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34

Part of 2000 National Olympiad First Round

Problems(1)

Turkish NMO First Round - 2000 P-34 (Number Theory)

Source:

7/8/2012
Which statement is not true for at least one prime pp?
<spanclass=′latex−bold′>(A)</span> If x2+x+3≡0(modp) has a solution, then x2+x+25≡0(modp) has a solution.<spanclass=′latex−bold′>(B)</span> If x2+x+3≡0(modp) does not have a solution, thenx2+x+25≡0(modp) has no solution<spanclass=′latex−bold′>(C)</span> If x2+x+25≡0(modp) has a solution, thenx2+x+3≡0(modp) has a solution.<spanclass=′latex−bold′>(D)</span> If x2+x+25≡0(modp) does not have a solution, thenx2+x+3≡0(modp) has no solution. <spanclass=′latex−bold′>(E)</span> None <span class='latex-bold'>(A)</span>\ \text{If } x^2+x+3 \equiv 0 \pmod p \text{ has a solution, then } \\ \qquad x^2+x+25 \equiv 0 \pmod p \text{ has a solution.} \\ \\ <span class='latex-bold'>(B)</span>\ \text{If } x^2+x+3 \equiv 0 \pmod p \text{ does not have a solution, then} \\ \qquad x^2+x+25 \equiv 0 \pmod p \text{ has no solution} \\ \\ \qquad<span class='latex-bold'>(C)</span>\ \text{If } x^2+x+25 \equiv 0 \pmod p \text{ has a solution, then} \\ \qquad x^2+x+3 \equiv 0 \pmod p \text{ has a solution}. \\ \\ \qquad<span class='latex-bold'>(D)</span>\ \text{If } x^2+x+25 \equiv 0 \pmod p \text{ does not have a solution, then} \\ \qquad x^2+x+3 \equiv 0 \pmod p \text{ has no solution. } \\ \\ \qquad<span class='latex-bold'>(E)</span>\ \text{None}
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