MathDB

P1

Part of Russian TST 2014

Problems(10)

Execenter and incircle problem

Source: Russian TST 2014, Day 9 P1 (Groups A & B)

1/8/2024
The inscribed circle of the triangle ABCABC{} touches the sides BC,CABC,CA and ABAB{} at A′,B′A',B' and C′C'{} respectively. Let IaI_a be the AA-excenter of ABC.ABC{}. Prove that IaA′I_aA' is perpendicular to the line determined by the circumcenters of IaBC′I_aBC' and IaCB′.I_aCB'.
geometryincircle
Dissection of regular octagon

Source: Russian TST 2014, Day 7 P1 (Group NG), P2 (Groups A & B)

4/21/2023
For what values of kk{} can a regular octagon with side-length kk{} be cut into 1×21 \times 2{} dominoes and rhombuses with side-length 1 and a 45∘45^\circ{} angle?
geometrycombinatorics
An asymmetric inequality

Source: Russian TST 2015, Day 7 P1 (Groups A & B)

4/21/2023
Let x,y,zx,y,z be positive real numbers. Prove that xy+yz+zx⩾z(x+y)y(y+z)+x(y+z)z(z+x)+y(z+x)x(x+y).\frac{x}{y}+\frac{y}{z}+\frac{z}{x}\geqslant\frac{z(x+y)}{y(y+z)}+\frac{x(y+z)}{z(z+x)}+\frac{y(z+x)}{x(x+y)}.
algebrainequalities
Divisibility problem

Source: Russian TST 2014, Day 8 P1 (Group NG), P2 (Groups A & B)

1/8/2024
Let pp{} be a prime number and x1,x2,…,xpx_1,x_2,\ldots,x_p be integers for which x1n+x2n+⋯+xpnx_1^n+x_2^n+\cdots+x_p^n is divisible by pp{} for any positive integer nn{}. Prove that x1−x2x_1-x_2 is divisible by p.p{}.
number theoryDivisibilityprime numbers
Two 1001-gons are overlapped

Source: Russian TST 2014, Day 8 P1 (Groups A & B)

1/8/2024
A regular 1001-gon is drawn on a board, the vertiecs of which are numbered with 1,2,…,1001.1,2,\ldots,1001. Is it possible to label the vertices of a cardboard 1001-gon with the numbers 1,2,…,10011,2,\ldots,1001 such that for any overlap between the two 1001-gons, there are two vertices with the same number one over the other? Note that the cardboard polygon can be inverted.
combinatorics
Finitely many lines coloured red and blue

Source: Russian TST 2014, Day 9 P1 (Group NG), P2 (Groups A & B)

1/8/2024
Finitely many lines are given, which pass through some point P.P{}. Prove that these lines can be coloured red and blue and one can find a point Q≠PQ\neq P such that the sum of the distances from QQ{} to the red lines is equal to the sum of the distance from QQ{} to the blue lines.
geometry
Set theory with common elements

Source: Russian TST 2014, Day 10 P1 (Group NG)

1/8/2024
Given are twenty-two different five-element sets, such that any two of them have exactly two elements in common. Prove that they all have two elements in common.
combinatoricsset theory
Algebra-ish problem

Source: Russian TST 2014, Day 10 P1 (Groups A & B) [thank you for the translation, Phorphyrion]

1/8/2024
Nine numbers a,b,c,…a, b, c, \dots are arranged around a circle. All numbers of the form a+bc,…a+b^c, \dots are prime. What is the largest possible number of different numbers among a,b,c,…a, b, c, \dots?
algebra
Geometric inequality with radii

Source: Russian TST 2014, Day 11 P1 (Group NG), P3 (Groups A & B)

1/8/2024
Let RR{} and rr{} be the radii of the circumscribed and inscribed circles of the acute-angled triangle ABCABC{} respectively. The point MM{} is the midpoint of its largest side BC.BC. The tangents to its circumscribed circle at BB{} and CC{} intersect at XX{}. Prove that rR⩾AMAX.\frac{r}{R}\geqslant\frac{AM}{AX}.
geometryInequality
Domino dissections of a board

Source: Russian TST 2014, Day 11 P1 (Groups A & B)

1/8/2024
On each non-boundary unit segment of an 8×88\times 8 chessboard, we write the number of dissections of the board into dominoes in which this segment lies on the border of a domino. What is the last digit of the sum of all the written numbers?
boarddominoescombinatorics