MathDB

P1

Part of Russian TST 2018

Problems(8)

Fermat pseudoprime

Source:

4/23/2020
Let k>1k>1 be the given natural number and p∈Pp\in \mathbb{P} such that n=kp+1n=kp+1 is composite number. Given that n∣2n−1−1.n\mid 2^{n-1}-1. Prove that n<2k.n<2^k.
number theory
Excenter lies on excircle

Source: Russian TST 2018, Day 4 P1

3/30/2023
Let II{} be the incircle of the triangle ABCABC. Let A1,B1A_1, B_1 and C1C_1 be the midpoints of the sides BC,CABC, CA and ABAB respectively. The point XX{} is symmetric to II{} with respect to A1A_1. The line ℓ\ell parallel to BCBC and passing through XX{} intersects the lines A1B1A_1B_1 and A1C1A_1C_1 at MM{} and NN{} respectively. Prove that one of the excenters of the triangle ABCABC lies on the A1A_1-excircle of the triangle A1MNA_1MN.
geometryexcircleexcenter
&lt;OAC=2 &lt;BPM wanted, AB = AC, &lt;PBA = &lt;PCB

Source: 2017 Baltic Way Shortlist G7 BW https://artofproblemsolving.com/community/c2659348_baltic_way_shortlist_2017_geometry

11/25/2021
Let ABCABC be an isosceles triangle with AB=ACAB = AC. Let P be a point in the interior of ABCABC such that PB>PCPB > PC and ∠PBA=∠PCB\angle PBA = \angle PCB. Let MM be the midpoint of the side BCBC. Let OO be the circumcenter of the triangle APMAPM. Prove that ∠OAC=2∠BPM\angle OAC=2 \angle BPM .
equal anglesgeometry
There are 2018 points on a sphere

Source: Russian TST 2018, Day 7 P1 (Group NG), P2 (Groups A &amp; B)

3/30/2023
There are 2018 points marked on a sphere. A zebra wants to paint each point white or black and, perhaps, connect some pairs of points of different colors with a segment. Find the residue modulo 5 of the number of ways to do this.
combinatoricscounting
Roots of polynomial composition

Source: Russian TST 2018, Day 7 P1 (Groups A &amp; B)

3/30/2023
Let f(x)=x2+2018x+1f(x) = x^2 + 2018x + 1. Let f1(x)=f(x)f_1(x)=f(x) and fk(x)=f(fk−1(x))f_k(x)=f(f_{k-1}(x)) for all k⩾2k\geqslant 2. Prove that for any positive integer nn{}, the equation fn(x)=0f_n(x)=0 has at least two distinct real roots.
polynomialrootsalgebra
Lattice-point free circles

Source: Russian TST 2018, Day 6 P1

3/30/2023
Find all positive rr{} satisfying the following condition: For any d>0d > 0, there exist two circles of radius rr{} in the plane that do not contain lattice points strictly inside them and such that the distance between their centers is dd{}.
combinatoricsgeometrycombinatorial geometry
Prove that 8b+1 is composite

Source: Russian TST 2018, Day 9 P1 (Groups A &amp; B)

3/30/2023
The natural numbers a>ba > b are such that a−b=5b2−4a2a-b=5b^2-4a^2. Prove that the number 8b+18b + 1 is composite.
number theory
Inequality

Source: Russian TST 2018, Day 10 P1 (Groups A &amp; B)

3/30/2023
Let a,b,ca,b,c{} be positive real numbers. Prove that 108⋅(ab+bc+ca)⩽(a+b+b+c+c+a)4.108\cdot(ab+bc+ca)\leqslant(\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a})^4.
algebrainequalities