MathDB

1

Part of 2011 Belarus Team Selection Test

Problems(8)

f(sin x )+ a f(cos x) = cos 2x

Source: 2011 Belarus TST 1.1

11/7/2020
Find all real aa such that there exists a function f:R→Rf: R \to R satisfying the equation f(sin⁡x)+af(cos⁡x)=cos⁡2xf(\sin x )+ a f(\cos x) = \cos 2x for all real xx.
I.Voronovich
algebrafunctional equationtrigonometryfunctional
1,2,...,2011 around circle such that 8 of 25 successive multiples of 5 and/or 7

Source: 2011 Belarus TST 2.1

11/8/2020
Is it possible to arrange the numbers 1,2,...,20111,2,...,2011 over the circle in some order so that among any 2525 successive numbers at least 88 numbers are multiplies of 55 or 77 (or both 55 and 77) ?
I. Gorodnin
combinatoricsnumber theorymultipledivisible
S <= 1/5 A, A is sum of distinct product of sides of convex pentagon

Source: 2011 Belarus TST 4.1

11/7/2020
Let AA be the sum of all 1010 distinct products of the sides of a convex pentagon, SS be the area of the pentagon. a) Prove thas S≤15AS \le \frac15 A. b) Does there exist a constant c<15c<\frac15 such that S≤cAS \le cA ?
I.Voronovich
geometrypentagongeometric inequality
least no of elements removing from {1,2,...,20} such any 2 sum perfect square

Source: 2011 Belarus TST 8.1

11/8/2020
Find the least possible number of elements which can be deleted from the set {1,2,...,20}\{1,2,...,20\} so that the sum of no two different remaining numbers is not a perfect square.
N. Sedrakian , I.Voronovich
number theoryPerfect SquaresPerfect Square
a^{p_i}= 1 mod p_{i+1} for different odd primes given

Source: 2011 Belarus TST 3.1

11/7/2020
Given natural number a>1a>1 and different odd prime numbers p1,p2,...,pnp_1,p_2,...,p_n with ap1≡1a^{p_1}\equiv 1 (mod p2p_2), ap2≡1a^{p_2}\equiv 1 (mod p3p_3), ..., apn≡1a^{p_n}\equiv 1(mod p1p_1). Prove that a) (a−1)⋮pi(a-1)\vdots p_i for some i=1,..,ni=1,..,n b) Can (a−1)(a-1) be divisible by pip_i for exactly one ii of i=1,...,ni=1,...,n?
I. Bliznets
number theoryoddprimesdivisible
2 circles passing through parallel chords of parabola

Source: 2011 Belarus TST 6.1

6/14/2020
ABAB and CDCD are two parallel chords of a parabola. Circle S1S_1 passing through points A,BA,B intersects circle S2S_2 passing through C,DC,D at points E,FE,F. Prove that if EE belongs to the parabola, then FF also belongs to the parabola.
I.Voronovich
conicsparabolacirclesChordsparallelgeometry
2KF+BC=BH +HC, orthocenter, incenter , projection related

Source: 2011 Belarus TST 7.1

6/14/2020
In an acute-angled triangle ABCABC, the orthocenter is HH. IHI_H is the incenter of △BHC\vartriangle BHC. The bisector of ∠BAC\angle BAC intersects the perpendicular from IHI_H to the side BCBC at point KK. Let FF be the foot of the perpendicular from KK to ABAB. Prove that 2KF+BC=BH+HC2KF+BC=BH +HC
A. Voidelevich
geometryincenterorthocenterperpendicular
are g(2010), g(2011) divisible by 11? sum of n-digits form digits 0,1,2,3 such

Source: 2011 Belarus TST 5.1

11/7/2020
Let g(n)g(n) be the number of all nn-digit natural numbers each consisting only of digits 0,1,2,30,1,2,3 (but not nessesarily all of them) such that the sum of no two neighbouring digits equals 22. Determine whether g(2010)g(2010) and g(2011)g(2011) are divisible by 1111.
I.Kozlov
number theoryDigitsSumdivisible