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Beyond Advanced · Vector Algebra

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1
In a triangle ABCABC, AB⃗=u⃗\vec{AB}=\vec u, AC⃗=v⃗\vec{AC}=\vec v, ∣u⃗∣2=13|\vec u|^2=13, ∣v⃗∣2=10|\vec v|^2=10, and u⃗⋅v⃗=5\vec u\cdot\vec v=5. Points P,QP,Q satisfy AP:PB=2:1AP:PB=2:1 internally and AQ:QC=3:1AQ:QC=3:1 externally. A point XX lies on the entire line PQPQ. What is the minimum of ∣AX⃗∣2|\vec{AX}|^2?
Not attempted
ABCPQ
A
1890329\frac{1890}{329}
B
945329\frac{945}{329}
C
529\frac{52}{9}
D
1890161\frac{1890}{161}
2
Let ABCABC be any nondegenerate triangle. Put P=(2B+C)/3P=(2B+C)/3, Q=(3C+A)/4Q=(3C+A)/4, and R=(4A+B)/5R=(4A+B)/5, with the expressions denoting position vectors. The lines AP,BQ,CRAP,BQ,CR form a triangle of pairwise intersections. What fraction of the area of ABCABC does this triangle occupy?
Not attempted
ABCPQR
A
5293740\frac{529}{3740}
B
5291870\frac{529}{1870}
C
529935\frac{529}{935}
D
17110\frac{17}{110}
3
Four vectors a⃗,b⃗,c⃗,d⃗\vec a,\vec b,\vec c,\vec d have lengths 2,3,5,82,3,5,8 respectively and sum to zero. What is the complete range of ∣a⃗+b⃗∣|\vec a+\vec b|?
Not attempted
A
[1,5][1,5]
B
[3,13][3,13]
C
[3,5][3,5]
D
[1,13][1,13]
4
Noncoplanar vectors a⃗,b⃗,c⃗\vec a,\vec b,\vec c are position vectors of A,B,CA,B,C. A point DD has position vector 2a⃗−3b⃗+2c⃗2\vec a-3\vec b+2\vec c. Let GG be the centroid of ABCABC. The line DGDG meets the plane through O,A,BO,A,B at XX, where OO is the origin. Which is OX⃗\vec{OX}?
Not attempted
OABCGD
A
a⃗\vec a
B
a⃗+b⃗2\frac{\vec a+\vec b}{2}
C
2b⃗−a⃗2\vec b-\vec a
D
b⃗\vec b
5
A unit vector has direction cosines l,m,nl,m,n satisfying l+m+n=1l+m+n=1 and lm=1/8lm=1/8. What is the largest possible value of nn?
Not attempted
A
2−64\frac{2-\sqrt6}{4}
B
2+64\frac{2+\sqrt6}{4}
C
12\frac12
D
−14-\frac14
6
For real tt, let a⃗=(t,1,1)\vec a=(t,1,1), b⃗=(1,t,1)\vec b=(1,t,1), c⃗=(1,1,t)\vec c=(1,1,t). Define f(t)f(t) as the least possible x2+y2+z2x^2+y^2+z^2 among solutions of xa⃗+yb⃗+zc⃗=(1,−1,0)x\vec a+y\vec b+z\vec c=(1,-1,0). Leave ff undefined when there is no solution. Which graph represents ff on [−3,3][-3,3]? The plotted vertical window is 0≤f≤80\le f\le8; branches continue outside it.
Not attempted
A
Candidate graph 1
tf-3-2-1012302468
B
Candidate graph 2
tf-3-2-1012302468
C
Candidate graph 3
tf-3-2-1012302468
D
Candidate graph 4
tf-3-2-1012302468
7
Three noncoplanar unit vectors a⃗,b⃗,c⃗\vec a,\vec b,\vec c satisfy a⃗⋅b⃗=b⃗⋅c⃗=1/2\vec a\cdot\vec b=\vec b\cdot\vec c=1/2 and c⃗⋅a⃗=0\vec c\cdot\vec a=0. A vector x⃗\vec x has equal scalar projections of length 11 on all three. What is ∣x⃗∣2|\vec x|^2?
