MathDB

4

Part of 2006 District Olympiad

Problems(6)

Vertices of a cube are assigned numbers

Source: Romanian District Olympiad 2006, Grade 8, Problem 4

3/11/2006
a) Prove that we can assign one of the numbers 11 or −1-1 to the vertices of a cube such that the product of the numbers assigned to the vertices of any face is equal to −1-1. b) Prove that for a hexagonal prism such a mapping is not possible.
geometry3D geometryprism
Isosceles triangle and right angle

Source: Romanian District Olympiad 2006, Grade 7, Problem 4

3/11/2006
Let ABCABC be a triangle with AB=ACAB=AC. Let DD be the midpoint of BCBC, MM the midpoint of ADAD and NN the foot of the perpendicular from DD to BMBM. Prove that ∠ANC=90∘\angle ANC = 90^\circ.
vector
Smallest prime larger than n and larger prime greater than n

Source: Romanian District Olympiad 2006, Grade 9, Problem 4

3/11/2006
For each positive integer n≥2n\geq 2 we denote with p(n)p(n) the largest prime number less than or equal to nn, and with q(n)q(n) the smallest prime number larger than nn. Prove that ∑k=2n1p(k)q(k)<12. \sum^n_{k=2} \frac 1{p(k)q(k)} < \frac 12.
inequalitiesalgebra proposedalgebra
Function with discontinuity in every point

Source: Romanian District Olympiad 2006, Grade 11, Problem 4

3/11/2006
We say that a function f:R→Rf: \mathbb R \to \mathbb R has the property (P)(P) if, for any real numbers xx, sup⁡t≤xf(x)=x. \sup_{t\leq x} f(x) = x. a) Give an example of a function with property (P)(P) which has a discontinuity in every real point. b) Prove that if ff is continuous and satisfies (P)(P) then f(x)=xf(x) = x, for all x∈Rx\in \mathbb R.
functionreal analysisreal analysis unsolved
Sets of irrational and rational numbers

Source: Romanian District Olympiad 2006, Grade 10, Problem 4

3/11/2006
a) Find two sets X,YX,Y such that X∩Y=∅X\cap Y =\emptyset, X∪Y=Q+⋆X\cup Y = \mathbb Q^{\star}_{+} and Y={a⋅b∣a,b∈X}Y = \{a\cdot b \mid a,b \in X \}. b) Find two sets U,VU,V such that U∩V=∅U\cap V =\emptyset, U∪V=RU\cup V = \mathbb R and V={x+y∣x,y∈U}V = \{x+y \mid x,y \in U \}.
quadraticsabstract algebravectorgroup theoryalgebra proposedalgebra
Find the smallest constant c such that the ineq. is true

Source: Romanian District Olympiad 2006, Grade 12, Problem 4

3/11/2006
Let F={f:[0,1]→[0,∞)∣f\mathcal F = \{ f: [0,1] \to [0,\infty) \mid f continuous }\} and nn an integer, n≥2n\geq 2. Find the smallest real constant cc such that for any f∈Ff\in \mathcal F the following inequality takes place ∫01f(xn)dx≤c∫01f(x)dx. \int^1_0 f \left( \sqrt [n] x \right) dx \leq c \int^1_0 f(x) dx.
inequalitiesintegrationfunctionreal analysisreal analysis unsolved