. T\ giac IA CJ la hinh binh hanh OJ= AI ==
2)Day la bai toan khong kho đi vcfi hoc sinh kha, gioi toan LiAi yla phai xet hét cac kha nang xay rạ
hét cac kha nang xay rạ
Cfiu IIỊ 1. A p dung bát d i n g thiJc Co-si cho hai só dUcfng, ta c6
x^ + 2xy + y^ + 1 + 2xy + 2 (1 + x^)(l + y^) + 2x.2y (x + y ) ( l + xy) < (X + yf + (1 + xyf _ y^ + x^y^ + x^ + 1 + 4xy 143
LUy^rt fliii flg IriAHc kl ! h i vA6 Idip 1U ba mi6n BSc, I rung. Nam mfln ToAn _ Nguy6n Diic T l r T
Vay (1 + x')(l + y^) > (x + y)(l + xy), vdi moi x, y diTdng
2) Ta CO P = + = 2 + ab (a - b)' a + b ab Tab 5 i 1 a + b 2 ^ 2 2(a + b) 2 1 ab 2(a + b) + — 5 2 P j ; , - = (\/a - Vb j Ta CO (^yja - Sf > 0.
Ap dung bat dang thtfc Co-si cho hai so diTdng, ta c6
11' 0':>
(Va + yfhf > 4VaVb ; > —!—. Do ' 2Vab a + b
4N/^Vb 1 8
do
ab 1 > 8
ab
1 2(a + b) 15
2(a + b) 2^fab 2(a + b) a + b 2(a + b) 2(a + b) > 0
Do do P > - . Dau "=" xay ra o ^/a = Vb a = b
Nhan xet: 1), 2) la cac bai loan baft dang thuTc, ciTc tri c6 the diang baft dang thtfc Co-si cho hai so difdng dc giaị 6 bai III (2), dau "=" xay ra la a = b nen giup nghi den. ,
â + b^ N/ab â + b^ . vab 1 5+ = 2 + + - , tir do CO diTdc Idi giai bai toan. ' A ^>,• -'-u--. '
ab a 4 b ab a + b 2 2
CSu IV. a) Ve AÍ la dudng phan giac cua tarn giac ABC
, ÍB AB Ta CO .Ma AB = 2AC.fn Ta CO .Ma AB = 2AC.fn rc AC . „ ÍB . „ , ÍB IB - Nen = 2. Tarc CO = — -2 r = 1. rc ic i r c D
Vay Al la tia phan giac BAC^
b) Ve AD la difclng phan giac ngoai cua tarn giac ABC. Ta c6 DAI = 90" va
DR AR DR IR
— = — = 2.Vi — = — = 2 , B v a C co dinh. Nen D, I c6 dinh DC AC DC IC
Vay A di dong tren diTdng Iron co dinh di/cfng kinh DI
Cty TNHH MTV D W H Khang Vi^t
2) a) Ta co A, I, J thang hang; (I) tiep xuc vdi AC tai T, tiép xuc vdi AB lai S; (J) tiep xiic vdi AC tai L.
Ta CO IT ± AC, JL 1 AC => IT // JL AI IF AAJL CO IT // JL AJ JL Ta CO I D I B C , J E 1 B C ^ I D / / J L AI IF AAJE CO IF // JE AJ JE JL = JE (= R) Dod61T = l F = ^ F e (I) , , b) Then tinh chát cua hai tiep tuyen c^t nhau, ta c6
AS = AT, BS = BD, CD = CT. > Do do 2BD = AB + BC - AC. , r ,. ( , ,, , *
Mat khac (adsbygoogle = window.adsbygoogle || []).push({});
2CE = 2CL = 2(AL - AC) = 2AL - 2AC
= AB + BC -1- AC - 2AC = AB -i- BC - AC. g ;
Nen 2BD = 2CE => BD = CẸ Do do MD = ME
Ml la dUcJng trung binh cua tarn giac DFE => MI // AE " .
AADE CO MD = ME, MI // AE
Suy ra MI di qua trung diem cua AD .
Nhan xet: Day la cac bai toan quen thuoc dói vdi hoc sinh gioi toan Idp 9.
CSu V. Chia 625 so tif nhien 1; 2; ...; 625 thanh nhom: nhom 1 gom 1 va 624, nhom 2 gom 2 va 623, nhom 48 gom 48 va 577, nhom 49 gom 50 va 575, nhom 50 gom 51 va 574 nhom 223 gom 224 va 401, nhom 224 gom 226 va 399, nhom 225 gom 227 va 398,..., nhom 310 gom 312 va 313, nhom 311
gom cac so chinh phiTdng 49; 225; 576; 400; 625 (310 nhom deu gom hai so
CO long bang 625) ^ ,,,, ,,; • Neu trong 312 só dúdc chon c6 it nhat mot so thuoc nhom 311. Bai loan
da dúdc chufng minh.
• Neu trong 312 só duTdc chon khong c6 so nao thuoc nhom 311. Theo nguyen t^c Di-rich-le c6 it nhaft hai só diTdc chpn thuoc cung mot trong 310 nhom daụ Hai so nay co lóng b^ng 625. Mau ihuan gia ihiét
^ay bai loan da dúdc chiJng minh.
Nh§n xet: Day la bai loan van dung nguyen t^c Di-rich-lc thi/c ra dc loan
chi can "ChiJng minh rang trong 311 so diTdc chon, bao gid cung co it nhat