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Prove or disprove: 100 consecutive positive integers can be placed around a circle so that the product of any two adjacent numbers is a perfect square.?. How many different values are p[r]
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(2)AITMO PROBLEMS – Team Contest
1 Refer to the diagram Quadrilateral ABCD with diagonals BD and AC bounds the rhombus EFGH as shown in the figure If BD k
AC = ,
||
EF AC, and FG BD|| , find Area
Area
ABCD EFGH Solution:
Let =λ
AB AE
, then EH BD BD
EH BA
BE = −λ =λ =λ
, ,
1 Similarly, EF =(1−λ)AC , but EH=EF, so that λ⋅BD=(1−λ)⋅AC and
λλ − = = AC BD
k , then
kH
1
=
λ
Hence ( ) ( )
k k k k EH BD EF AC EH EF BD AC S S EFGH ABCD 1 1 1 1 1 2 sin sin 2 + = + + + = − = ⋅ = ⋅ ⋅ ⋅ ⋅ = λ λ ϑ ϑ
2 Prove or disprove: 100 consecutive positive integers can be placed around a circle so that the product of any two adjacent numbers is a perfect square
Solution:
3 How many triples (a, b, c) of positive integers are there such that a, b and c are primes and 2 a −b =c?
Solution:
The unique ordered triple is (3, 2, 5)
4 Find the smallest positive integer k such that !k ends with 500 zeros [Note: k!=k k( −1)(k−2 1) ( )( )( )]
Solution: k=2005
5 Let x = a + b – c, y = a + c – b and z = b + c – a, where a, b and c are prime numbers Given that x2 = y and ( z− y) is the square of a prime number, find the value of the product abc
Solution:
abc=3*23*29=2001
6 The real numbers x1 , x2 , x3 , x4 , x5 , x6 are arbitrarily chosen within the interval (0,1)
Prove that ( 2)( 3)( 4)( 5)( 6)( 1)
1 16
x −x x −x x −x x −x x −x x −x ≤
Solution:
7 The number 222+1 has exactly one prime factor greater than 1000 Find it
Solution:
(2 1) (2 )(2 ) 2113*1985 2113*397*5
1
222+ = 11+ 2− 12 = 11+ + 11+ − = =
(3)8 We can assign one of the integers 1, 2, 3, …, (with no repetitions) to each of the seven regions in the diagram so that numbers in adjacent regions (having a common edge) differ by or more
How many different values are possible for region g?
Solution:
7
9 Three motorists A, B, and C often travel on a certain highway, and each motorist always travels at a constant speed A is the fastest of the three and C is the slowest One day when the three travel in the same direction, B overtakes C Five minutes later, A overtakes C In another three minutes, A overtakes
B
On another occasion when they again travel in the same direction, A overtakes B first Nine minutes later, A overtakes C
When will B overtake C?
Solution:
15 minutes
10 Divide the diagram into pentominoes so that the sum of the digits within each part is 10 The pentomino shapes are shown below They can be rotated or reflected Each must be used exactly once in the problem
Solution:
3 2 1
1 2
3 2 2
2 1 2 1
4 2 1
2 1 2 1
Example
a b c
d e f