Not attempted
A
33
B
32\frac32
C
22
D
43\frac43
8
Noncoplanar vectors a⃗,b⃗,c⃗\vec a,\vec b,\vec c satisfy [a⃗ b⃗ c⃗]=2[\vec a\ \vec b\ \vec c]=2, where [p⃗ q⃗ r⃗]=p⃗⋅(q⃗×r⃗)[\vec p\ \vec q\ \vec r]=\vec p\cdot(\vec q\times\vec r). Put u⃗=a⃗+b⃗\vec u=\vec a+\vec b, v⃗=b⃗+c⃗\vec v=\vec b+\vec c, w⃗=c⃗+a⃗\vec w=\vec c+\vec a. A vector x⃗\vec x satisfies x⃗⋅(v⃗×w⃗)=6\vec x\cdot(\vec v\times\vec w)=6, x⃗⋅(w⃗×u⃗)=−2\vec x\cdot(\vec w\times\vec u)=-2, and x⃗⋅(u⃗×v⃗)=10\vec x\cdot(\vec u\times\vec v)=10. What is the coefficient of b⃗\vec b when x⃗\vec x is expressed in a⃗,b⃗,c⃗\vec a,\vec b,\vec c?
Not attempted
A
22
B
44
C
−1-1
D
11
9
Unit vectors a⃗,b⃗\vec a,\vec b obey a⃗⋅b⃗=1/2\vec a\cdot\vec b=1/2. A unit vector x⃗\vec x satisfies x⃗⋅a⃗=x⃗⋅b⃗=1/2\vec x\cdot\vec a=\vec x\cdot\vec b=1/2. What is the largest possible value of x⃗⋅(a⃗×b⃗)\vec x\cdot(\vec a\times\vec b)?
Not attempted
A
12\frac1{\sqrt2}
B
32\frac{\sqrt3}{2}
C
12\frac12
D
23\sqrt{\frac23}
10
Position vectors a⃗,b⃗,c⃗\vec a,\vec b,\vec c form a triangle and satisfy ∣a⃗∣=∣b⃗∣=∣c⃗∣=R|\vec a|=|\vec b|=|\vec c|=R, a⃗+2b⃗+3c⃗=0⃗\vec a+2\vec b+3\vec c=\vec0. Which value of ∣a⃗−b⃗∣2/R2|\vec a-\vec b|^2/R^2 is possible?
Not attempted
A
00
B
No such triangle exists
C
11
D
44
11
Nonzero vectors a⃗,b⃗\vec a,\vec b satisfy ∣a⃗∣=2|\vec a|=2, ∣b⃗∣=3|\vec b|=3. Set f=∣a⃗+b⃗∣+∣a⃗−b⃗∣f=|\vec a+\vec b|+|\vec a-\vec b|. If ∣a⃗×b⃗∣≥33|\vec a\times\vec b|\ge3\sqrt3, what is the minimum of f2f^2?
Not attempted
A
5252
B
26+41026+4\sqrt{10}
C
26+213326+2\sqrt{133}
D
26+214226+2\sqrt{142}
12
A unit vector x⃗=(x,y,z)\vec x=(x,y,z) satisfies x+y+z=0x+y+z=0. What is the maximum of (x−y)2+2(y−z)2(x-y)^2+2(y-z)^2?
Not attempted
A
66
B
3+23+\sqrt2
C
44
D
3+33+\sqrt3
13
In a nondegenerate triangle ABCABC, PP lies on ABAB with AP:PB=2:3AP:PB=2:3, and QQ lies on ACAC with AQ:QC=3:2AQ:QC=3:2. The lines BQBQ and CPCP meet at XX. What is the ratio of the area of PXQPXQ to that of ABCABC?
Not attempted
ABCPQX
A
36475\frac{36}{475}
B
72475\frac{72}{475}
C
3695\frac{36}{95}
D
18475\frac{18}{475}
14
Fixed vectors satisfy ∣a⃗∣=2|\vec a|=2, ∣b⃗∣2=5|\vec b|^2=5, and a⃗⋅b⃗=2\vec a\cdot\vec b=2. For every real tt, let x⃗\vec x solve x⃗+ta⃗×x⃗=b⃗\vec x+t\vec a\times\vec x=\vec b and set f(t)=∣x⃗∣2f(t)=|\vec x|^2. Which graph represents ff on [−2,2][-2,2]? The window is 0≤f≤60\le f\le6.
