Z7-1-3 MO 2018

Grandfather prepared a pile of hazelnuts for his six grandchildren with the instruction that they should divide them somehow. First came Adam, who took half for himself, took one more nut, and left. The same was done by the second Bob, the third Cyril, the fourth Dano, and also the fifth Edo. Only Ferko sadly looked at the empty table; no nut was left for him. How many nuts were originally on the pile?

Final Answer:

n =  62

Step-by-step explanation:


E = (n-A-B-C-D)/2+1
D = (n-A-B-C)/2+1
C = (n-A-B)/2+1
B = (n-A)/2+1
A = n/2+1
n = A+B+C+D+E

e = (n-A-B-C-D)/2+1
D = (n-A-B-C)/2+1
C = (n-A-B)/2+1
B = (n-A)/2+1
A = n/2+1
n = A+B+C+D+e

A+B+C+D+2e-n = 2
A+B+C+2D-n = 2
A+B+2C-n = 2
A+2B-n = 2
2A-n = 2
A+B+C+D+e-n = 0

Pivot: Row 1 ↔ Row 5
2A-n = 2
A+B+C+2D-n = 2
A+B+2C-n = 2
A+2B-n = 2
A+B+C+D+2e-n = 2
A+B+C+D+e-n = 0

Row 2 - 1/2 · Row 1 → Row 2
2A-n = 2
B+C+2D-0.5n = 1
A+B+2C-n = 2
A+2B-n = 2
A+B+C+D+2e-n = 2
A+B+C+D+e-n = 0

Row 3 - 1/2 · Row 1 → Row 3
2A-n = 2
B+C+2D-0.5n = 1
B+2C-0.5n = 1
A+2B-n = 2
A+B+C+D+2e-n = 2
A+B+C+D+e-n = 0

Row 4 - 1/2 · Row 1 → Row 4
2A-n = 2
B+C+2D-0.5n = 1
B+2C-0.5n = 1
2B-0.5n = 1
A+B+C+D+2e-n = 2
A+B+C+D+e-n = 0

Row 5 - 1/2 · Row 1 → Row 5
2A-n = 2
B+C+2D-0.5n = 1
B+2C-0.5n = 1
2B-0.5n = 1
B+C+D+2e-0.5n = 1
A+B+C+D+e-n = 0

Row 6 - 1/2 · Row 1 → Row 6
2A-n = 2
B+C+2D-0.5n = 1
B+2C-0.5n = 1
2B-0.5n = 1
B+C+D+2e-0.5n = 1
B+C+D+e-0.5n = -1

Pivot: Row 2 ↔ Row 4
2A-n = 2
2B-0.5n = 1
B+2C-0.5n = 1
B+C+2D-0.5n = 1
B+C+D+2e-0.5n = 1
B+C+D+e-0.5n = -1

Row 3 - 1/2 · Row 2 → Row 3
2A-n = 2
2B-0.5n = 1
2C-0.25n = 0.5
B+C+2D-0.5n = 1
B+C+D+2e-0.5n = 1
B+C+D+e-0.5n = -1

Row 4 - 1/2 · Row 2 → Row 4
2A-n = 2
2B-0.5n = 1
2C-0.25n = 0.5
C+2D-0.25n = 0.5
B+C+D+2e-0.5n = 1
B+C+D+e-0.5n = -1

Row 5 - 1/2 · Row 2 → Row 5
2A-n = 2
2B-0.5n = 1
2C-0.25n = 0.5
C+2D-0.25n = 0.5
C+D+2e-0.25n = 0.5
B+C+D+e-0.5n = -1

Row 6 - 1/2 · Row 2 → Row 6
2A-n = 2
2B-0.5n = 1
2C-0.25n = 0.5
C+2D-0.25n = 0.5
C+D+2e-0.25n = 0.5
C+D+e-0.25n = -1.5

Row 4 - 1/2 · Row 3 → Row 4
2A-n = 2
2B-0.5n = 1
2C-0.25n = 0.5
2D-0.125n = 0.25
C+D+2e-0.25n = 0.5
C+D+e-0.25n = -1.5

Row 5 - 1/2 · Row 3 → Row 5
2A-n = 2
2B-0.5n = 1
2C-0.25n = 0.5
2D-0.125n = 0.25
D+2e-0.125n = 0.25
C+D+e-0.25n = -1.5

Row 6 - 1/2 · Row 3 → Row 6
2A-n = 2
2B-0.5n = 1
2C-0.25n = 0.5
2D-0.125n = 0.25
D+2e-0.125n = 0.25
D+e-0.125n = -1.75

Row 5 - 1/2 · Row 4 → Row 5
2A-n = 2
2B-0.5n = 1
2C-0.25n = 0.5
2D-0.125n = 0.25
2e-0.0625n = 0.125
D+e-0.125n = -1.75

Row 6 - 1/2 · Row 4 → Row 6
2A-n = 2
2B-0.5n = 1
2C-0.25n = 0.5
2D-0.125n = 0.25
2e-0.0625n = 0.125
e-0.0625n = -1.875

Row 6 - 1/2 · Row 5 → Row 6
2A-n = 2
2B-0.5n = 1
2C-0.25n = 0.5
2D-0.125n = 0.25
2e-0.0625n = 0.125
-0.03125n = -1.9375


n = -1.9375/-0.03125 = 62
e = 0.125+0.0625n/2 = 0.125+0.0625 · 62/2 = 2
D = 0.25+0.125n/2 = 0.25+0.125 · 62/2 = 4
C = 0.5+0.25n/2 = 0.5+0.25 · 62/2 = 8
B = 1+0.5n/2 = 1+0.5 · 62/2 = 16
A = 2+n/2 = 2+62/2 = 32

A = 32
B = 16
C = 8
D = 4
e = 2
n = 62

Our linear equations calculator calculates it.

Try another example


Help us improve! If you spot a mistake, please let let us know. Thank you!







Tips for related online calculators
Need help calculating a sum, simplifying, or multiplying fractions? Try our fraction calculator.
Do you have a linear equation or system of equations and are looking for a solution? Or do you have a quadratic equation?

You need to know the following knowledge to solve this word math problem:

algebrabasic operations and conceptsnumbersthemes, topicsGrade of the word problem

Related math problems and questions: