, is

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