P:1800: Difference between revisions

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Created page with "'''P:1800 (Ioan Ovidiu Pop, Coroieni)''' ''Aflați numărul de telefon <math>\overline{07abcdefgh}</math>, format din zece cifre, nu neapărat distincte, pentru care numerele <math>a+c</math>, <math>b+c</math>, <math>d+e</math>, <math>c+d</math>, <math>a+b+e</math>, <math>c+d+f</math>, <math>b+c+g</math> și <math>d+e+h</math> sunt opt numere consecutive așezate în ordine crescătoare.'' '''Soluție''' Cele opt numere consecutive așezate în ordine crescătoare..."
 
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'''[[P:1800]] (Ioan Ovidiu Pop, Coroieni)'''
'''[[P:1800]] (Ioan Ovidiu Pop, Coroieni)'''


''Aflați numărul de telefon <math>\overline{07abcdefgh}</math>, format din zece cifre, nu neapărat distincte, pentru care numerele <math>a+c</math>, <math>b+c</math>, <math>d+e</math>, <math>c+d</math>, <math>a+b+e</math>, <math>c+d+f</math>, <math>b+c+g</math> și <math>d+e+h</math> sunt opt numere consecutive așezate în ordine crescătoare.''
''Aflați numărul de telefon <math>\overline{07abcdefgh}</math>, format din zece cifre, nu neapărat distincte, pentru care numerele <math>a+c</math>, <math>b+c</math>, <math>d+e</math>, <math>c+d</math>, <math>a+b+e</math>, <math>c+d+f</math>, <math>b+c+g</math> și <math>d+e+h</math> sunt opt numere consecutive, așezate în ordine crescătoare.''


'''Soluție'''
'''Soluție'''
Cele opt numere consecutive așezate în ordine crescătoare sunt
Cele opt numere consecutive așezate în ordine crescătoare sunt


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Din <math>d+e+2 = a+b+e</math> rezultă <math>d+2=a+b</math>, deci <math>a+3+2=a+a+1</math>, ceea ce implică <math>a=4</math>. Atunci <math>b=5</math> și <math>d=7</math>.
Din <math>d+e+2 = a+b+e</math> rezultă <math>d+2=a+b</math>, deci <math>a+3+2=a+a+1</math>, ceea ce implică <math>a=4</math>. Atunci <math>b=5</math> și <math>d=7</math>.


Din <math>d+e+1 = c+d</math> rezultă <math>c=e+1</math>.
Din <math>d+e+1 = c+d</math> rezultă <math>c=e+1</math>, deci numerele de telefon
 
<math>0745170255, 0745271255, 0745372255, 0745473255, 0745575255, 0745675255, 0745776255, 0745877255, 0745978255</math>
 
sunt soluții acceptate.


'''Verificare'''


Răspuns: <math>\overline{07abcdefgh} = 0745877255</math>.
Pentru soluțiile obținute se verifică faptul că numerele ''<math>a+c</math>, <math>b+c</math>, <math>d+e</math>, <math>c+d</math>, <math>a+b+e</math>, <math>c+d+f</math>, <math>b+c+g</math>'' și ''<math>d+e+h</math>'' sunt opt numere consecutive.
{| class="wikitable"
|+
!
!a+c
!b+c
!d+e
!c+d
!a+b+e
!c+d+f
!b+c+g
!d+e+h
|-
|c=1
e=0
|5
|6
|7
|8
|9
|10
|11
|12
|-
|c=2
e=1
|6
|7
|8
|9
|10
|11
|12
|13
|-
|c=3
e=2
|7
|8
|9
|10
|11
|12
|13
|14
|-
|c=4
e=3
|8
|9
|10
|11
|12
|13
|14
|15
|-
|c=5
e=4
|9
|10
|11
|12
|13
|14
|15
|16
|-
|c=6
e=5
|10
|11
|12
|13
|14
|15
|16
|17
|-
|c=7
e=6
|11
|12
|13
|14
|15
|16
|17
|18
|-
|c=8
e=7
|12
|13
|14
|15
|16
|17
|18
|19
|-
|c=9
e=8
|13
|14
|15
|16
|17
|18
|19
|20
|}

Latest revision as of 17:14, 19 September 2025

P:1800 (Ioan Ovidiu Pop, Coroieni)

Aflați numărul de telefon Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{07abcdefgh}} , format din zece cifre, nu neapărat distincte, pentru care numerele Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a+c} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b+c} , , , , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c+d+f} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b+c+g} și Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d+e+h} sunt opt numere consecutive, așezate în ordine crescătoare.

Soluție

Cele opt numere consecutive așezate în ordine crescătoare sunt

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a+c < b+c < d+e < c+d < a+b+e < c+d+f < b+c+g < d+e+h. }


Din Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c+d+2 = c+d+f} rezultă Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f=2} .

Din Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b+c+5 = b+c+g} rezultă Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g=5} .

Din Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d+e+5 = d+e+h} rezultă Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h=5} .

Din Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a+c+1 = b+c} rezultă Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=a+1} .

Din Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a+c+3 = c+d} rezultă Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d=a+3} .

Din Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d+e+2 = a+b+e} rezultă Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d+2=a+b} , deci Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a+3+2=a+a+1} , ceea ce implică Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a=4} . Atunci Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=5} și Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d=7} .

Din Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d+e+1 = c+d} rezultă Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c=e+1} , deci numerele de telefon

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0745170255, 0745271255, 0745372255, 0745473255, 0745575255, 0745675255, 0745776255, 0745877255, 0745978255}

sunt soluții acceptate.

Verificare

Pentru soluțiile obținute se verifică faptul că numerele Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a+c} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b+c} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d+e} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c+d} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a+b+e} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c+d+f} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b+c+g} și Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d+e+h} sunt opt numere consecutive.

a+c b+c d+e c+d a+b+e c+d+f b+c+g d+e+h
c=1

e=0

5 6 7 8 9 10 11 12
c=2

e=1

6 7 8 9 10 11 12 13
c=3

e=2

7 8 9 10 11 12 13 14
c=4

e=3

8 9 10 11 12 13 14 15
c=5

e=4

9 10 11 12 13 14 15 16
c=6

e=5

10 11 12 13 14 15 16 17
c=7

e=6

11 12 13 14 15 16 17 18
c=8

e=7

12 13 14 15 16 17 18 19
c=9

e=8

13 14 15 16 17 18 19 20
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