MathDB

4

Part of 1997 Romania National Olympiad

Problems(4)

parallelogram if it has 2 // sides and PM/AP=1/4 - 1997 Romania NMO VII p4

Source:

8/13/2024
The quadrilateral ABCDABCD has two parallel sides. Let MM and NN be the midpoints of [DC][DC] and [BC][BC], and PP the common point of the lines AMAM and DNDN. If PMAP=14\frac{PM}{AP}=\frac{1}{4}, prove that ABCDABCD is a parallelogram.
geometryparallelogram
distance of projection of a point on a plane from another plane

Source: 1997 Romania NMO VIII p4

8/13/2024
Let SS be a point outside of the plane of the parallelogram ABCDABCD, such that the triangles SABSAB, SBCSBC, SCDSCD and SADSAD are equivalent.
a) Prove that ABCDABCD is a rhombus.
b) If the distance from SS to the plane (A,B,C,D)(A, B, C, D) is 1212, BD=30BD = 30 and AC=40AC = 40, compute the distance from the projection of the point SS on the plane (A,B,C,D)(A, B, C, D) to the plane (S,B,C)(S,B,C) .
geometry3D geometrydistance
Hard problem (i think)

Source: Romanian MO

3/8/2006
Let two bijective and continuous functionsf,g:R→Rf,g: \mathbb{R}\to\mathbb{R} such that : (f∘g−1)(x)+(g∘f−1)(x)=2x\left(f\circ g^{-1}\right)(x)+\left(g\circ f^{-1}\right)(x)=2x for any real xx. Show that If we have a value x0∈Rx_{0}\in\mathbb{R} such that f(x0)=g(x0)f(x_{0})=g(x_{0}), then f=gf=g.
functioninductionreal analysisreal analysis unsolved
sequence of integral

Source: Romania 1997

8/27/2005
Suppose that (fn)n∈N(f_n)_{n\in N} be the sequence from all functions fn:[0,1]→R+f_n:[0,1]\rightarrow \mathbb{R^+} s.t. f0f_0 be the continuous function and ∀x∈[0,1],∀n∈N,fn+1(x)=∫0x11+fn(t)dt\forall x\in [0,1] , \forall n\in \mathbb {N} , f_{n+1}(x)=\int_0^x \frac {1}{1+f_n (t)}dt. Prove that for every x∈[0,1]x\in [0,1] the sequence of (fn(x))n∈N(f_n(x))_{n\in N} be the convergent sequence and calculate the limitation.
calculusintegrationfunctionreal analysisreal analysis unsolved