id	topic_id	thread_id	type	formula
q_253	A.31	3082747	question	"V!p	→	n	- V!p	V!q	nn	- →	V!q	n	n V!q	0!	n	nn"
q_254	A.31	3082747	question	"V!p	0!	n	-"
q_255	A.31	3082747	question	"V!q	0!	n	-"
q_256	A.31	3082747	question	"V!p	→	n	- V!p	V!q	nn	- →	V!q	n	n V!q	0!	n	nn"
q_257	A.31	3082747	question	"V!p	0!	n	-"
q_258	A.31	3082747	question	"V!q	0!	n	-"
q_259	A.31	3082747	question	"V!p	0!	n	-"
q_260	A.31	3082747	question	"V!q	0!	n	-"
q_261	A.31	3082747	question	"V!p	→	n	- V!p	V!q	nn	- →	V!q	n	n V!q	0!	n	nn"
q_730	A.78	3237032	title	"∑	V!∞	o	- V!∞	0!	n	o ∑	V!a	n	- ∑	V!n	nb	- V!a	V!n	b	n V!n	0!	n	nb V!a	0!	n	n ∑	V!n	u	- ∑	=	un	- V!n	=	n	u V!n	N!1	nn	u =	N!1	n	un N!1	0!	n	unn"
q_731	A.78	3237032	title	"∑	V!∞	o	- V!∞	0!	n	o ∑	V!a	n	- ∑	V!n	nb	- ∑	N!2	na	- V!a	N!2	a	n N!2	0!	n	na V!a	V!n	b	n V!n	0!	n	nb V!a	0!	n	n ∑	V!n	u	- ∑	=	un	- V!n	=	n	u V!n	N!1	nn	u =	N!1	n	un N!1	0!	n	unn"
q_732	A.78	3237032	question	"V!ε	V!&	n	- V!ε	V!g	nn	- V!&	V!g	n	n V!&	V!t	nn	n V!g	V!t	n	nn V!g	;	nn	nn V!t	;	n	nnn V!t	N!0	nn	nnn ;	N!0	n	nnnn N!0	0!	n	nnnnn"
q_733	A.78	3237032	question	"∑	V!a	n	- ∑	V!n	nb	- V!a	V!n	b	n V!n	0!	n	nb V!a	0!	n	n"
q_734	A.78	3237032	question	"V!n	V!&	n	- V!n	V!g	nn	- V!&	V!g	n	n V!&	V!t	nn	n V!g	V!t	n	nn V!g	;	nn	nn V!t	;	n	nnn V!t	V!m	nn	nnn ;	V!m	n	nnnn ;	≥	nn	nnnn V!m	≥	n	nnnnn V!m	V!N	nn	nnnnn ≥	V!N	n	6n V!N	0!	n	7n"
q_735	A.78	3237032	question	"|	∑	n	- |	V!k	nu	- |	V!n	no	- |	|	nn	- ∑	V!n	o	n V!n	0!	n	no ∑	|	n	n ∑	V!a	nn	n |	V!a	n	nn |	V!k	nb	nn |	|	nn	nn V!a	|	n	nnn V!a	|	nn	nnn |	|	n	nnnn |	=	nn	nnnn |	=	n	nnnnn |	∑	nn	nnnnn =	∑	n	6n =	V!k	nu	6n =	V!n	no	6n =	|	nn	6n ∑	V!n	o	7n V!n	0!	n	7n1o ∑	|	n	7n ∑	V!a	nn	7n |	V!a	n	8n |	V!k	nb	8n |	|	nn	8n V!a	|	n	9n V!a	V!&	nn	9n |	V!&	n	10n |	V!l	nn	10n V!&	V!l	n	11n V!&	V!t	nn	11n V!l	V!t	n	12n V!l	;	nn	12n V!t	;	n	13n V!t	O!root	nn	13n ;	O!root	n	14n ;	V!ε	nw	14n O!root	V!ε	w	15n V!ε	0!	n	15n1w O!root	0!	n	15n V!a	V!k	b	9n V!k	0!	n	9n1b ∑	V!k	u	7n ∑	=	un	7n V!k	=	n	7n1u V!k	V!m	nn	7n1u =	V!m	n	7n1u1n =	+	nn	7n1u1n V!m	+	n	7n1u2n V!m	N!1	nn	7n1u2n +	N!1	n	7n1u3n N!1	0!	n	7n1u4n V!a	V!k	b	nnn V!k	0!	n	nnnb ∑	V!k	u	n ∑	=	un	n V!k	=	n	nu V!k	V!m	nn	nu =	V!m	n	nun =	+	nn	nun V!m	+	n	nunn V!m	N!1	nn	nunn +	N!1	n	nunnn N!1	0!	n	1n1u4n"
