MathDB

2

Part of 1998 Romania National Olympiad

Problems(4)

n + k^2 is perfect square - 1998 Romania NMO VII p2

Source:

8/14/2024
Show that there is no positive integer nn such that n+k2n + k^2 is a perfect square for at least nn positive integer values of kk.
number theoryPerfect Square
2 polynomias of 1998 degree

Source: 1998 Romania NMO VIII p2

8/14/2024
Let P(x)=a1998X1998+a1997X1997+...+a1X+a0P(x) = a_{1998}X^{1998} + a_{1997}X^{1997} +...+a_1X + a_0 be a polynomial with real coefficients such that P(0)≠P(−1)P(0) \ne P (-1), and let a,ba, b be real numbers. Let Q(x)=b1998X1998+b1997X1997+...+b1X+b0Q(x) = b_{1998}X^{1998} + b_{1997}X^{1997} +...+b_1X + b_0 be the polynomial with real coefficients obtained by taking bk=aak+bb_k = aa_k + b ,∀k=0,1,2,...,1998\forall k = 0, 1,2,..., 1998. Show that if Q(0)=Q(−1)≠0Q(0) = Q (-1) \ne 0 , then the polynomial QQ has no real roots.
algebrapolynomial
Inequality in a cyclic quadrilateral

Source:

11/20/2017
Let ABCDABCD be a cyclic quadrilateral. Show that ∣AC‾−BD‾∣≤∣AB‾−CD‾∣\vert \overline{AC} - \overline{BD} \vert \le \vert \overline{AB}-\overline{CD} \vert and determine when does equality hold.
geometrycyclic quadrilateralinequalities
| z | < =a if | z + a | <= a and |z^2 + a | <= a

Source: 1998 Romania NMO X p2

8/14/2024
Let a≥1a \ge1 be a real number and zz be a complex number such that ∣z+a∣≤a| z + a | \le a and ∣z2+a∣≤a|z^2+ a | \le a. Show that ∣z∣≤a| z | \le a.
complex numbersinequalitiesalgebra