MathDB

P2

Part of Russian TST 2018

Problems(4)

Many angle conditions

Source: Russian TST 2018, Day 5 P2

3/30/2023
Inside the acute-angled triangle ABCABC, the points PP{} and QQ{} are chosen so that ∠ACP=∠BCQ\angle ACP = \angle BCQ and ∠CBP=∠ABQ\angle CBP =\angle ABQ. The point ZZ{} is the projection of PP{} onto the line BCBC. The point Q′Q' is symmetric to QQ{} with respect to ZZ{}. The points KK{} and LL{} are chosen on the rays ABAB and ACAC respectively, so that Q′K∥QCQ'K \parallel QC and Q′L∥QBQ'L \parallel QB. Prove that ∠KPL=∠ BPC\angle KPL=\angle BPC.
geometryangles
Bear vs crocodile

Source: Russian TST 2018, Day 4 P2

3/30/2023
Let F\mathcal{F} be a finite family of subsets of some set XX{}. It is known that for any two elements x,y∈Xx,y\in X there exists a permutation π\pi of the set XX such that π(x)=y\pi(x)=y, and for any A∈FA\in\mathcal{F} π(A):={π(a):a∈A}∈F.\pi(A):=\{\pi(a):a\in A\}\in\mathcal{F}.A bear and crocodile play a game. At a move, a player paints one or more elements of the set XX in his own color: brown for the bear, green for the crocodile. The first player to fully paint one of the sets in F\mathcal{F} in his own color loses. If this does not happen and all the elements of XX have been painted, it is a draw. The bear goes first. Prove that he doesn't have a winning strategy.
combinatoricsset theorygameTSTRussian TSTGame Theory
Hypercube graph with edge weights

Source: Russian TST 2018, Day 8 P2 (Group NG), P3 (Groups A & B)

3/30/2023
There are 2n2^n airports, numbered with binary strings of length nn{}. Any two stations whose numbers differ in exactly one digit are connected by a flight that has a price (which is the same in both directions). The sum of the prices of all nn{} flights leaving any station does not exceed 1. Prove that one can travel between any two airports by paying no more than 1.
combinatoricsgraph theory
Tangent circles

Source: Russian TST 2018, Day 9 P2 (Group NG), P4 (Groups A & B)

3/30/2023
The point KK{} is the middle of the arc BACBAC of the circumcircle of the triangle ABCABC. The point II{} is the center of its inscribed circle ω\omega. The line KIKI intersects the circumcircle of the triangle ABCABC at TT{} for the second time. Prove that the circle passing through the midpoints of the segments BC,BTBC, BT and CTCT is tangent to the circle which is symmetric to ω\omega with respect to BCBC.
geometrytangency