x * y / z | |
x / (y * z) x / y / z | |
(a**3 + b**3) / (b*c) | |
(a[i+1] + b[i-1]) / (2*x*y) | |
(-b + sqrt(b*b - 4*a*c)) / (2*a) | |
(x<0) | sign(x) * abs(x) ** (1/5) |
0.49 * exp(a*a - b*b) + ln(cos(a*a)) ** 3 | |
x/(1 + x*x/(3 + (2*x)**3)) |
:
5x + 1 a + sin x ((a + b)/c**3 | 5 x sin x |
7.21. ?
, , , < (), <= ( ), > (), >= ( ), = (), <> ( ), , , .
, .
a | int(a) = 0 |
a | mod(a, 2) = 0 |
a | mod(a, 2) = 1 |
k | mod(a, 7) = 0 |
a, b | (a>0) (b>0) |
a, b | ((a>0) (b<=0)) ((a<=0) (b>0)) |
a, b, c | (a<0) (b<0) (c<0) |
x a < x < b | (x>a) (x<b) |
x [1, 3] | (x>=1) (x<=3) |
a b | ((mod(a, 2)=0) (mod(b, 2)=0) ((mod(a, 2)=1) (mod(b, 2)=1)) |
(x, y) r (a, b) | (x-a)**2 + (y-b)**2 < r*r |
ax^2 + bx + c = 0 | b*b - 4*a*c < 0 |
(x, y) | ((x>0) (y>0)) ((x<0) (y>0)) |
(x, y) | (x*x + y*y > 1) ((x*x + y*y <= 1) (x<0) (y>0)) |
a b | a = -b |
a b | a*b = 1 |
a b, c, d | a > (b+c+d) / 3 |
a b, c, d | a >= (b+c+d) ** (1/3) |
F1 F2 | F1 F2 |
F1 F2 | F1 F2 |
F1 F2 | F1 F2 |
F1 , F2 | F1 F2 |
F1 F2 | (F1 F2) (F2 F1) |
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7.1. :
a) | e) | ||
) | ) | ||
) | ) | ||
) | ) | ||
) | ) |
7.2. :
) a / b ** 2; ) a+b/c+1; ) 1/a*b/c; ) a**b**c/2; ) (a**b)**c/2; ) a/b/c/d*p*q; ) x**y**z/a/b; ) 4/3*3.14*r**3; ) b/sqrt(a*a+b); ) d*c/2/R+a**3; | ) 5*arctg(x)-arctg(y)/4; ) lg(u*(1/3)+sqrt(v)+z); ) ln(y*(-sqrt(abs(x)))); ) abs(x**(y/x)-(y/x)**(1/3)); ) sqrt((x1-x2)**2+(y1-y2)**2); ) exp(abs(x-y))*(tg(z)**2+1)**x; c) lg(sqrt(exp(x-y))+x**abs(y)+z); ) sqrt(exp(a*x)*sin(x)**n)/cos(x)**2; ) sqrt(sin(arctg(u))**2+abs(cos(v))); ) abs(cos(x)+cos(y))**(1+sin(y)**2); |
7.3. x=1:
) abs(x-3)/ln(exp(3))*2/lg(10000);
: abs(1-3)=2; ln(exp(3))=3; lg(10000)=4; 2/3*2/4=0.33;
) sign(sqrt(sqrt(x+15)))*2**2**2;
) int(-2.1)*int(-2.9)/int(2.9)+x;
) -sqrt(x+3)**2**(sign(x+0.5)*3)+tg(0);
) lg(x)+cos(x**2-1)*sqrt(x+8)-div(2,5);
) sign(x-2)*sqrt(int(4.3))/abs(min(2,-1));
) div(10,x+2)*mod(10,x+6)/max(10,x)*mod(2,5).
7.4. , :
) a, b, c (a, b, c >0) p;
: sqrt(p*(p-a)*(p-b)*(p-c));
) a, b, c, d;
) (x,y) (0,0);
) x ;
) ( );
) ( );
) ,
a1x+b1y+c1=0 a2x+b2y+c2=0 ( ).
7.5. :
) x*x+y*y<=9 x=1, y=-2
: ;
) b*b-4*a*c<0 a=2, b=1, c=-2;
) (a>=1) (a<=2) a=1.5;
) (a<1) (a>1.2) a=1.5;
) (mod(a,7)=1) (div(a,7)=1) a=8;
) ((a>b) (a<9) (*=4)) a=5, b=4.