Not attempted
A
Candidate graph 1
tf-2-10120246
B
Candidate graph 2
tf-2-10120246
C
Candidate graph 3
tf-2-10120246
D
Candidate graph 4
tf-2-10120246
15
Vectors u⃗,v⃗\vec u,\vec v satisfy ∣u⃗∣2=5|\vec u|^2=5, ∣v⃗∣2=2|\vec v|^2=2, u⃗⋅v⃗=1\vec u\cdot\vec v=1. For real tt, let a⃗=u⃗+tv⃗\vec a=\vec u+t\vec v and b⃗=tu⃗+v⃗\vec b=t\vec u+\vec v. Among the values of tt for which a⃗⊥b⃗\vec a\perp\vec b, what is ∣a⃗×b⃗∣2|\vec a\times\vec b|^2?
Not attempted
A
405405
B
4052\frac{405}{2}
C
It has two distinct possible values
D
99
16
A parallelogram has diagonal vectors p⃗,q⃗\vec p,\vec q with ∣p⃗∣=5|\vec p|=5, ∣q⃗∣=3|\vec q|=3. Its adjacent side lengths have product 55. What is its area?
Not attempted
ABCD
A
66
B
55
C
32\frac32
D
33
17
Three unit vectors have pairwise dot products a⃗⋅b⃗=b⃗⋅c⃗=t\vec a\cdot\vec b=\vec b\cdot\vec c=t and c⃗⋅a⃗=2t2−1\vec c\cdot\vec a=2t^2-1. For which real tt can they be noncoplanar?
Not attempted
A
No real tt
B
−1<t<1-1<t<1
C
0<t<10<t<1
D
t≠0t\ne0
18
Noncoplanar vectors have [a⃗ b⃗ c⃗]=−3[\vec a\ \vec b\ \vec c]=-3. Put u⃗=b⃗×c⃗\vec u=\vec b\times\vec c, v⃗=c⃗×a⃗\vec v=\vec c\times\vec a, w⃗=a⃗×b⃗\vec w=\vec a\times\vec b. A tetrahedron has vertices u⃗+v⃗\vec u+\vec v, v⃗+w⃗\vec v+\vec w, w⃗+u⃗\vec w+\vec u, and u⃗+v⃗+w⃗\vec u+\vec v+\vec w. What is its volume?
Not attempted
A
33
B
32\frac32
C
99
D
12\frac12
19
Let OABCOABC be a tetrahedron of volume VV. Points P,Q,RP,Q,R lie on the entire lines OA,OB,OCOA,OB,OC respectively with OP⃗=sOA⃗\vec{OP}=s\vec{OA}, OQ⃗=tOB⃗\vec{OQ}=t\vec{OB}, OR⃗=uOC⃗\vec{OR}=u\vec{OC}, where s,t,u>0s,t,u>0. The plane PQRPQR passes through the centroid of OABCOABC. What is the least possible volume of OPQROPQR in terms of VV? The schematic shows one allowed intercept plane, with Q=BQ=B and R=CR=C; these coincidences are not constraints.
Not attempted
OABCP
A
V64\frac V{64}
B
3V4\frac{3V}{4}
C
27V64\frac{27V}{64}
D
9V16\frac{9V}{16}
20
Unit vectors satisfy a⃗⋅b⃗=1/3\vec a\cdot\vec b=1/3 and a⃗⋅c⃗=2/3\vec a\cdot\vec c=2/3. If [a⃗ b⃗ c⃗]2=4/9[\vec a\ \vec b\ \vec c]^2=4/9, how many distinct values can b⃗⋅c⃗\vec b\cdot\vec c take?
