MathDB

P1

Part of 2024 Israel TST

Problems(5)

Existence of tangent circle given <BAC=60

Source: 2024 Israel TST Test 1 P1

8/29/2023
Triangle ABCABC with ∠BAC=60∘\angle BAC=60^\circ is given. The circumcircle of ABCABC is Ω\Omega, and the orthocenter of ABCABC is HH. Let SS denote the midpoint of the arc BCBC of Ω\Omega which doesn't contain AA. Point PP was chosen on Ω\Omega so that ∠HPS=90∘\angle HPS=90^\circ. Prove that there exists a circle that goes through PP and SS and is tangent to lines ABAB, ACAC.
geometrytangencyTSTcircumcircle
Power diophantine

Source: 2024 Israel TST Test 2 P1

11/7/2023
Solve in positive integers: xy2+1+yx2+1=2zx^{y^2+1}+y^{x^2+1}=2^z
number theoryDiophantine equationExponential equation
Exsimilicenters are collinear

Source: 2024 Israel TST Test 3 P1

1/29/2024
Let ABCABC be a triangle and let DD be a point on BCBC so that ADAD bisects the angle ∠BAC\angle BAC. The common tangents of the circles (BAD)(BAD), (CAD)(CAD) meet at the point A′A'. The points B′B', C′C' are defined similarly. Show that A′A', B′B', C′C' are collinear.
geometryTangentsAngle Bisectors
Subgraphs with even cross-edges

Source: 2024 Israel TST Test 6 P1

3/20/2024
Let GG be a connected (simple) graph with nn vertices and at least nn edges. Prove that it is possible to color the vertices of GG red and blue, so that the following conditions hold:
i. There is at least one vertex of each color, ii. There is an even number of edges connecting a red vertex to a blue vertex, and iii. If all such edges are deleted, one is left with two connected graphs.
combinatoricsTSTgraph theoryColoring
Maximal factorial dividing product

Source: 2024 Israel TST Test 8 P1

5/10/2024
For each positive integer nn let ana_n be the largest positive integer satisfying (an)!∣∏k=1n⌊nk⌋(a_n)!\left| \prod_{k=1}^n \left\lfloor \frac{n}{k}\right\rfloor\right. Show that there are infinitely many positive integers mm for which am+1<ama_{m+1}<a_m.
factorialnumber theoryfloor function