7.6. , :
) x [ a, b ]
: (x>=a) (x<=b);
) x [ a, b ];
) x [ a, b ] [ c, d ];
) x [ a, b ] [ c, d ];
) k ;
) k , ;
) ai,j ;
) a1x+b1y+c1=0 a2x+b2y+c2=0 ;
) a, b, c , b;
) a, b, c, d ;
) a, b, c ;
) a, b, c ;
) a1, b1, c1 a2, b2, c2 ;
) (x,y) A (0,5), B (5,0) C (1,0);
) (x,y) , A (0,5), B (1,0) C (5,0);
) a, b, c d .
7.7. (x,y) , . , , .
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) (x<=0) (y>=0) : | ) ((x-2)**2+y*y<=4) (y>x/2) : |
) (x>=0) (y<=0) ) x+y>=0 ) (x+y>0) (y<0) ) abs(x)+abs(y)>=1 | ) (x*x+y*y<1) (y>x*x); ) (y>=x) (y+x>=0) (y<=1); ) (abs(x)<=1) (y<2); ) (x**2+y**2<4) (x**2+y**2>1); |
7.8. , "" , (x, y) .
7.9. a =3, b =5, c =7. :
) a:=a+1; b:=a+b; c:=a+b; a:=sqrt(a)
: a =3+1=4, b =4+5=9, c =4+9=13, a = { } 4 =2.
: =2, b =9, c =13;
) :=a*b+2; b:=b+1; a:=c-b**2; b:=b*a;
) b:=b+a; c:=c+b; b:=1/b*c;
) p:=c; c:=b; b:=a; a:=p; c:=a*b*c*p;
) c:=a**(b-3); b:=b-3; a:=(c+1)/2*b; c:=(a+b)*a;
) x:=a; a:=b; b:=c; c:=x; a:=sqrt(a+b+c+x-2);
) b:=(a+c)**2; a:=lg(b**2)**2; c:=c*a*b.
7.10. :
) X=(x1, x2) : x1 , 2 ;
: c:=x[1]; x[1]:=x[1]+x[2]; x[2]:=c*x[2]
) X=(x1, x2);
) A(N) i (1<i<N) , , 0.5;
) u = max(x, y, z) + min(x-z, y+z, y, z);
7.11. :
a) | ||
) | ||
) | ||
) | ||
) | ||
) | ||
) | r (r>0) (a,b) |
7.12. y(x), :
) x<=-1 y:=1/x**2 x<=2 y:=x*x y:=4 | ) x<-0.5 y:=1/abs(x) x<1 y:=2 y:=1/(x-0.5) |
) x<0 y:=1 x<3.14 y:=cos(x) y:=-1 | |
) x<-5 y:=-5 x<0 y:=x x<3 y:=2*x y:=6 | ) abs(x)>2 y:=x*x x<0 y:=-2*x x>=1 y:=4 y:=4*x*x |
7.13. S :
) S:=128 i 1 4 S:=div(S,2) |
: S=8 | ) S:=0 i 1 2 j 2 3 S:=S+i+j |
: S=16 | ||||||||||||||||||||||||||||||
) S:=1; a:=1 i 1 3 S:=S+i*(i+1)*a a:=a+2 | ) i 1 3 S:=0 j 2 3 S:=S+i+j | ||||||||||||||||||||||||||||||||
) S:=1; a:=1 i 1 3 S:= S+i j o 2 3 S:= S+j | ) i 1 2 S:= 0 j o 2 3 k o 1 2 S:= S+i+j+k |
7.14. S :
) i:=0; S:=0 i<3 i:=i+1; S:=S+i*i | ) S:=0; N:=125 N>0 S:=S+mod(N,10) | S N:=div(N,10) | N | ||||||||||||||||||||||||||||||||||||||
: S=14 |
: S=8
| ||||||||||||||||||||||||||||||||||||||
) S:=0; i:=1 i>1 S:=S+1/i i:=i-1 | ) :=1; b:=1; S:=0; a<=5 a:=a+b; b:=b+a; S:=S+a+b | ||||||||||||||||||||||||||||||||||||||
) S:=0; i:=1; j:=5 i<j S:=S+i*j i:=i+1 j:=j-1 | ) a:=1; b:=1 a+b<10 a:=a+1 b:=b+a S:=a+b |
7.15. ( .. , .. .. " ", 1989):
) a, b c; ( ) , :
=180o-(+). |
. , arccos arcsin .