Not attempted
A
00
B
11
C
44
D
22
21
A unit vector a⃗\vec a and a vector b⃗\vec b satisfy ∣b⃗∣2=7|\vec b|^2=7 and a⃗⋅b⃗=1\vec a\cdot\vec b=1. A vector x⃗\vec x solves x⃗+a⃗×x⃗=b⃗\vec x+\vec a\times\vec x=\vec b. What is ∣x⃗×b⃗∣2|\vec x\times\vec b|^2?
Not attempted
A
1212
B
2424
C
66
D
2828
22
Nonparallel unit vectors a⃗,b⃗\vec a,\vec b satisfy a⃗⋅b⃗=1/2\vec a\cdot\vec b=1/2. For what real λ\lambda does the equation x⃗+λa⃗×(b⃗×x⃗)=a⃗\vec x+\lambda\vec a\times(\vec b\times\vec x)=\vec a have infinitely many solutions?
Not attempted
A
22
B
No real λ\lambda
C
−2-2
D
00
23
Nonparallel vectors a⃗,b⃗\vec a,\vec b have ∣a⃗∣2=2|\vec a|^2=2, ∣b⃗∣2=3|\vec b|^2=3, a⃗⋅b⃗=1\vec a\cdot\vec b=1. A vector x⃗\vec x obeys a⃗×x⃗=a⃗×b⃗\vec a\times\vec x=\vec a\times\vec b and b⃗⋅x⃗=0\vec b\cdot\vec x=0. What is ∣x⃗∣2|\vec x|^2?
Not attempted
A
33
B
2121
C
1515
D
99
24
Nonparallel unit vectors have a⃗⋅b⃗=1/2\vec a\cdot\vec b=1/2. Define f(t)=∣x⃗∣2f(t)=|\vec x|^2 when x⃗+ta⃗×(b⃗×x⃗)=a⃗\vec x+t\vec a\times(\vec b\times\vec x)=\vec a has a solution, and leave it undefined otherwise. Which graph represents ff on [−2,4][-2,4]? The vertical window is 0≤f≤120\le f\le12; branches continue outside it.
Not attempted
A
Candidate graph 1
tf-2-101234024681012
B
Candidate graph 2
tf-2-101234024681012
C
Candidate graph 3
tf-2-101234024681012
D
Candidate graph 4
tf-2-101234024681012
25
Vectors a⃗,b⃗,c⃗\vec a,\vec b,\vec c have lengths 2,3,42,3,4 and a⃗+b⃗+c⃗=0\vec a+\vec b+\vec c=0. Set p⃗=2a⃗+b⃗\vec p=2\vec a+\vec b and q⃗=b⃗+2c⃗\vec q=\vec b+2\vec c. Which statements hold?
Not attempted
A
p⃗⋅q⃗=−31\vec p\cdot\vec q=-31
B
p⃗=q⃗\vec p=\vec q
C
p⃗×q⃗=0\vec p\times\vec q=0
D
∣p⃗∣2=31|\vec p|^2=31
26
In triangle ABCABC take P=(B+C)/2P=(B+C)/2, Q=(2C+A)/3Q=(2C+A)/3, R=(3A+B)/4R=(3A+B)/4. Let X=AP∩BQX=AP\cap BQ. Which statements are true? Positions are affine combinations.
Not attempted
ABCPQRX
A
C,R,XC,R,X are collinear
B
AX:XP=4:1AX:XP=4:1
C
XX lies strictly inside ABCABC
D
area⁡(ABX)/area⁡(ABC)=2/5\operatorname{area}(ABX)/\operatorname{area}(ABC)=2/5
27
Four position vectors satisfy a⃗+b⃗+c⃗+d⃗=0\vec a+\vec b+\vec c+\vec d=0 and ∣a⃗∣=∣b⃗∣=∣c⃗∣=∣d⃗∣=1|\vec a|=|\vec b|=|\vec c|=|\vec d|=1. Put p⃗=a⃗+b⃗\vec p=\vec a+\vec b, q⃗=a⃗+c⃗\vec q=\vec a+\vec c, r⃗=a⃗+d⃗\vec r=\vec a+\vec d. Which statements always hold?
