S:E15313: Difference between revisions

Revision as of 03:14, 11 December 2024

E:15313 (Cristina Vijdeluc și Mihai Vijdeluc)

Determinați cifra 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} pentru care fracția ${\frac {a^{2}\cdot {\overline {aa}}+a^{2}(a^{2}+1)}{\overline {202a}}}$ este echiunitară.

Soluție:

Fracția este echiunitară dacă $a^{2}\cdot {\overline {aa}}+a^{2}(a^{2}+1)={\overline {202a}}$ 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^4 + 11a^3 + a^2 = \overline{202a}} . 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 2020 \leq \overline{202a} \leq 2029 } . Dacă 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 \leq 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 a^4 + 11a^3 + a^2 \leq 976 < 2020 } , iar dacă 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 \geq 6} , 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 a^4 + 11a^3 + a^2 \geq 3708 > 2029 } . Pentru 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 = 5 } , 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 5^4 + 11 \cdot 5^3 + 5^2 = 2025 } .

@@ Line 1: / Line 1: @@
-'''S:E15313 (Cristina Vijdeluc și Mihai Vijdeluc, Baia Mare)'''
+'''E:15313 (Cristina Vijdeluc și Mihai Vijdeluc)'''
 ''Determinați cifra <math>a</math> pentru care fracția <math>\frac{a^2 \cdot \overline{aa} + a^2(a^2+1)}{\overline{202a}}</math> este echiunitară.''
@@ Line 5: / Line 5: @@
 '''Soluție:'''
-''Fracția este echiunitară dacă <math>a^2 \cdot \overline{aa} + a^2(a^2+1) = \overline{202a}</math> sau <math>a^4 + 11a^3 + a^2 = \overline{202a}</math>. Avem <math> 2020 \leq \overline{202a} \leq 2029 </math>. Dacă <math> a \leq 4 </math>, atunci <math> a^4 + 11a^3 + a^2 \leq 976 < 2020 </math>, iar dacă <math>a \geq 6</math>, atunci <math>a^4 + 11a^3 + a^2 \geq 3708 > 2029 </math>. Pentru <math>a = 5 </math>, avem <math>5^4 + 11 \cdot 5^3 + 5^2 = 2025 </math>. ''
+Fracția este echiunitară dacă <math>a^2 \cdot \overline{aa} + a^2(a^2+1) = \overline{202a}</math> sau <math>a^4 + 11a^3 + a^2 = \overline{202a}</math>. Avem <math> 2020 \leq \overline{202a} \leq 2029 </math>. Dacă <math> a \leq 4 </math>, atunci <math> a^4 + 11a^3 + a^2 \leq 976 < 2020 </math>, iar dacă <math>a \geq 6</math>, atunci <math>a^4 + 11a^3 + a^2 \geq 3708 > 2029 </math>. Pentru <math>a = 5 </math>, avem <math>5^4 + 11 \cdot 5^3 + 5^2 = 2025 </math>.