q_736	A.78	3237032	question	"V!n	V!&	n	- V!n	V!g	nn	- V!&	V!g	n	n V!&	V!t	nn	n V!g	V!t	n	nn V!g	;	nn	nn V!t	;	n	nnn V!t	V!m	nn	nnn ;	V!m	n	nnnn ;	≥	nn	nnnn V!m	≥	n	nnnnn V!m	V!N	nn	nnnnn ≥	V!N	n	6n V!N	0!	n	7n"
q_737	A.78	3237032	question	"|	∑	n	- |	V!k	nu	- |	V!n	no	- |	V!a	nn	- ∑	V!n	o	n V!n	0!	n	no ∑	V!a	n	n ∑	V!k	nb	n ∑	N!2	na	n ∑	|	nn	n V!a	N!2	a	nn N!2	0!	n	nna V!a	|	n	nn V!a	=	nn	nn |	=	n	nnn |	∑	nn	nnn =	∑	n	nnnn =	V!k	nu	nnnn =	V!n	no	nnnn =	V!a	nn	nnnn ∑	V!n	o	nnnnn V!n	0!	n	5n1o ∑	V!a	n	nnnnn ∑	V!k	nb	nnnnn ∑	N!2	na	nnnnn ∑	≤	nn	nnnnn V!a	N!2	a	6n N!2	0!	n	6n1a V!a	≤	n	6n V!a	M!()1x1	nn	6n ≤	M!()1x1	n	7n ≤	∑	nw	7n ≤	N!2	na	7n ≤	V!&	nn	7n M!()1x1	N!2	a	8n N!2	0!	n	8n1a M!()1x1	V!&	n	8n M!()1x1	V!l	nn	8n V!&	V!l	n	9n V!&	V!t	nn	9n V!l	V!t	n	10n V!l	;	nn	10n V!t	;	n	11n V!t	O!root	nn	11n ;	O!root	n	12n ;	V!ε	nw	12n ;	N!2	na	12n ;	=	nn	12n O!root	N!2	a	13n N!2	0!	n	13n1a O!root	=	n	13n O!root	V!ε	nn	13n =	V!ε	n	14n V!ε	0!	n	15n O!root	V!ε	w	13n V!ε	0!	n	13n1w M!()1x1	∑	w	8n M!()1x1	V!k	wu	8n M!()1x1	V!n	wo	8n M!()1x1	|	wn	8n ∑	V!n	o	8n1w V!n	0!	n	8n1w1o ∑	|	n	8n1w ∑	V!a	nn	8n1w |	V!a	n	8n1w1n |	V!k	nb	8n1w1n |	|	nn	8n1w1n V!a	|	n	8n1w2n |	0!	n	8n1w3n V!a	V!k	b	8n1w2n V!k	0!	n	8n1w2n1b ∑	V!k	u	8n1w ∑	=	un	8n1w V!k	=	n	8n1w1u V!k	V!m	nn	8n1w1u =	V!m	n	8n1w1u1n =	+	nn	8n1w1u1n V!m	+	n	8n1w1u2n V!m	N!1	nn	8n1w1u2n +	N!1	n	8n1w1u3n N!1	0!	n	8n1w1u4n V!a	V!k	b	6n V!k	0!	n	6n1b ∑	V!k	u	nnnnn ∑	=	un	nnnnn V!k	=	n	5n1u V!k	V!m	nn	5n1u =	V!m	n	5n1u1n =	+	nn	5n1u1n V!m	+	n	5n1u2n V!m	N!1	nn	5n1u2n +	N!1	n	5n1u3n N!1	0!	n	5n1u4n V!a	V!k	b	nn V!k	0!	n	nnb ∑	V!k	u	n ∑	=	un	n V!k	=	n	nu V!k	V!m	nn	nu =	V!m	n	nun =	+	nn	nun V!m	+	n	nunn V!m	N!1	nn	nunn +	N!1	n	nunnn N!1	0!	n	1n1u4n"