Not attempted
A
∣p⃗∣2+∣q⃗∣2+∣r⃗∣2=4|\vec p|^2+|\vec q|^2+|\vec r|^2=4
B
p⃗,q⃗,r⃗\vec p,\vec q,\vec r are pairwise perpendicular in the dot-product sense
C
At least one of p⃗,q⃗,r⃗\vec p,\vec q,\vec r is zero
D
∣[p⃗ q⃗ r⃗]∣=∣p⃗∣∣q⃗∣∣r⃗∣|[\vec p\ \vec q\ \vec r]|=|\vec p||\vec q||\vec r|
28
A unit vector has direction cosines l,m,nl,m,n and l+m+n=0l+m+n=0. Which assertions hold?
Not attempted
A
The largest possible ∣l∣|l| is 2/3\sqrt{2/3}
B
lm+mn+nl=−1/2lm+mn+nl=-1/2
C
(l−m)2+(m−n)2+(n−l)2=3(l-m)^2+(m-n)^2+(n-l)^2=3
D
l3+m3+n3=0l^3+m^3+n^3=0
29
For real tt let a⃗=(t,1,1)\vec a=(t,1,1), b⃗=(1,t,1)\vec b=(1,t,1), c⃗=(1,1,t)\vec c=(1,1,t). Consider xa⃗+yb⃗+zc⃗=(1,1,1)x\vec a+y\vec b+z\vec c=(1,1,1). Which statements are correct?
Not attempted
A
For t∉{1,−2}t\notin\{1,-2\} the unique solution has x=y=zx=y=z
B
At t=−2t=-2 there are no solutions
C
There is at least one solution for every real tt
D
At t=1t=1 there are infinitely many solutions
30
Three unit vectors a⃗,b⃗,c⃗\vec a,\vec b,\vec c satisfy a⃗⋅b⃗=a⃗⋅c⃗=1/2\vec a\cdot\vec b=\vec a\cdot\vec c=1/2 and b⃗⋅c⃗=t\vec b\cdot\vec c=t. Which assertions are correct?
Not attempted
A
t=−1/2t=-1/2 forces b⃗+c⃗=a⃗\vec b+\vec c=\vec a
B
They are independent for every −1/2<t<1-1/2<t<1
C
The possible interval is [−1/2,1][-1/2,1]
D
The volume is greatest at t=0t=0
31
For nonparallel unit vectors a⃗,b⃗\vec a,\vec b with a⃗⋅b⃗=c\vec a\cdot\vec b=c, define Pa⃗(x⃗)=(x⃗⋅a⃗)a⃗P_{\vec a}(\vec x)=(\vec x\cdot\vec a)\vec a and similarly Pb⃗P_{\vec b}. Which statements hold for every x⃗\vec x?
Not attempted
A
Pa⃗Pb⃗(x⃗)−Pb⃗Pa⃗(x⃗)=c x⃗×(a⃗×b⃗)P_{\vec a}P_{\vec b}(\vec x)-P_{\vec b}P_{\vec a}(\vec x)=c\,\vec x\times(\vec a\times\vec b)
B
The two compositions commute for all inputs iff c=0c=0
C
Pa⃗(x⃗)+Pb⃗(x⃗)=x⃗P_{\vec a}(\vec x)+P_{\vec b}(\vec x)=\vec x
D
The difference above is perpendicular to x⃗\vec x
32
Points A,B,CA,B,C have position vectors a⃗,b⃗,c⃗\vec a,\vec b,\vec c. Positive weights 2,3,52,3,5 define F(X)=2∣x⃗−a⃗∣2+3∣x⃗−b⃗∣2+5∣x⃗−c⃗∣2F(X)=2|\vec x-\vec a|^2+3|\vec x-\vec b|^2+5|\vec x-\vec c|^2. Put g⃗=(2a⃗+3b⃗+5c⃗)/10\vec g=(2\vec a+3\vec b+5\vec c)/10. Which assertions hold?
