Tiêu đề 3b-Dap-an-De-thi-Toan-chuyen-CSP-2024.pdf ✅

B0 Gt,q.o DUC vA EAo rAo rRrIoNG EAr Hec srJ PHAM uA xQr xY rru ruvEN srNn r.6p ro TRtIdNc rHpr cuuvtx o4r Hqc srJ pH4M NAn zoz+ of cniNu rnrtc DAP AN.HIJdNG DAN GIAI u6x rrrt: ToAN (cHUYEN) CAu NQi dung Oi6m Bni l 3,0 a 1,5 Di6u kiQn: -2024 < a < 2024, - 2024 < b < 2024. Ta c6: a+2024+ cr( a+2024- b+2024) +( I 2025-ct - 202s-b)+( 2024-b - 2024 - a)=a <+(r-D).( I .dffi)=oo a +2024+ b + 2024 2025-a+ 2025 - b Do I 1 >0 nOn 2024-a + 2024- b 2025- a + 2025-b I I I + >0. a+2024 + b +2024 2024- a + 2024- b 2025-a + 2025- b Do d6, tt (*) suy ra a : D. Vdi a: b thi M = a''o + a'o" - b2o24 - b2025 =0. b 1,5 Gid st t6n tai c6c s6 hiru ti duong a, b sao "no Ji * JO = J^l-zou Ta c6: Ji *Ji= Jzou + a+b+zJi6 =JM =2JA -Jzou-(a+b) + 4ab -2024 + (a + b)' -2(a + DJzou (+) Tri (*) vi a > 0, b > 0, ta suy ra: 2@ + DJZOU = 2024 +l@ + b)' - 4ab) 2024+[(a+b)' -4abl (**). 2(a + b) Tt (**), ta nhfln duo. c m6u thuAn vl ^!2024 li s6 vd ti, cdn 2024 +[(a + b)1 - 4ab] ld s6 hfu ti. Vfy kh6ng t6n tai c6c sti htru ti duong a, , sao cho.6 + rE = ,l^FnU J2025 - a - J2a4 - o = J b + 20% + J2ox 4 - J2a4 4 -Jzou-- 2(a+ b)
Bni 2 2,0 a 1,0 Ta c6: 16*2024=2024*16= (2024 -16)*16 - (2024 - 2.16)'*16 =... = (2024-126.16)*16 b 1,0 X6t quy tic * cho tuong img m6i c6p a6 "grf" duo.g (r, b) "di TJCLN(r, D). Tuc td c *b = TICLN (a, b). Bii 3 A P s B DM c 3,0 1,0 Do tri gi6c BCEF nQitidp ndn AFQ = 193. Suy ra s b: Do LICLN(a,6)= t/CLN(a -b,g nAn q*S-(a-b)*b. Viy quy tic * xric tlinh nhu tr6n th6a m6n y6u cAu cria bdi to6n. a
Yiy AP, = AQ, = AD,AH, 1,0 Gqi S h giao l fti f(n)>2. N6u /(n) = 2 thi 3n'? + n +l c6 dang l0'..01, trong tt6 chir s6 dAu vA cht s5 cutii bing l,c6cchts6cdnlai(n6uc6)phnibing 0. Dod6 3n'+n+l=10' +1 voi t nguy0n duong =3r?2 tn=llk =n(3n+l)-lOL. Vi (n;3n + l) = 1 ndn ta c6 c6c trudng hqp sau: Trudng hqp t: n=1,3n+l= l0' =4 = l0' : Didu ndy ld v6 li. Tntdnghsp 2: n=2r,3n+l=5r +3.2r +l=51. N6u k=1 thi 3.2+1=5: Ei€u ndy liv6 li. Nilu & > 2 thi 5r > 4t --2' .2' > 4.2' --3.2' +2t >3.2' +l'. Di6u nAy li v6 li. V{y ta lu6n c6 f(n)>z,r&c f(n)>3. M{t kJrdc, ta c6:/(8)=3. Vi thi! gi6 tri nhd nh6t cria f(n) ld 3. b 2 Bii 4
1,0 Ta tl6nh d6u r vdo mOt s6 6 vu6ng don vi cia bing da aho nhu sau x J. x x ta x x j. x x J. x v J. x t u J( x J. t , , x Phdn bdng cdn lai gtim cdc 6 ru6ng don vi kh6ng duqc d6nh d6u ld hqp cta t hinh vu6ng 2 x 2, ta ggi c6c hinh vu6ng ndy ld hinh vu6ng "tlgp". Ta ki hi€u t hlnh vu6ng "tlep" niy ld HbH2,...,He. Xdt mQt c6ch l6t th6a mln y€u cAu. Voi m6i hhh vu6ng "de.p; Il,, a ggi d, h s5 domino mdu den phri it nhdt mQt 6 vu6ng d1, Yi =1,2,...,9, suy ra I>9. Do kh6ng th€ c6 domino mau den duop l6t ndo t4i phu l€n cd hai hinh ru6ng "ilgp" kh6c nhau n€n s6 domino mdu den cdn dtng l6n hon holc blng 7. Viy sti domino miu den cAn sri dgng di! lat dugc theo y€u ciu lon hon hodc bdng 9. Ta chi ra c6ch t6t voi 9 domino miu den vd 23 domino mdu trang th6a mdn il6 bdi. Btrdc l:Ta l6t 9 domino mdu tlen nhu hinh v€. Btroc 2: Ta l6t kin phin cdn tai cta bdng bing 23 domino mdu tring. Vay gi6 tri nh6 nh6t cta & lA 9. II I I IIII IIII IIII HET Bii 5 x [r.] l r-.1 Itl I ffi |ll -T-T