q_738	A.78	3237032	question	"∑	V!a	n	- ∑	V!n	nb	- ∑	N!2	na	- V!a	N!2	a	n N!2	0!	n	na V!a	V!n	b	n V!n	0!	n	nb V!a	0!	n	n"
q_1027	A.101	3163489	question	"V!n	N!2	a	- N!2	0!	n	a V!n	≤	n	- V!n	N!2	nn	- ≤	N!2	n	n ≤	V!n	na	n N!2	V!n	a	nn V!n	0!	n	nna N!2	0!	n	nn"
q_1028	A.101	3163489	question	"∀	V!n	n	- ∀	∈	nn	- V!n	∈	n	n V!n	V!N	nn	n ∈	V!N	n	nn ∈	,	nn	nn V!N	,	n	nnn V!N	V!n	nn	nnn ,	V!n	n	nnnn ,	V!&	nn	nnnn V!n	V!&	n	nnnnn V!n	V!g	nn	nnnnn V!&	V!g	n	6n V!&	V!t	nn	6n V!g	V!t	n	7n V!g	;	nn	7n V!t	;	n	8n V!t	N!3	nn	8n ;	N!3	n	9n N!3	0!	n	10n"
q_1029	A.101	3163489	question	"V!n	=	n	- V!n	N!4	nn	- =	N!4	n	n N!4	0!	n	nn"
q_1030	A.101	3163489	question	"V!k	0!	n	-"
q_1031	A.101	3163489	question	"V!n	=	n	- V!n	V!k	nn	- =	V!k	n	n V!k	0!	n	nn"
q_1032	A.101	3163489	question	"V!k	N!2	a	- N!2	0!	n	a V!k	≤	n	- V!k	N!2	nn	- ≤	N!2	n	n ≤	V!k	na	n N!2	V!k	a	nn V!k	0!	n	nna N!2	0!	n	nn"
q_1033	A.101	3163489	question	"M!()1x1	N!2	a	- N!2	0!	n	a M!()1x1	≤	n	- M!()1x1	N!2	nn	- ≤	N!2	n	n ≤	V!k	na	n N!2	V!k	a	nn N!2	+	an	nn V!k	+	n	nna V!k	N!1	nn	nna +	N!1	n	nnan N!1	0!	n	nnann N!2	0!	n	nn M!()1x1	V!k	w	- M!()1x1	+	wn	- V!k	+	n	w V!k	N!1	nn	w +	N!1	n	wn N!1	0!	n	wnn"
q_1034	A.101	3163489	question	"=	V!k	n	- =	N!2	na	- =	≤	nn	- V!k	N!2	a	n N!2	0!	n	na V!k	≤	n	n V!k	N!2	nn	n ≤	N!2	n	nn ≤	V!k	na	nn N!2	V!k	a	nnn V!k	0!	n	nnna N!2	0!	n	nnn"
q_1035	A.101	3163489	question	"=	N!2	n	- =	V!k	nn	- N!2	V!k	n	n N!2	N!2	na	n N!2	≤	nn	n V!k	N!2	a	nn N!2	0!	n	nna V!k	≤	n	nn V!k	N!2	nn	nn ≤	N!2	n	nnn ≤	V!k	na	nnn N!2	V!k	a	nnnn N!2	+	an	nnnn V!k	+	n	nnnna V!k	N!1	nn	nnnna +	N!1	n	4n1a1n N!1	0!	n	4n1a2n N!2	0!	n	nnnn"
q_1036	A.101	3163489	question	"M!()1x1	N!2	a	- N!2	0!	n	a M!()1x1	≤	n	- M!()1x1	N!2	nn	- ≤	N!2	n	n ≤	V!k	nn	n N!2	V!k	n	nn N!2	N!2	na	nn V!k	N!2	a	nnn N!2	0!	n	nnna V!k	0!	n	nnn M!()1x1	V!k	w	- M!()1x1	+	wn	- V!k	+	n	w V!k	N!1	nn	w +	N!1	n	wn N!1	0!	n	wnn"