Not attempted
A
Translating all four points by the same vector leaves FF unchanged
B
min⁡F=35AB2+32BC2+CA2\min F=\frac35 AB^2+\frac32 BC^2+CA^2
C
GG is the unique minimizer
D
Every nonempty level set above the minimum is a sphere centered at GG
33
Unit vectors a⃗,b⃗,c⃗\vec a,\vec b,\vec c satisfy a⃗+b⃗+c⃗=0\vec a+\vec b+\vec c=0. For an arbitrary unit vector x⃗\vec x, set S=(x⃗⋅a⃗)2+(x⃗⋅b⃗)2+(x⃗⋅c⃗)2S=(\vec x\cdot\vec a)^2+(\vec x\cdot\vec b)^2+(\vec x\cdot\vec c)^2. Which statements are correct?
Not attempted
A
S≥1S\ge1 for every unit x⃗\vec x
B
There are exactly two unit vectors with S=0S=0
C
0≤S≤3/20\le S\le3/2
D
S=3/2S=3/2 iff x⃗\vec x is in the plane of a⃗,b⃗\vec a,\vec b
34
Nonparallel nonzero vectors a⃗,b⃗\vec a,\vec b and vectors x⃗,y⃗\vec x,\vec y satisfy a⃗×x⃗=a⃗×y⃗\vec a\times\vec x=\vec a\times\vec y. In addition, ∣x⃗∣=∣y⃗∣|\vec x|=|\vec y|. Which assertions are necessarily true?
Not attempted
A
Either x⃗=y⃗\vec x=\vec y or a⃗⋅(x⃗+y⃗)=0\vec a\cdot(\vec x+\vec y)=0
B
x⃗=y⃗\vec x=\vec y
C
x⃗×y⃗\vec x\times\vec y must be zero
D
b⃗⋅x⃗=b⃗⋅y⃗\vec b\cdot\vec x=\vec b\cdot\vec y
35
Nonzero vectors a⃗,b⃗\vec a,\vec b satisfy ∣a⃗+b⃗∣=∣a⃗−b⃗∣|\vec a+\vec b|=|\vec a-\vec b|. Let p⃗=a⃗+2b⃗\vec p=\vec a+2\vec b and q⃗=2a⃗−b⃗\vec q=2\vec a-\vec b. Which statements hold?
Not attempted
A
p⃗⊥q⃗\vec p\perp\vec q for all allowed vectors
B
∣p⃗×q⃗∣=5∣a⃗∣∣b⃗∣|\vec p\times\vec q|=5|\vec a||\vec b|
C
∣p⃗∣2+∣q⃗∣2=5(∣a⃗∣2+∣b⃗∣2)|\vec p|^2+|\vec q|^2=5(|\vec a|^2+|\vec b|^2)
D
p⃗,q⃗\vec p,\vec q can be parallel
36
For a closed polygon with consecutive position vectors r⃗1,…,r⃗n\vec r_1,\ldots,\vec r_n, define A⃗=12∑i=1nr⃗i×r⃗i+1\vec A=\frac12\sum_{i=1}^n\vec r_i\times\vec r_{i+1}, where r⃗n+1=r⃗1\vec r_{n+1}=\vec r_1. The polygon need not be planar. Which assertions hold?
Not attempted
r₁r₂r₃r₄
A
∣A⃗∣|\vec A| always equals the sum of the unsigned areas of triangles from any chosen origin
B
A⃗⋅n⃗\vec A\cdot\vec n is the signed area of the projection onto a plane normal to n⃗\vec n
C
Reversing the traversal changes A⃗\vec A to −A⃗-\vec A
D
A⃗\vec A is invariant under translating the origin
37
Let a⃗,b⃗,c⃗\vec a,\vec b,\vec c be noncoplanar and D=[a⃗ b⃗ c⃗]D=[\vec a\ \vec b\ \vec c]. Define a⃗∗=(b⃗×c⃗)/D\vec a^*=(\vec b\times\vec c)/D and cyclically b⃗∗,c⃗∗\vec b^*,\vec c^*. Which assertions are correct?
Not attempted
A
All three reciprocal vectors have unit length
B
[a⃗∗ b⃗∗ c⃗∗]=1/D[\vec a^*\ \vec b^*\ \vec c^*]=1/D
C
x⃗=∑u⃗∈{a⃗,b⃗,c⃗}(x⃗⋅u⃗)u⃗∗\vec x=\sum_{\vec u\in\{\vec a,\vec b,\vec c\}}(\vec x\cdot\vec u)\vec u^*
D
Taking the reciprocal frame twice recovers a⃗,b⃗,c⃗\vec a,\vec b,\vec c
38
Unit vectors a⃗,b⃗,c⃗\vec a,\vec b,\vec c have equal pairwise dot product tt. Which assertions are correct?
