Răspuns: [tex]\bf \overline{abcd}\in[/tex] 5230, 5320, 5140, 5410, 3142, 3412, 3502, 3052, 2143, 2413, 2503, 2053, 1234, 1324, 1504, 1054, 4231, 4321, 4051, 4501
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Salutare !
[tex]\bf~[/tex]
[tex]\bf Fie ~\overline{abcd}~numerele ~cautate[/tex]
[tex]\bf a,b,c,d - cifre[/tex]
[tex]\bf Cifrele~ sunt: 0,1,2,3,4,5,6,7,8,9[/tex]
[tex]\bf a\neq b\neq c\neq d[/tex]
[tex]\bf a\neq 0[/tex]
[tex]\bf a+d = b+c =5 \Rightarrow \red{a,b,c,d \in\big \{0,1,2,3,4,5\big\}}[/tex]
[tex]\bf Caz ~I)~~ \underline{a = 1} \Rightarrow 1 +d = 5\Rightarrow \underline{d = 4}[/tex]
[tex]\bf \underline{b = 2} \Rightarrow 2 +c = 5\Rightarrow \underline{c = 3}[/tex]
[tex]\bf \underline{b = 3} \Rightarrow 3 +c = 5\Rightarrow \underline{c = 2}[/tex]
[tex]\bf \underline{b = 5} \Rightarrow 5 +c = 5\Rightarrow \underline{c = 0}[/tex]
[tex]\bf \underline{b = 0} \Rightarrow 0 +c = 5\Rightarrow \underline{c = 5}[/tex]
[tex]\pink{\boxed{\bf \overline{abcd}\in \big\{1234,1324,1504,1054\big\}}}[/tex]
[tex]\bf~[/tex]
[tex]\bf Caz ~II)~~ \underline{a = 2} \Rightarrow 2 +d = 5\Rightarrow \underline{d = 3}[/tex]
[tex]\bf \underline{b = 1} \Rightarrow 1 +c = 5\Rightarrow \underline{c = 4}[/tex]
[tex]\bf \underline{b = 4} \Rightarrow 4 +c = 5\Rightarrow \underline{c = 1}[/tex]
[tex]\bf \underline{b = 5} \Rightarrow 5 +c = 5\Rightarrow \underline{c = 0}[/tex]
[tex]\bf \underline{b = 0} \Rightarrow 0 +c = 5\Rightarrow \underline{c = 5}[/tex]
[tex]\purple{\boxed{\bf \overline{abcd}\in \big\{2143,2413,2503,2053\big\}}}[/tex]
[tex]\bf ~[/tex]
[tex]\bf Caz ~III)~~ \underline{a = 3} \Rightarrow 3 +d = 5\Rightarrow \underline{d = 2}[/tex]
[tex]\bf \underline{b = 1} \Rightarrow 1 +c = 5\Rightarrow \underline{c = 4}[/tex]
[tex]\bf \underline{b = 4} \Rightarrow 4 +c = 5\Rightarrow \underline{c = 1}[/tex]
[tex]\bf \underline{b = 5} \Rightarrow 5 +c = 5\Rightarrow \underline{c = 0}[/tex]
[tex]\bf \underline{b = 0} \Rightarrow 0 +c = 5\Rightarrow \underline{c = 5}[/tex]
[tex]\blue{\boxed{\bf \overline{abcd}\in \big\{3142,3412,3502,3052\big\}}}[/tex]
[tex]\bf~[/tex]
[tex]\bf Caz ~IV)~~ \underline{a = 4} \Rightarrow 4 +d = 5\Rightarrow \underline{d = 1}[/tex]
[tex]\bf \underline{b = 2} \Rightarrow 2 +c = 5\Rightarrow \underline{c = 3}[/tex]
[tex]\bf \underline{b = 3} \Rightarrow 3 +c = 5\Rightarrow \underline{c = 2}[/tex]
[tex]\bf \underline{b = 5} \Rightarrow 5 +c = 5\Rightarrow \underline{c = 0}[/tex]
[tex]\bf \underline{b = 0} \Rightarrow 0 +c = 5\Rightarrow \underline{c = 5}[/tex]
[tex]\green{\boxed{\bf \overline{abcd}\in \big\{4231,4321,4051,4501\big\}}}[/tex]
[tex]\bf~[/tex]
[tex]\bf Caz ~V)~~ \underline{a = 5} \Rightarrow 5 +d = 5\Rightarrow \underline{d = 0}[/tex]
[tex]\bf \underline{b = 2} \Rightarrow 2 +c = 5\Rightarrow \underline{c = 3}[/tex]
[tex]\bf \underline{b = 3} \Rightarrow 3 +c = 5\Rightarrow \underline{c = 2}[/tex]
[tex]\bf \underline{b = 1} \Rightarrow 1 +c = 5\Rightarrow \underline{c = 4}[/tex]
[tex]\bf \underline{b = 4} \Rightarrow 4 +c = 5\Rightarrow \underline{c = 1}[/tex]
[tex]\red{\boxed{\bf \overline{abcd}\in \big\{5230,5320,5140,5410\big\}}}[/tex]
[tex]\bf~[/tex]
[tex]\bf Din~cazurile ~analizate~ nr.~ naturale~ce~ respecta~ conditiile~ problemei:[/tex]
[tex] \bf \overline{abcd}\in[/tex] 5230, 5320, 5140, 5410, 3142, 3412, 3502, 3052, 2143, 2413, 2503, 2053, 1234, 1324, 1504, 1054, 4231, 4321, 4051, 4501
==pav38==
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