MathDB

3

Part of 1998 Belarus Team Selection Test

Problems(6)

sequence of moves changes pair of integers ( x,y) to (x+t, y-s)

Source: 1998 Belarus TST 1.3

12/25/2020
Let s,ts,t be given nonzero integers, (x,y)(x,y) be any (ordered) pair of integers. A sequence of moves is performed as follows: per move (x,y)(x,y) changes to (x+t,y−s)(x+t, y-s). The pair (x,y) is said to be good if after some (may be, zero) number of moves described a pair of integers arises that are not relatively prime. a) Determine whether (s,t)(s,t) is itself a good pair; bj Prove that for any nonzero ss and tt there is a pair (x,y)(x,y) which is not good.
number theorycombinatorics
area computational with hexagon, parallelogram and equilateral

Source: Ukrainian TST 1999 p11 - Belarus TST 1998 2.3

2/13/2020
Let ABCDEFABCDEF be a convex hexagon such that BCEFBCEF is a parallelogram and ABFABF an equilateral triangle. Given that BC=1,AD=3,CD+DE=2BC = 1, AD = 3, CD+DE = 2, compute the area of ABCDEFABCDEF
geometryparallelogramhexagonareaEquilateral
f(x,y) = x^3 + (3y^2+1)x^2 + (3y^4 - y^2 + 4 y - 1)x + (y^6-y^4 + 2y^3)

Source: 1998 Belarus TST 4.3

12/25/2020
a) Let f(x,y)=x3+(3y2+1)x2+(3y4−y2+4y−1)x+(y6−y4+2y3)f(x,y) = x^3 + (3y^2+1)x^2 + (3y^4 - y^2 + 4 y - 1)x + (y^6-y^4 + 2y^3). Prove that if for some positive integers a,ba, b the number f(a,b)f(a, b) is a cube of an integer then f(a,b)f(a, b) is also a square of an integer.
b) Are there infinitely many pairs of positive integers (a,b)(a, b) for which f(a,b)f(a, b) is a square but not a cube ?
number theoryPerfect Squareperfect cube
d_7^2+d_{10}^2=(n/d_{22})^2

Source: 1998 Belarus TST 5.3

12/25/2020
Let 1=d1<d2<d3<...<dk=n1=d_1<d_2<d_3<...<d_k=n be all different divisors of positive integer nn written in ascending order. Determine all nn such that d72+d102=(n/d22)2.d_7^2+d_{10}^2=(n/d_{22})^2.
number theoryDivisors
g(g(x)) = g(x)+2x , continuous

Source: 1998 Belarus TST 7.3

12/25/2020
Find all continuous functions f:R→Rf: R \to R such that g(g(x))=g(x)+2xg(g(x)) = g(x)+2x for all real xx.
continuousfunctional equationfunctionalalgebra
sequence of triangles each has sidelenghts equal to angles of previous one

Source: 1998 Belarus TST 8.3

12/25/2020
For any given triangle A0B0C0A_0B_0C_0 consider a sequence of triangles constructed as follows: a new triangle A1B1C1A_1B_1C_1 (if any) has its sides (in cm) that equal to the angles of A0B0C0A_0B_0C_0 (in radians). Then for △A1B1C1\vartriangle A_1B_1C_1 consider a new triangle A2B2C2A_2B_2C_2 (if any) constructed in the similar พay, i.e., △A2B2C2\vartriangle A_2B_2C_2 has its sides (in cm) that equal to the angles of A1B1C1A_1B_1C_1 (in radians), and so on. Determine for which initial triangles A0B0C0A_0B_0C_0 the sequence never terminates.
geometrycombinatoricssidelengthsangles