MathDB

1

Part of 2009 Belarus Team Selection Test

Problems(8)

AB=CD, 2 equal circles and 2 tangent circles and a line (both tangent + secant)

Source: Belarus TST 2009 1.1

6/13/2020
Two equal circles S1S_1 and S2S_2 meet at two different points. The line ℓ\ell intersects S1S_1 at points A,CA,C and S2S_2 at points B,DB,D respectively (the order on ℓ\ell: A,B,C,DA,B,C,D) . Define circles Γ1\Gamma_1 and Γ2\Gamma_2 as follows: both Γ1\Gamma_1 and Γ2\Gamma_2 touch S1S_1 internally and S2S_2 externally, both Γ1\Gamma_1 and Γ2\Gamma_2 line ℓ\ell, Γ1\Gamma_1 and Γ2\Gamma_2 lie in the different halfplanes relatively to line ℓ\ell. Suppose that Γ1\Gamma_1 and Γ2\Gamma_2 touch each other. Prove that AB=CDAB=CD.
I. Voronovich
circlesgeometrytangent circlesequal segmentstangent
sum 1/(a+b)b >= 9/2(ab+bc+ca) for a,b,c>0

Source: 2009 Belarus TST 2.1

11/8/2020
Prove that any positive real numbers a,b,c satisfy the inequlaity 1(a+b)b+1(b+c)c+1(c+a)a≥92(ab+bc+ca)\frac{1}{(a+b)b}+\frac{1}{(b+c)c}+\frac{1}{(c+a)a}\ge \frac{9}{2(ab+bc+ca)}
I.Voronovich
inequalitiesalgebraBelarus
f(x-f(y))=xf(y)-yf(x)+g(x)

Source: 2009 Belarus TST 3.1

11/8/2020
Find all functions f:R→Rf: R \to R and g:R→Rg:R \to R such that f(x−f(y))=xf(y)−yf(x)+g(x)f(x-f(y))=xf(y)-yf(x)+g(x) for all real numbers x,yx,y.
I.Voronovich
algebrafunctional equationfunctional
min c such that S(MKNL)<c S(ABCD)

Source: Belarus TST 2009 5.1

6/13/2020
Let M,NM,N be the midpoints of the sides AD,BCAD,BC respectively of the convex quadrilateral ABCDABCD, K=AN∩BMK=AN \cap BM, L=CM∩DNL=CM \cap DN. Find the smallest possible c∈Rc\in R such that S(MKNL)<c⋅S(ABCD)S(MKNL)<c \cdot S(ABCD) for any convex quadrilateral ABCDABCD.
I. Voronovich
geometrygeometric inequalityareasareamidpoints
both roots of x^2+(2-3n^2)x+(n^2-1)^2=0 are perfect squares

Source: 2009 Belarus TST 4.1

11/8/2020
Prove that there exist many natural numbers n so that both roots of the quadratic equation x2+(2−3n2)x+(n2−1)2=0x^2+(2-3n^2)x+(n^2-1)^2=0 are perfect squares.
S. Kuzmich
Perfect SquaresPerfect Squarenumber theory
circumcenter of ABC lies on the line CL, median, bisector related

Source: Belarus TST 2009 6.1

6/13/2020
In a triangle ABC,AMABC, AM is a median, BKBK is a bisectrix, L=AM∩BKL=AM\cap BK. It is known that BC=a,AB=c,a>cBC=a, AB=c, a>c. Given that the circumcenter of triangle ABCABC lies on the line CLCL, find ACAC
I. Voronovich
geometryCircumcenterangle bisectormedian
If $\phi(n)|n-1$, then prove this problem :-)

Source: Korea NMO 1998

8/20/2011
Denote by ϕ(n)\phi(n) for all n∈Nn\in\mathbb{N} the number of positive integer smaller than nn and relatively prime to nn. Also, denote by ω(n)\omega(n) for all n∈Nn\in\mathbb{N} the number of prime divisors of nn. Given that ϕ(n)∣n−1\phi(n)|n-1 and ω(n)≤3\omega(n)\leq 3. Prove that nn is a prime number.
number theoryrelatively primenumber theory unsolved
binary operation on R, (a/b)/c = (a/c) / (b/c) and (a/b)*c = (a*c) / (b*c)

Source: 2009 Belarus TST 8.1

11/8/2020
On R a binary algebraic operation ''*'' is defined which satisfies the following two conditions: i) for all a,b∈Ra,b \in R, there exists a unique x∈Rx \in R such that x∗a=bx *a=b (write x=b/ax=b/a) ii) (a∗b)∗c=(a∗c)∗(b∗c)(a*b)*c= (a*c)* (b*c) for all a,b,c∈Ra,b,c \in R a) Is this operation necesarily commutative (i.e. a∗b=b∗aa*b=b*a for all a,b∈Ra,b \in R) ? b) Prove that (a/b)/c=(a/c)/(b/c)(a/b)/c = (a/c) / (b/c) and (a/b)∗c=(a∗c)/(b∗c)(a/b)*c = (a*c) / (b*c) for all a,b,c∈Ra,b,c \in R.
A. Mirotin
algebraBinary operationOperation