E:16382: Difference between revisions

Revision as of 12:23, 16 January 2024

E:16382 (Cristina Vijdeluc și Mihai Vijdeluc)

Afișați numerele întregi pozitive 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{abcd}} cu proprietateaFailed 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^7 + e^d + ad = \overline{a000}.} Soluție Deoarece 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 4^7 > 9999} 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 1^7 + a^b + a^c + a^d < 1000} , avem 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 = 2} sau 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} . Egalitatea din enunț corespunde scrierii numărului 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{a000}} în baza 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} .

Cum 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 2000 = 11111010000_2} , 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=2} nu convine.

De asemenea, 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 3000 = 10101001000_2} corespunde egalității din enunț 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 3000 = 3^7 + 3^5 + 3^4 + 3^2 \cdot 2^2.} Prin urmare, avem soluțiile 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 = 6} , 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 = 4} , 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 = 1} și permutările acestora.

Se obțin numerele întregi pozitive: 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 \{{3641}, {3614}, {3461}, {3461}, {3164}, {3146}\}.}

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 '''E:16382 (Cristina Vijdeluc și Mihai Vijdeluc)'''
-Afișați numerele întregi pozitive $\overline{abcd}$ cu proprietatea
+Afișați numerele întregi pozitive <math>\overline{abcd}</math> cu proprietatea<math display="block">a^7 + e^d + ad = \overline{a000}.</math>'''Soluție'''
+Deoarece <math>4^7 > 9999</math> și <math>1^7 + a^b + a^c + a^d < 1000</math>, avem <math>a = 2</math> sau <math>a = 3</math>. Egalitatea din enunț corespunde scrierii numărului <math>\overline{a000}</math> în baza <math>a</math>.
-\[ a^7 + e^d + ad = \overline{a000}. \]
+Cum <math>2000 = 11111010000_2</math>, <math>a=2</math> nu convine.
+De asemenea, <math>3000 = 10101001000_2</math> corespunde egalității din enunț <math display="block">3000 = 3^7 + 3^5  + 3^4 + 3^2 \cdot 2^2.</math> Prin urmare, avem soluțiile <math>b = 6</math>, <math>c = 4</math>, <math>d = 1</math> și permutările acestora.
+Se obțin numerele întregi pozitive: <math display="block"> \{{3641}, {3614}, {3461}, {3461}, {3164}, {3146}\}.</math>