:
( a,b,c, UgolA,UgolB,UgolC) RadGr,UgolARad | RadGr . | UgolARad A ( ) RadGr:=180/3.14 UgolARad:=ArcCos((b*b+c*c-a*a)/(2*b*c)) UgolA:=UgolARad*RadGr UgolB:=ArcSin(b*sin(UgolARad)/a)*RadGr UgolC:=180-(UgolA+UgolB)) a, b C ( ) ; c, A B ( ) , :
2 = a2 + b2 - 2ab cos C.
. c, ;
) a, b c; A ( ), :
) a A ( ) ; , :
V=Soc H/2; | ||
) R r A ( ) ; , :
e) a, A; , d; :
7.16. :
) , a, b, c ;
:
( a,b,c, Otvet) | a>0, b>0, c>0, a+b>c, a+c>b, b+c>a | Otvet = , | Otvet = , (a=b) (a=c) (b=c) Otvet:= Otvet:=) a, b c;
) , ;
) a b , c d ; , ;
) ; , ;
) , (x, y) , , r1 r2;
) a, b c.
7. . .
7.1.
) (x+y)/(x-1/2)-(x-z)/(x*y); ) (1+z)*(x+y/z)/(a-1/(1+x*x)); ) x**(n*(m+2)) + x**(n**m); ) (a+b)**n/(1+a/(a**m-b**(m-n))); ) (a[i]**(2*l) + b[j+1]**(2*k)) * (3**n-x*x*y)/(z-(d[i,j+1]+1)/(z+ y/sqrt(t*t+x*y*z))); ) sqrt(abs(sin(x)**2))/(3.01*x - exp(2*x)); ) abs(cos(x**3) - sin(y)**2) / (abs(ln(x))**(1/4) + x*y); ) ln(y**(-sqrt(abs(x+1)))) * sin(arctg(z))**2; ) r[i,j]**abs(x-y) - 0.15*abs(sin(exp(-z**8))); ) a**((x+y)/2) - ((x-1)/(abs(y)+1))**(1/3)*exp(-(y+u/2)).7.2. ) ; ) ; ) ; ) ; ) ; ) ; ) ; ) ; ) ; ) ; ) ; ) ; ) ; ) ; ) ; ) ; ) ; ) ; ) ; ) .
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7.3. ) 16; ) 5,5; ) -256; ) 3; ) -2; ) 1.8.
7.4. ) : (a+b+c+d)/4; : (a*b*c*d)**(1/4); ) sqrt(x*x+y*y); ) sin(x*3.14/180); ) 6*a*a; ) sqrt(3)*a/2; ) : (c1*b2-c2*b1)/(b1*a2-b2*a1); : (c2*a1-c1*a2)/(b1*a2-b2*a1).
7.5. ) ; ) ; ) ; ) ; ) ;
7.6.
) (x < a) (x > b); ) ((x>=a) (x<=b)) ((x>=c) (x<=d)); ) ((x < a) (x > b)) ((x < c) (x > d)); ) mod(k,2)=1; ) (mod(k,5)=0) (k > 99) (k < 1000); ) (mod(i,2)=1) (mod(j,2)=0); ) a1*b2=a2*b1; ) (c < a) (b > a); ) (a=-b) (a=-c) (a=-d) (b=-c) (b=-d) (c=-d); ) ((mod(a,2)=0) (mod(b,2)=0)) ((mod(a,2)=0) (mod(c,2)=0)) ((mod(b,2)=0) (mod(,2)=0)); ) (a>0) (b>0) (c>0) (a+b>c) (a+c>b) (b+c>a); ) ((a1*b2=a2*b1) (a1*c2=a2*c1)) ((a1*c2=a2*b1) (a1*b2=a2*c1)) ((a1*c2=b2*b1) (a1*a2=b2*c1)) ((a1*a2=b2*b1) (a1*c2=b2*c1)) ((a1*a2=c2*b1) (a1*b2=c2*c1)) ((a1*b2=c2*b1) (a1*a2=c2*c1)); ) (y>5-5*x) (y<5-x) (y>0); ) (y<5-5*x)) (y>5-x) (y<0); ) (a=b) (c=d) (b=c).7.7. ) ) ) ) ) ) ) )
7.8.