q_1037	A.101	3163489	question	"M!()1x1	N!2	a	- N!2	0!	n	a M!()1x1	≤	n	- M!()1x1	N!2	nn	- ≤	N!2	n	n ≤	V!k	na	n N!2	V!k	a	nn N!2	+	an	nn V!k	+	n	nna V!k	N!1	nn	nna +	N!1	n	nnan N!1	0!	n	nnann N!2	0!	n	nn M!()1x1	V!k	w	- M!()1x1	+	wn	- V!k	+	n	w V!k	N!1	nn	w +	N!1	n	wn N!1	0!	n	wnn"
q_1038	A.101	3163489	question	"M!()1x1	N!2	a	- N!2	0!	n	a M!()1x1	≤	n	- M!()1x1	N!2	nn	- ≤	N!2	n	n ≤	V!k	nn	n N!2	V!k	n	nn N!2	N!2	na	nn V!k	N!2	a	nnn N!2	0!	n	nnna V!k	0!	n	nnn M!()1x1	V!k	w	- M!()1x1	+	wn	- V!k	+	n	w V!k	N!1	nn	w +	N!1	n	wn N!1	0!	n	wnn"
q_1039	A.101	3163489	question	"=	V!k	n	- =	N!2	na	- =	+	nn	- V!k	N!2	a	n N!2	0!	n	na V!k	+	n	n V!k	N!2	nn	n +	N!2	n	nn +	V!k	nn	nn N!2	V!k	n	nnn N!2	+	nn	nnn V!k	+	n	nnnn V!k	N!1	nn	nnnn +	N!1	n	nnnnn +	≤	nn	nnnnn N!1	≤	n	6n N!1	N!2	nn	6n ≤	N!2	n	7n ≤	V!k	nn	7n N!2	V!k	n	8n N!2	N!2	na	8n V!k	N!2	a	9n N!2	0!	n	9n1a V!k	0!	n	9n"
q_1040	A.101	3163489	question	"=	N!2	n	- =	V!k	nn	- N!2	V!k	n	n N!2	+	nn	n V!k	+	n	nn V!k	N!1	nn	nn +	N!1	n	nnn +	≤	nn	nnn N!1	≤	n	nnnn N!1	V!k	nn	nnnn ≤	V!k	n	nnnnn ≤	N!2	na	nnnnn V!k	N!2	a	6n N!2	0!	n	6n1a V!k	0!	n	6n"
q_1041	A.101	3163489	question	"=	N!2	n	- =	V!k	nn	- N!2	V!k	n	n N!2	+	nn	n V!k	+	n	nn V!k	N!1	nn	nn +	N!1	n	nnn +	≤	nn	nnn N!1	≤	n	nnnn N!1	N!2	nn	nnnn ≤	N!2	n	nnnnn ≤	V!k	na	nnnnn N!2	V!k	a	6n V!k	0!	n	6n1a N!2	0!	n	6n"
q_1042	A.101	3163489	question	"M!()1x1	N!2	a	- N!2	0!	n	a M!()1x1	≤	n	- M!()1x1	N!2	nn	- ≤	N!2	n	n ≤	V!k	nn	n N!2	V!k	n	nn N!2	N!2	na	nn V!k	N!2	a	nnn N!2	0!	n	nnna V!k	0!	n	nnn M!()1x1	V!k	w	- M!()1x1	+	wn	- V!k	+	n	w V!k	N!1	nn	w +	N!1	n	wn N!1	0!	n	wnn"
q_1043	A.101	3163489	question	"V!n	=	n	- V!n	N!4	nn	- =	N!4	n	n N!4	0!	n	nn"
q_1044	A.101	3163489	question	"V!j	,	n	- V!j	V!k	nn	- ,	V!k	n	n ,	=	nn	n V!k	=	n	nn V!k	V!j	nn	nn =	V!j	n	nnn V!j	0!	n	nnnn"
q_1045	A.101	3163489	question	"N!2	V!k	n	- N!2	+	nn	- V!k	+	n	n V!k	N!1	nn	n +	N!1	n	nn +	≤	nn	nn N!1	≤	n	nnn N!1	N!2	nn	nnn ≤	N!2	n	nnnn ≤	V!k	na	nnnn N!2	V!k	a	nnnnn V!k	0!	n	5n1a N!2	0!	n	nnnnn"