MathDB

P2

Part of Russian TST 2015

Problems(5)

Infinite family of circles passing through a point

Source: Russian TST 2015, Day 7 P2

4/21/2023
In the isosceles triangle ABCABC where AB=ACAB = AC, the point II{} is the center of the inscribed circle. Through the point AA{} all the rays lying inside the angle BACBAC are drawn. For each such ray, we denote by XX{} and YY{} the points of intersection with the arc BICBIC and the straight line BCBC respectively. The circle γ\gamma passing through XX{} and YY{}, which touches the arc BICBIC at the point XX{} is considered. Prove that all the circles γ\gamma pass through a fixed point.
geometrycircles
Strange angle equality

Source: Russian TST 2015, Day 8 P2 (Group NG), P3 (Groups A & B)

4/21/2023
The triangle ABCABC is given. Let P1P_1 and P2P_2 be points on the side ABAB such that P2P_2 lies on the segment BP1BP_1 and AP1=BP2AP_1 = BP_2. Similarly, Q1Q_1 and Q2Q_2 are points on the side BCBC such that Q2Q_2 lies on the segment BQ1BQ_1 and BQ1=CQ2BQ_1 = CQ_2. The segments P1Q2P_1Q_2 and P2Q1P_2Q_1 intersect at the point RR{}, and the circles P1P2RP_1P_2R and Q1Q2RQ_1Q_2R intersect a second time at the point SS{} lying inside the triangle P1Q1RP_1Q_1R. Let MM{} be the midpoint of the segment ACAC. Prove that the angles P1RSP_1RS and Q1RMQ_1RM are equal.
geometryangles
concyclic wanted, altitude and perpendiculars related

Source: Ukraine TST 2015 p6

5/1/2020
Given an acute triangle ABC,HABC, H is the foot of the altitude drawn from the point AA on the line BC,PBC, P and K≠HK \ne H are arbitrary points on the segments AHAH andBC BC respectively. Segments ACAC and BPBP intersect at point B1B_1, lines ABAB and CPCP at point C1C_1. Let XX and YY be the projections of point HH on the lines KB1KB_1 and KC1KC_1, respectively. Prove that points A,P,XA, P, X and YY lie on one circle.
geometryConcyclicperpendicularaltitude
A sum and a product give the same remainder

Source: Russian TST 2015, Day 10 P2 (Group NG)

4/21/2023
Let p⩾5p\geqslant 5 be a prime number. Prove that the set {1,2,…,p−1}\{1,2,\ldots,p - 1\} can be divided into two nonempty subsets so that the sum of all the numbers in one subset and the product of all the numbers in the other subset give the same remainder modulo pp{}.
number theoryprime numbers
Four variable inequality

Source: Russian TST 2015, Day 10 P2 (Group NG), P3 (Groups A & B)

4/21/2023
Let a,b,c,da,b,c,d be positive real numbers satisfying a2+b2+c2+d2=1a^2+b^2+c^2+d^2=1. Prove that a3+b3+c3+d3+abcd(1a+1b+1c+1d)⩽1.a^3+b^3+c^3+d^3+abcd\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}\right)\leqslant 1.
algebrainequalities