MathDB

3

Part of 2018 Junior Balkan Team Selection Tests - Romania

Problems(6)

AE = AC, if <ACP =<ABC, line reflection, circumcircle related

Source: 2018 Romania JBMO TST 1.3

6/1/2020
Let ABCABC be a triangle with AB>ACAB > AC. Point P∈(AB)P \in (AB) is such that ∠ACP=∠ABC\angle ACP = \angle ABC. Let DD be the reflection of PP into the line ACAC and let EE be the point in which the circumcircle of BCDBCD meets again the line ACAC. Prove that AE=ACAE = AC.
geometrygeometric transformationreflectioncircumcircleequal angles
Special point D.

Source: Unknown

5/27/2019
Let DD be a unique point on segment BCBC, in ABCABC. If AD2=BD⋅CDAD^2 = BD \cdot CD, show that AB+AC=2BCAB + AC = \sqrt{2}BC.
geometry
A + A \ne A x A , (A + A) \cap N = (A \cdot A) \cap N

Source: 2018 Romania JBMO TST 3.3

6/19/2020
Let A={a=q+1q/q∈Q∗,q>0}A =\left\{a = q + \frac{1}{q }/ q \in Q^*,q > 0 \right\}, A+A={a+b∣a,b∈A}A + A = \{a + b |a,b \in A\},A⋅A={a⋅b∣a,b∈A}A \cdot A =\{a \cdot b | a, b \in A\}. Prove that: i) A+A≠A⋅AA + A \ne A \cdot A ii) (A+A)∩N=(A⋅A)∩N(A + A) \cap N = (A \cdot A) \cap N.
Vasile Pop
Setsalgebra
projective candidate, lines AB, CD , FG are either parallel or concurrent.

Source: 2018 Romania JBMO TST 4.3

6/1/2020
Let ABCDABCD be a cyclic quadrilateral. The line parallel to BDBD passing through AA meets the line parallel to ACAC passing through BB at EE. The circumcircle of triangle ABEABE meets the lines ECEC and EDED, again, at FF and GG, respectively. Prove that the lines AB,CDAB, CD and FGFG are either parallel or concurrent.
cyclic quadrilateralgeometryconcurrencyparallelcircumcircle
a heap and 330 stones game

Source: 2018 Romania JBMO TST 5.3

6/19/2020
Alina and Bogdan play the following game. They have a heap and 330330 stones in it. They take turns. In one turn it is allowed to take from the heap exactly 11, exactly nn or exactly mm stones. The player who takes the last stone wins. Before the beginning Alina says the number nn, (1<n<101 < n < 10). After that Bogdan says the number mm, (m≠n,1<m<10m \ne n, 1 < m < 10). Alina goes first. Which of the two players has a winning strategy? What if initially there are 2018 stones in the heap?
adapted from a Belarus Olympiad problem
gamewinning strategycombinatoricsgame strategy
circumcircle is tangent to AB

Source: All-Russian olympiad 2015, class 9, day 2, problem 7

11/5/2015
Given an acute triangle ABCABC with AB<ACAB < AC.Let Ω\Omega be the circumcircle of ABC ABC and MM be centeriod of triangle ABCABC.AHAH is altitude of ABCABC.MHMH intersect with Ω\Omega at A′A'.prove that circumcircle of triangle A′HBA'HB is tangent to ABAB.
A.I.Golovanov, A. Yakubov
geometry proposedgeometrycircumcircle