MathDB

3

Part of 2023 India IMO Training Camp

Problems(3)

Hard triangle geo (impossible?)

Source: India TST 2023 Day 3 P3

7/9/2023
In triangle ABCABC, with orthocenter HH and circumcircle Γ\Gamma, the bisector of angle BACBAC meets BC‾\overline{BC} at KK. Point QQ lies on Γ\Gamma such that AQ‾⊥QK‾\overline{AQ} \perp \overline{QK}. Circumcircle of △AQH\triangle AQH meets AC‾\overline{AC} at YY and AB‾\overline{AB} at ZZ. Let BY‾\overline{BY} and CZ‾\overline{CZ} meet at TT. Prove that TH‾⊥KA‾\overline{TH} \perp \overline{KA}
geometry
Sum of reciprocals of product of pairs is 1

Source: India TST 2023 Practice Test 1 P3

7/9/2023
Prove that for all integers k>2k>2, there exists kk distinct positive integers a1,…,aka_1, \dots, a_k such that ∑1≤i<j≤k1aiaj=1.\sum_{1 \le i<j \le k} \frac{1}{a_ia_j} =1.
Proposed by Anant Mudgal
algebra
Sum of fourth powers is prime

Source: India TST 2023 Practice Test 2 P3

7/9/2023
Let nn be any positive integer, and let S(n)S(n) denote the number of permutations τ\tau of {1,…,n}\{1,\dots,n\} such that k4+(τ(k))4k^4+(\tau(k))^4 is prime for all k=1,…,nk=1,\dots,n. Show that S(n)S(n) is always a square.
combinatoricsnumber theory