Not attempted
A
They are noncoplanar exactly for −1/2<t<1-1/2<t<1
B
t=−1/2t=-1/2 forces a⃗+b⃗+c⃗=0\vec a+\vec b+\vec c=0
C
The largest volume is attained at t=1/2t=1/2
D
They exist exactly for −1/2≤t≤1-1/2\le t\le1
39
Noncoplanar position vectors a⃗,b⃗,c⃗\vec a,\vec b,\vec c define A,B,CA,B,C. A fourth point has d⃗=pa⃗+qb⃗+rc⃗\vec d=p\vec a+q\vec b+r\vec c. Which assertions are correct?
Not attempted
A
p+q+r=0p+q+r=0 forces DD into plane ABCABC
B
If p,q,r>0p,q,r>0 and p+q+r<1p+q+r<1, then DD is inside OABCOABC
C
A,B,C,DA,B,C,D are coplanar iff p+q+r=1p+q+r=1
D
V(ABCD)/V(OABC)=∣p+q+r−1∣V(ABCD)/V(OABC)=|p+q+r-1|
40
Let a⃗≠0\vec a\ne0 and let x⃗+a⃗×x⃗=b⃗\vec x+\vec a\times\vec x=\vec b. Which statements always hold?
Not attempted
A
There is exactly one solution for every b⃗\vec b
B
x⃗⋅(b⃗−x⃗)=0\vec x\cdot(\vec b-\vec x)=0
C
∣x⃗∣≤∣b⃗∣|\vec x|\le|\vec b|
D
x⃗=b⃗+(a⃗⋅b⃗)a⃗+a⃗×b⃗1+∣a⃗∣2\vec x=\frac{\vec b+(\vec a\cdot\vec b)\vec a+\vec a\times\vec b}{1+|\vec a|^2}
41
For fixed nonparallel unit vectors a⃗,c⃗\vec a,\vec c, determine all b⃗\vec b for which a⃗×(b⃗×c⃗)=(a⃗×b⃗)×c⃗\vec a\times(\vec b\times\vec c)=(\vec a\times\vec b)\times\vec c. Which assertions hold?
Not attempted
A
The solution set is b⃗=t(a⃗×c⃗)\vec b=t(\vec a\times\vec c), t∈Rt\in\mathbb R
B
b⃗=a⃗+c⃗\vec b=\vec a+\vec c is always a solution
C
There are exactly two unit choices for b⃗\vec b
D
The common nested product must be zero
42
Let a⃗,b⃗\vec a,\vec b be nonparallel unit vectors with a⃗⋅b⃗=c\vec a\cdot\vec b=c. Consider x⃗+λa⃗×(b⃗×x⃗)=d⃗\vec x+\lambda\vec a\times(\vec b\times\vec x)=\vec d for real λ\lambda. Which statements are correct?
Not attempted
A
At the exceptional value, every consistent equation has infinitely many solutions
B
If c≠0c\ne0 and λ=1/c\lambda=1/c, a solution exists iff d⃗\vec d is parallel to b⃗\vec b
C
If c=0c=0, the solution is unique for every real λ\lambda
D
Every singular case is inconsistent
43
Let A=(0,0,0)A=(0,0,0), B=(4,0,0)B=(4,0,0), C=(0,3,0)C=(0,3,0). A point PP is inside or on triangle ABCABC, with affine position p⃗=αa⃗+βb⃗+γc⃗\vec p=\alpha\vec a+\beta\vec b+\gamma\vec c, where α+β+γ=1\alpha+\beta+\gamma=1 and all three coefficients are nonnegative. It satisfies α=2γ\alpha=2\gamma. Find 102102 times the minimum of ∣PA⃗∣2+2∣PB⃗∣2+3∣PC⃗∣2|\vec{PA}|^2+2|\vec{PB}|^2+3|\vec{PC}|^2, correct to two decimals.
