MathDB

2

Part of 1996 Romania National Olympiad

Problems(5)

\sqrt{\frac{x-7}{1990}}+ ..=\sqrt{\frac{x-1989}{7}}+....1996 Romania NMO VII p2

Source:

8/13/2024
Find all real numbers xx for which the following equality holds :
x−71989+x−61990+x−51991=x−19897+x−19906+x−19915\sqrt{\frac{x-7}{1989}}+\sqrt{\frac{x-6}{1990}}+\sqrt{\frac{x-5}{1991}}=\sqrt{\frac{x-1989}{7}}+\sqrt{\frac{x-1990}{6}}+\sqrt{\frac{x-1991}{5}}
algebraRadicals
cute and easy

Source: Romania 1996

8/31/2005
Find all polynomials pn(x)=anxn+an−1xn−1+...+a1x+a0p_n(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0 (n≥2n\geq 2) with real and non-zero coeficients s.t. pn(x)−p1(x)p2(x)...pn−1(x)p_n(x)-p_1(x)p_2(x)...p_{n-1}(x) be a constant polynomial. ;)
algebrapolynomialalgebra proposed
Inequality

Source: Romania 1996

1/27/2010
a,b,c,d∈[0,1] a,b,c,d \in [0,1] and x,y,z,t∈[0,12] x,y,z,t \in [0, \frac{1}{2}] and a+b+c+d=x+y+z+t=1 a+b+c+d=x+y+z+t=1.prove that: (i) (i) ax+by+cz+dt ax+by+cz+dt ≥ \geq min(a+b2,b+c2,c+d2,d+a2,a+c2,b+d2) min( {\frac{a+b}{2} , \frac{b+c}{2} , \frac{c+d}{2} , \frac{d+a}{2} , \frac{a+c}{2} , \frac{b+d}{2} )} (ii) (ii) ax+by+cz+dt ax+by+cz+dt ≥ \geq 54abcd 54abcd
inequalitiesinequalities unsolved
another hard problem

Source: Romania National Olympiad 1996

8/29/2005
Suppose that f:[a,b]→R f: [a,b]\rightarrow \mathbb{R} be a monotonic function and for every x1,x2∈[a,b] x_1,x_2\in [a,b] that x1<x2 x_1<x_2 ,there exist c∈(a,b) c\in (a,b) such that ∫x1x2f(x)dx=f(c)(x1−x2) \int _{x_1}^{x_2}f(x)dx=f(c)(x_1-x_2) a) Show that f f be the continuous function on interval (a,b) (a,b) b) Suppose that f f is integrable function on interval [a,b] [a,b] but f f isn't a monotonic function then ,is it the result of part a) right?
functionintegrationreal analysisreal analysis unsolved
fixed sum MM_1 + MM_2 + MM_3 in tetrahedron

Source: 1996 Romania NMO X p2

8/13/2024
Let ABCDABCD a tetrahedron and MM a variable point on the face BCDBCD. The line perpendicular to (BCD)(BCD) in MM . intersects the planes(ABC) (ABC), (ACD)(ACD), and (ADB)(ADB) in M1M_1, M2M_2, and M3M_3. Show that the sum MM1+MM2+MM3MM_1 + MM_2 + MM_3 is constant if and only if the perpendicular dropped from AA to (BCD)(BCD) passes through the centroid of triangle BCDBCD.
geometry3D geometryfixedtetrahedron