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03 Dec 2020, 08:05
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
27% (02:18) correct
73%
(02:17)
wrong
based on 52
sessions
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How many different triangles exist in which no two sides are of the same length, each side is of integral unit length and the perimeter of the triangle is less than 14 units?
A. 3
B. 4
C. 5
D. 6
E. 7
A. 3
B. 4
C. 5
D. 6
E. 7
03 Dec 2020, 08:44
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Bunuel
The minimum value of the LARGEST side => a+a+1>a+2....a>1 ( Sum of the smaller two sides is greater than the third side)
So largest side = a+2=2+2=4, and minimum perimeter = 2+3+4=9
The perimeter can be 13 max.
In this triangle of perimeter 13, the largest side has to be <13/2 or \(\leq{6}\).
Possibilities
If the largest side is 6
6, 4, 3
6, 5, 2
6, 6, 1......Eliminate as sides cannot be equal
If the largest side is 5
5, 5, 3......Eliminate as sides cannot be equal
5, 4, 4......Eliminate as sides cannot be equal
If the perimeter is 12, the largest side has to be <12/2 or \(<{6}\).
If the largest side is 5
5, 5, 2......Eliminate as sides cannot be equal
5, 4, 3
If the largest side is 4
4, 4, 4......Eliminate as sides cannot be equal
If the perimeter is 11, the largest side has to be <11/2 or \(\leq{5}\).
If the largest side is 5
5, 5, 1......Eliminate as sides cannot be equal
5, 4, 2
5, 3, 3...Eliminate
If the largest side is 4
4, 4, 3......Eliminate as sides cannot be equal
If the perimeter is 10, the largest side has to be <10/2 or \(<{5}\).
If the largest side is 4
4, 4, 2......Eliminate as sides cannot be equal
4, 3, 3.....Eliminate
If the perimeter is 9, the largest side has to be <9/2 or \(\leq{4}\).
If the largest side is 4
4, 3, 2......
No more possibilities as the largest side will become lesser than 4.
Total colored cases = 5
C
03 Dec 2020, 10:09
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Bunuel
Let the sides be a, b, c
Thus: a + b + c < 14;
None of a, b, c are equal to each other
Also, sum of the lengths of any 2 sides must exceed the length of the 3rd side; i.e. max length of a side must be less than half the perimeter
Case 1: a + b + c = 13 =>
# max length = 6: other 2 sides = 3, 4 or 2, 5 ---> 2 triangles
# max length = 5: other 2 sides = 4, 4 (not allowed) or 3, 5 (also not allowed)
# max length cannot be 4 or less since it is less than the mean 13/3 = 4.33
Case 2: a + b + c = 12 =>
# max length = 5: other 2 sides = 3, 4 ---> 1 triangle
# max length cannot be 4 or less since 4 is the mean 12/3 = 4
Case 3: a + b + c = 11 =>
# max length = 5: other 2 sides = 2, 4 ---> 1 triangle
# max length = 4: other 2 sides = 3, 4 (not allowed)
# max length cannot be 3 or less since it is less than the mean 11/3 = 3.67
Case 4: a + b + c = 10 =>
# max length = 4: other 2 sides = 3, 3 (not allowed) or 2, 4 (also not allowed)
# max length cannot be 3 or less since it is less than the mean 10/3 = 3.33
Case 5: a + b + c = 9 =>
# max length = 4: other 2 sides = 3, 2 ---> 1 triangle
# max length cannot be 3 or less since it is less than the mean 9/3 = 3
Case 6: a + b + c = 8 or lower =>
The minimum sum of 3 unequal numbers that can form a triangle = 2 + 3 + 4 = 9
(Think why 1 + 2 + 3 = 6 does not work)
Thus, none of these cases would work
Total triangles = 2 + 1 + 1 + 1 = 5
Answer C
