MathDB

P2

Part of 2024 Israel TST

Problems(5)

Quasi-harmonic function on tree

Source: 2024 Israel TST Test 3 P2

1/29/2024
A positive integer NN is given. Panda builds a tree on NN vertices, and writes a real number on each vertex, so that 11 plus the number written on each vertex is greater or equal to the average of the numbers written on the neighboring vertices. Let the maximum number written be MM and the minimal number written mm. Mink then gives Panda M−mM-m kilograms of bamboo. What is the maximum amount of bamboo Panda can get?
functionaveragesTreescombinatorics
Realizing graphs with boxes

Source: 2024 Israel TST Test 1 P2

8/29/2023
Let n>1n>1 be an integer. Given a simple graph GG on nn vertices v1,v2,…,vnv_1, v_2, \dots, v_n we let k(G)k(G) be the minimal value of kk for which there exist nn kk-dimensional rectangular boxes R1,R2,…,RnR_1, R_2, \dots, R_n in a kk-dimensional coordinate system with edges parallel to the axes, so that for each 1≤i<j≤n1\leq i<j\leq n, RiR_i and RjR_j intersect if and only if there is an edge between viv_i and vjv_j in GG.
Define MM to be the maximal value of k(G)k(G) over all graphs on nn vertices. Calculate MM as a function of nn.
TSTcombinatoricsgraph theoryanalytic geometry
Center of (AID), KI perp TI_A

Source: 2024 Israel TST Test 2 P2

11/7/2023
In triangle ABCABC the incenter is II. The center of the excircle opposite AA is IAI_A, and it is tangent to BCBC at DD. The midpoint of arc BACBAC is NN, and NINI intersects (ABC)(ABC) again at TT. The center of (AID)(AID) is KK. Prove that TIA⊥KITI_A\perp KI.
geometrymixtilinear incircleexcircleincentercircumcircle
Root of unity mod x^n-1

Source: 2024 Israel TST Test 6 P2

3/20/2024
Let nn be a positive integer. Find all polynomials Q(x)Q(x) with integer coefficients so that the degree of Q(x)Q(x) is less than nn and there exists an integer m≥1m\geq 1 for which xn−1∣Q(x)m−1x^n-1\mid Q(x)^m-1
algebrapolynomialTSTpolynomial division
Reflections of tangents, line OI

Source: 2024 Israel TST Test 8 P2

5/10/2024
Triangle ABCABC is inscribed in circle Ω\Omega with center OO. The incircle of ABCABC is tangent to BCBC, ACAC, ABAB at DD, EE, FF respectively, and its center is II. The reflection of the tangent line to Ω\Omega at AA with respect to EFEF will be denoted ℓA\ell_A. We similarly define ℓB\ell_B, ℓC\ell_C. Show that the orthocenter of the triangle with sides ℓA\ell_A, ℓB\ell_B, ℓC\ell_C lies on OIOI.
geometrygeometric transformationreflectionincircle