MathDB

Problem 3

Part of 2024 Ukraine National Mathematical Olympiad

Problems(4)

Kinda cute imo!

Source: Ukrainian Mathematical Olympiad 2024. Day 1, Problem 8.3

3/19/2024
Points XX and YY are chosen inside an acute triangle ABCABC so that:
∠AXB=∠CYB=180∘−∠ABC, ∠ABX=∠CBY\angle AXB = \angle CYB = 180^\circ - \angle ABC, \text{ } \angle ABX = \angle CBY
Show that the points XX and YY are equidistant from the center of the circumscribed circle of △ABC\triangle ABC.
Proposed by Anton Trygub
geometry
Circle inequality

Source: Ukrainian Mathematical Olympiad 2024. Day 1, Problem 9.3

3/19/2024
20242024 positive real numbers with sum 11 are arranged on a circle. It is known that any two adjacent numbers differ at least in 22 times. For each pair of adjacent numbers, the smaller one was subtracted from the larger one, and then all these differences were added together. What is the smallest possible value of this resulting sum?
Proposed by Oleksiy Masalitin
algebrainequalities
When bisector meets altitudes

Source: Ukrainian Mathematical Olympiad 2024. Day 1, Problem 10.3

3/19/2024
Altitudes AHA,BHB,CHCAH_A, BH_B, CH_C of triangle ABCABC intersect at HH, and let MM be the midpoint of the side ACAC. The bisector BLBL of △ABC\triangle ABC intersects HAHCH_AH_C at point KK. The line through LL parallel to HMHM intersects BHBBH_B in point TT. Prove that TK=TLTK = TL.
Proposed by Anton Trygub
geometryorthocenter
Almost means

Source: Ukrainian Mathematical Olympiad 2024. Day 1, Problem 11.3

3/19/2024
Let's define almost mean of numbers a1,a2,…,ana_1, a_2, \ldots, a_n as a1+a2+…+ann+1\frac{a_1 + a_2 + \ldots + a_n}{n+1}. Oleksiy has positive real numbers b1,b2,…,b2023b_1, b_2, \ldots, b_{2023}, not necessarily distinct. For each pair (i,j)(i, j) with 1≤i,j≤20231 \leq i, j \leq 2023, Oleksiy wrote on a board almost mean of numbers bi,bi+1,…,bjb_i, b_{i+1}, \ldots, b_j. Prove that there are at least 4545 distinct numbers on the board.
Proposed by Anton Trygub
algebracombinatoricsposet