MathDB

1

Part of 1998 Romania National Olympiad

Problems(4)

f(f(1)))= f(f(2))= f(f(3)), f(x) = ax^2 +bx + c

Source: 1998 Romania NMO IX p1

8/14/2024
Find the integer numbers a,b,ca, b, c such that the function f:R→Rf: R \to R, f(x)=ax2+bx+cf(x) = ax^2 +bx + c satisfies the equalities : f(f(1)))=f(f(2))=f(f(3))f(f(1) ))= f (f(2 ) )= f(f (3 ))
algebraquadraticstrinomial
sum x_i^2 +n^3<=(2n-1) sum x_i +n^2 - 1998 Romania NMO VII p1

Source:

8/14/2024
Let nn be a positive integer and x1,x2,...,xnx_1,x_2,...,x_n be integer numbers such that x12+x22+...+xn2+n3≤(2n−1)(x1+x2+...+xn)+n2x_1^2+x_2^2+...+x_n^2+ n^3 \le (2n - 1)(x_1+x_2+...+x_n ) + n^2 . Show that : a) x1,x2,...,xnx_1,x_2,...,x_n are non-negative integers b) the number x1+x2+...+xn+n+1x_1+x_2+...+x_n+n+1 is not a perfect square.
inequalitiesalgebraPerfect Squarenumber theory
x + y = a, x^3 + y^3 >= a

Source: 1998 Romania NMO VIII p1

8/14/2024
Let aa be a real number and A={(x,y)∈R×R∣ x+y=a}A = \{(x, y) \in R \times R | \, x + y = a\}, B={(x,y)∈R×R∣ x3+y3<a}B = \{(x,y) \in R \times R | \, x^3 + y^3 < a\} . Find all values of aa such that A∩B=∅A \cap B = \emptyset .
algebrasystem of equationsinequalities
oh!! Integral

Source: Romania 1998

8/23/2005
Suppose that a,b∈R+a,b\in\mathbb{R}^+ which a+b<1a+b<1 and f:[0,+∞)→[0,+∞)f:[0,+\infty) \rightarrow [0,+\infty) be the increasing function s.t. ∀x≥0,∫0xf(t)dt=∫0axf(t)dt+∫0bxf(t)dt\forall x\geq 0 ,\int _0^x f(t)dt=\int _0^{ax} f(t)dt+\int _0^{bx} f(t)dt. Prove that ∀x≥0,f(x)=0\forall x\geq 0 , f(x)=0
calculusintegrationfunctionreal analysisreal analysis unsolved