Problem B. 4574. (November 2013)

B. 4574. Solve the simultaneous equations

x+y+z&=1,

ax+by+cz=d,

a²x+b²y+c²z=d²,

where a, b, c, d are distinct real parameters.

Matlap, Kolozsvár, 2013

(4 pont)

Deadline expired on December 10, 2013.

Sorry, the solution is available only in Hungarian. Google translation

Végeredmény: Az egyenletrendszer akkor oldható meg egyértelműen, ha az a,b,c paraméterek különbözőek. A megoldás:

$x = \frac{(d-b)(d-c)}{(a-b)(a-c)}, \quad y = \frac{(d-a)(d-c)}{(b-a)(b-c)}, \quad z = \frac{(d-a)(d-b)}{(c-a)(c-b)}.$

Megjegyzés. A megoldást a Cramer-szabállyal felírva, mindhárom ismeretlen két. úgynevezett Vandermonde-determináns hányadosa lesz. pl,

$x = \frac{ \left|\matrix{1&1&1\cr d&b&c\cr d^2&b^2&c^2}\right|}{ \left|\matrix{1&1&1\cr a&b&c\cr a^2&b^2&c^2}\right|} = \frac{(b-d)(c-d)(c-b)}{(b-a)(c-a)(c-b)}.$

Statistics:

121 students sent a solution.

4 points: 94 students.

3 points: 12 students.

2 points: 3 students.

1 point: 4 students.

0 point: 6 students.

Unfair, not evaluated: 2 solutionss.

Problems in Mathematics of KöMaL, November 2013

121 students sent a solution.
4 points:	94 students.
3 points:	12 students.
2 points:	3 students.
1 point:	4 students.
0 point:	6 students.
Unfair, not evaluated:	2 solutionss.

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