Baltimore County ARML Team Formula Sheet, v2.1 (08 Apr 2008) By Raymond Cheong. Difference of squares Difference of cubes Sum of cubes.

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1 Bltimoe Couty ARML Tem Fomul Seet, v. (08 Ap 008) By Rymo Ceog POLYNOMIALS Ftoig Diffeee of sques Diffeee of ues Sum of ues Ay itege O iteges ( )( ) 3 3 ( )( ) 3 3 ( )( ) ( )(... ) ( )(... ) Biomil expsio ( ) C C C... C C 0 Reltiosip etwee oots oeffiiets Qutis 4 x x 0 x ( x )( x ) x ( ) x egtive sum pout of oots of oots Cuis ( x )( x )( x 3 ) Geel x ( ) x ( ) x egtive sum of oots sum of oots tke egtive t time pout of oots x x x... x 0 0 (Viet s (... ) Teoem) 3... p (sum of oots tke p t time) p 0 ( )... Rtiol oot teoem Let x x x... x 0 0, wee ll oeffiiets e iteges. All tiol oots (if tey exist) e of te fom / wee e ftos of 0, espetively.

2 SEQUENCES Aitmeti sequees Coseutive tems ve te sme iffeee:,,, 3,..., ( ) ( tems) sum = (# of tems)(vege of fist lst tem) sum = (# of tems)(vege of ll tems) sum = (# of tems)(mei of ll tems) ( ) 3... fist iteges ( ) = fist o iteges Geometi sequees Coseutive tems ve te sme tio: 3,,,,..., ( tems) ( ) fiite sum =... ifiite sum =..., < sum of powes of = = + Ote sequees ( )( ) Sum of sques = ( ) Sum of ues =... 4 LOGARITHMS Bsi popeties Defiitio: log mes tt log log 0 log log log log m log m log log log m log log m log log log log

3 NUMBER THEORY Moul itmeti (mo m) mes tt leve te sme emie we ivie y m If (mo m) (mo m), te fo y itege : (mo m) (mo m) (mo m) (mo m) Femt s Little Teoem: If p is pime is eltively pime p to p te (mo p) Nume of ftos m If te pime ftoiztio of p p... p m, te s ( )( )...( m ) positive ftos. Defiitio of se A ume wit igits, -,,, 0 i se mes tt Divisiility ules Let k... 0 i se 0. lst igit ( 0 ) is eve 3 sum of igits ( ) is ivisile y 3 5 lst igit ( 0 ) is 0 o is ivisile y 7 (use itetively) 9 sum of igits ( ) is ivisile y 9 0 lst igit ( 0 ) is is ivisile y k ume fome y lst k igits e ivisile y k 0 k lst k igits e 0 Remie ules Let k... 0 i se 0., 5, 0 lst igit s sme emie 3, 9 sum of igits s sme emie s sme emie k, 0 k ume fome y lst k igits s sme emie

4 Comitois Pemuttio: ume of wys to oose items fom istit ojets wee iffeet oeigs e istit P!! Comitio: ume of wys to oose items fom istit ojets wee oe oes ot mtte C!!( )! Ptitio: ume of wys to goup ietil ojets ito m istit is, wit zeo items i i llowe Wo egemet: Nume of wys to ege te lettes of wo wit A A s, B B s,, Z Z s, lettes i totl. m m!!!!...! A B C Z Psl s tigle Oes ow te igt left sies. E ety is te sum of te two eties ove it. Sum of te t ow = E ety is omitio. Eties of te t ow give te oeffiiets of te t oe iomil expsio (Te ows e umee off sttig fom 0.) Pime ftoiztio of yes 936 = 44 = = 45 = 3 4 5

5 TRIANGLE GEOMETRY Ae B e = C A e = si C e = s( s )( s )( s ) wee s (semipeimete) s Equiltel tigle Ae = 3 s 4 Pytgoe Teoem Commo tiples: 3-4-5, 5--3, 7-0-, , multiples (e.g ) m -, m, m + wee m, e iteges is Pytgoe Tiple Tigoometi lws of tigles B C A Lw of sies: si A si B sic Lw of osies: os C Lw of tgets: t At Bt C t A t B t C Agle isetos

6 Meis Meis ivie e ote ito : segmets Te 6 little tigles ll ve equl e. Fmous tigle teoems m Stewt s Teoem m m e f Cev s Teoem e f f e Meelus Teoem e f

7 QUADRILATERAL GEOMETRY Pllelogm Def.: opposite sies e pllel. Popeties: - Opposite sies ve equl legts - Opposite gles e oguet C - Digols iset e ote - Digols fom two pis of simil tigles Ae = = si C Romus Def.: pllelogm w/ ll equl sies Popeties: - Digols p q e pepeiul. - Digols fom 4 oguet tigles q p Ae = ½ pq Tpezoi Def.: pi of opposite sies e pllel Popeties: Te two tigles fome y te igols ses e simil. Ae = ( ) Retgle Def.: pllelogm wit ll igt gles Popeties: Higly symmeti Ae = Sque Def.: etgle wit ll sies equl Popeties: Higly symmeti Ae = s s Ote Cevo (left): Symmeti ove Kite (igt): Pepeiul igols, oe of wi isets te ote

8 Bitis Flg Teoem P Ay poit P (isie, outsie, o, ove, o elow etgle): Cyli quiltels pq (Ptolemy s Teoem) q p e = ( s )( s )( s )( s ) wee s (semipeimete) 3D & POLYGON GEOMETRY Retgul pllelpipe Volume = Itel igol = Geelize ylie Volume = K (K is te e of te se) K Geelize pymi Volume = 3 K (K is te e of te se) K Spee 4 3 Volume = 3 Sufe e = 4 Polygos Sum of iteio gles = 80( ) (-sie ovex polygo) s Ae = (-sie egul polygo) 4 t(80 / )

9 CIRCLE GEOMETRY Bsi popeties of iles O (x 0,y 0 ) Ae = Ciumfeee = Equtio: ( x x ) ( y y ) 0 0 Powe of poit teoem, I e igm, = Agles fome y os, sets, tgets Agle is lf te fome

10 TRIGONOMETRY Defiitios is = 80º si os si t os s si se os ot t Commo gles 0º 30º 45º 60º 90º 80º si os t o - 0 Pol vs. Ctesi ooites P (x,y) x os t y/ x y si x y Pytgoe Teoem si os ot s t se Doule gle fomuls si si os os os si si os t t t Sum iffeee fomuls si( ) si os os si si( ) si os os si os( ) os os si si os( ) os os si si t t t t t( ) t( ) t t t t

ALGEBRA. ( ) is a point on the line ( ) + ( ) = + ( ) + + ) + ( Distance Formula The distance d between two points x, y

ALGEBRA. ( ) is a point on the line ( ) + ( ) = + ( ) + + ) + ( Distance Formula The distance d between two points x, y ALGEBRA Popeties of Asoute Vue Fo e umes : 0, 0 + + Tige Iequity Popeties of Itege Epoets Ris Assume tt m e positive iteges, tt e oegtive, tt eomitos e ozeo. See Appeies B D fo gps fute isussio. + ( )