IOQM 2022 Part 1 #IOQM2022 #maths

IOQM 2022 (Q1 to Q10) A triangle ABC with AC= 20 is inscribed in a circle. A tangent t to circle is drawn through B. The distance of t from A is 25 and that from C is 16. If S denotes the area of the triangle ABC, find the largest integer not exceeding S/20. 𝐼𝑛 𝑎 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙𝑜𝑔𝑟𝑎𝑚 𝐴𝐵𝐶𝐷, 𝑡ℎ𝑒 𝑝𝑜𝑖𝑛𝑡 𝑃 𝑜𝑛 𝑎 𝑠𝑒𝑔𝑚𝑒𝑛𝑡 𝐴𝐵 𝑖𝑠 𝑡𝑎𝑘𝑒𝑛 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝐴𝑃/𝐴𝐵 𝑖𝑠 61/2022 𝑎𝑛𝑑 𝑎 𝑝𝑜𝑖𝑛𝑡 𝑄 𝑜𝑛 𝑡ℎ𝑒 𝑠𝑒𝑔𝑚𝑒𝑛𝑡 𝐴𝐷 𝑖𝑠 𝑡𝑎𝑘𝑒𝑛 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝐴𝑄/𝐴𝐷 𝑖𝑠 61/2065. 𝐼𝑓 𝑃𝑄 𝑖𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡𝑠 𝐴𝐶 𝑎𝑡 𝑇, 𝑓𝑖𝑛𝑑 𝐴𝐶/𝐴𝑇 𝑡𝑜 𝑡ℎ𝑒 𝑛𝑒𝑎𝑟𝑒𝑠𝑡 𝑖𝑛𝑡𝑒𝑔𝑒𝑟. In a trapezoid ABCD, the internal bisector of angle A intersects the base BC (or its extension) at the point E. Inscribed in the triangle ABE is a circle touching the side AB at M and side BE at the point P. Find the angle DAE in degrees, if AB:MP = 2. Starting with a positive integer M written on the board, Alice plays the following game: in each move, if x is the number on the board, she replaces it with 3x + 2. Similarly, starting with a positive integer N written on the board, Bob plays the following game: in each move, if x is the number on the board, he replaces it with 2x + 27. Given that Alice and Bob reach the same number after playing 4 moves each, find the smallest value of M + N. 𝐿𝑒𝑡 𝑚 𝑏𝑒 𝑡ℎ𝑒 𝑠𝑚𝑎𝑙𝑙𝑒𝑠𝑡 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑚^2 + (𝑚 + 1)^2 + … + (𝑚 + 10)^2 𝑖𝑠 𝑡ℎ𝑒 𝑠𝑞𝑢𝑎𝑟𝑒 𝑜𝑓 𝑎 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑛. 𝐹𝑖𝑛𝑑 𝑚 + 𝑛. 𝑳𝒆𝒕 𝒂, 𝒃 𝒃𝒆 𝒑𝒐𝒔𝒊𝒕𝒊𝒗𝒆 𝒊𝒏𝒕𝒆𝒈𝒆𝒓𝒔 𝒔𝒂𝒕𝒊𝒔𝒇𝒚𝒊𝒏𝒈 𝒂^𝟑– 𝒃^𝟑 – 𝒂𝒃=𝟐𝟓. 𝑭𝒊𝒏𝒅 𝒕𝒉𝒆 𝒍𝒂𝒓𝒈𝒆𝒔𝒕 𝒑𝒐𝒔𝒔𝒊𝒃𝒍𝒆 𝒗𝒂𝒍𝒖𝒆 𝒐𝒇 𝒂^𝟐 + 𝒃^𝟑. 𝑭𝒊𝒏𝒅 𝒕𝒉𝒆 𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒐𝒓𝒅𝒆𝒓𝒆𝒅 𝒑𝒂𝒊𝒓𝒔 (𝒂, 𝒃)𝒔𝒖𝒄𝒉 𝒕𝒉𝒂𝒕 𝒂, 𝒃 ∈ (𝟏𝟎, 𝟏𝟏, …, 𝟐𝟗, 𝟑𝟎) 𝒂𝒏𝒅 𝑮𝑪𝑫 (𝒂, 𝒃) + 𝑳𝑪𝑴 (𝒂, 𝒃) = 𝒂 + 𝒃. 𝑆𝑢𝑝𝑝𝑜𝑠𝑒 𝑡ℎ𝑒 𝑝𝑟𝑖𝑚𝑒 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑝 𝑎𝑛𝑑 𝑞 𝑠𝑎𝑡𝑖𝑠𝑓𝑦 𝑞^2 + 3𝑝=197𝑝^2+𝑞. 𝑊𝑟𝑖𝑡𝑒 𝑞/𝑝 𝑎𝑠 𝑙 𝑚/𝑛, 𝑤ℎ𝑒𝑟𝑒 𝑙, 𝑚, 𝑛 𝑎𝑟𝑒 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠, 𝑚 less than 𝑛 𝑎𝑛𝑑 𝐺𝐶𝐷(𝑚, 𝑛) = 1. 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑙 + 𝑚 + 𝑛. Consider the 10-digit number M = 9876543210. We obtain a new 10-digit number from M according to the following rule: we can choose one or more disjoint pairs of adjacent digits in M and interchange the digits in these chosen pairs, keeping the remaining digits in their own places. For example, from M = 9876543210, by interchanging the 2 underlined pairs, and keeping the others in their places, we get M1 9786453210 . Note that any number of (disjoint) pairs can be interchanged. Find the number of new numbers that can be so obtained from M.