) (y>=1-x) ((y<=0) (x<=0)); ) (y<1) (y>=x) (y>=-x) ( : (y<1) (y>=abs(x))); ) (abs(x)<=1) (abs(y)<1); ) ((x-1)*(x-1)+y*y<=4) (y<=3-x) (y>=x-3); ) (abs(x)<=3) (abs(y)<=3) (x*x+y*y>=9) ((x>=0) (y<=0)); ) (abs(x)+abs(y)<=2) (sign(x)<>sign(y)) (x*x+y*y<=4) ((y>=2-x) (y<=-x-2)); ) ((y>=x*x) (y<=-x*x)) ((x>=y*y) (x<=-y*y)); ) (((x+2)*(x+2)+y*y<=4) ((x+2)*(x+2)+y*y>=1)) (x>=-1); ) (((y<=0)=(y>=-x)) ((x>=0)=(y>=x))) (x*x+y*y<=1).7.9. ) a=-19; b=-114; c=17; ) a=3; b=1,875; c=15; ) a=7; b=3; c=735; ) a=10; b=2; c=120; ) a=4; b=7; c=3; ) a=16; b=100, c=11200.
7.10. ) c:=x[1]; x[1]:=x[2]; x[2]:=c; ) a[i]:=(a[i-1]+a[i+1])/2; a[i+1]:=0; a[i-1]:=a[i-1]+0.5; ) u:=max(max(x, y), z) + min(min(x-z,y+z), min(y,z)).
7.11.
) x <= -100 ) x*x+y*y <= 1 y:=sign(x)*abs(x)**(1/7) z:=x*x+y*y x < 100 y>=x y:=sign(x)*abs(x)**(1/3) z:=x+y y:=sqrt(x) z:=0.5 ) x < 0 ) z:=lg(-x) =0: z:=1 z:=sqrt(x+1) =1: z:=x =2: z:=3*x*x - 1/2 z>=0 =3: z:=x*x*x - 3*x/2 F:=2*z+1 z:=2*x**4 - 3*x/2 F:=sin(z) ) abs(x)+abs(y) < r ) x>1 z:=sqrt(x*x+y*y) y>1 z:=max(abs(x), abs(y)) v:=x+y v:=x-y ) (x-a)**2 +(y-b)**2 < r*r y>0 z:=abs(x)+abs(y) v:=y-x z:=x+y v:=-x-y7.12. ) ) ) )
7.13. ) 81; ) 21; ) 11; ) 44.
7.14. ) 0; ) 13; ) 52; ) 14.
7.15.
) 1( a,b,UgolC, c, UgolA, UgolB, S) a, b, UgolC c:=sqrt(a*a+b*b-2*a*b*cos(UgolC)) UgolA:=arcsin(a*sin(UgolC)/c) UgolB:=arcsin(b*sin(UgolC)/c) S:=b*c*sin(UgolA)/2 c, UgolA, UgolB, S ) 2( a,b,c, Radius,UgolA) p a,b,c p:=(a+b+c)/2 UgolA:=2*arctg(sqrt((p-b)*(p-c)/(p*(p-a))))*180/3.14 Radius:=a*b*c/(4*sqrt(p*(p-a)*(p-b)*(p-c))) Radius, UgolA ) ( a,UgolAGrad, V, S) H,SBase,UgolARad | H - ; SBase - a,UgolAGrad UgolARad:=UgolAGrad*3.14/180 SBase:=a*a*sqrt(3)/4 H:=a*sqrt(3)/6*tg(UgolARad) V:=SBase*H/3 S:=SBase*(1+1/cos(UgolARad)) V, S ) ( RBig,RSmall,Ugol, V, S) H,L RBig,RSmall,Ugol H:=(RBig-RSmall)*tg(Ugol) L:=(RBig-RadSmall)/cos(Ugol) V:=1/3*3.14*H*(RSmall**2 + RBig**2 + RSmall*RBig) S:=3.14*L*(RBig+RSmall) V, S ) ( a,UgolA, V, S, Stion) H a,UgolA H:=a*sqrt(2)/2*tg(UgolA) V:=1/3*a*a*H Stion:=a*H*sqrt(2)/2 S:=a*a*(1+sqrt(2*tg(UgolA)**2+1)) V, S, Stion7.16.
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