Not attempted
ABCT
Your answer: —
Correct answer: 3953
44
In triangle ABCABC, points P,QP,Q divide BC,CABC,CA internally with BP:PC=2:3BP:PC=2:3 and CQ:QA=3:4CQ:QA=3:4. Put X=AP∩BQX=AP\cap BQ. The line CXCX meets ABAB at RR. Find 35 area⁡(PQR)/area⁡(ABC)35\,\operatorname{area}(PQR)/\operatorname{area}(ABC).
Not attempted
ABCPQRX
Your answer: —
Correct answer: 8
45
Vectors a⃗,b⃗,c⃗,d⃗\vec a,\vec b,\vec c,\vec d in space have lengths 1,2,3,41,2,3,4 and sum to zero. Find the greatest possible value of ∣a⃗−c⃗∣2+∣b⃗−d⃗∣2|\vec a-\vec c|^2+|\vec b-\vec d|^2. The figure is one allowed planar arrangement with successive edge vectors OA⃗=a⃗\vec{OA}=\vec a, AB⃗=b⃗\vec{AB}=\vec b, BC⃗=c⃗\vec{BC}=\vec c, CO⃗=d⃗\vec{CO}=\vec d; the vectors in the question may lie in space.
Not attempted
OABC
Your answer: —
Correct answer: 52
46
A tetrahedron has vertices O,A,B,CO,A,B,C. A point PP in its interior has V(PABC):V(POBC):V(POCA):V(POAB)=1:2:3:4V(PABC):V(POBC):V(POCA):V(POAB)=1:2:3:4. Put p⃗=OP⃗\vec p=\vec{OP}. Let XX be the intersection of line APAP with plane OBCOBC, and let YY divide OXOX externally in ratio OY:YX=3:2OY:YX=3:2. Find 8V(AYBC)/V(OABC)8V(AYBC)/V(OABC).
Not attempted
OABCP
Your answer: —
Correct answer: 13
47
Direction cosines l,m,nl,m,n satisfy l+m+n=1l+m+n=1. Find 2727 times the greatest possible value of ∣lmn∣|lmn|.
Not attempted
Your answer: —
Correct answer: 4
48
For real tt, the vectors a⃗=(1,t,0)\vec a=(1,t,0), b⃗=(0,1,t)\vec b=(0,1,t), c⃗=(t,0,1)\vec c=(t,0,1) are used in xa⃗+yb⃗+zc⃗=(1,2,3)x\vec a+y\vec b+z\vec c=(1,2,3). For all parameters giving a unique solution, put S=x+y+zS=x+y+z. Find the unique real tt for which S=3S=3.
Not attempted
Your answer: —
Correct answer: 1
49
Let a⃗=(1,1,0)\vec a=(1,1,0), b⃗=(0,1,1)\vec b=(0,1,1) and c⃗=(1,0,1)\vec c=(1,0,1). A unit vector x⃗=pa⃗+qb⃗+rc⃗\vec x=p\vec a+q\vec b+r\vec c satisfies p+q+r=0p+q+r=0. Find the largest possible value of 6(p−2q+r)26(p-2q+r)^2.
Not attempted
Your answer: —
Correct answer: 36
50
Let a⃗,b⃗,c⃗\vec a,\vec b,\vec c be noncoplanar. Put u⃗=a⃗+tb⃗\vec u=\vec a+t\vec b, v⃗=b⃗+tc⃗\vec v=\vec b+t\vec c, w⃗=c⃗+ta⃗\vec w=\vec c+t\vec a. Among real tt for which these vectors are independent, write a⃗+b⃗+c⃗=xu⃗+yv⃗+zw⃗\vec a+\vec b+\vec c=x\vec u+y\vec v+z\vec w. If x2+y2+z2=12x^2+y^2+z^2=12, find the sum of all possible tt.
Not attempted
Your answer: —
Correct answer: -2
51
Unit vectors a⃗,b⃗\vec a,\vec b make an angle with cosine 1/31/3. For unit x⃗\vec x define F=(x⃗⋅a⃗)(x⃗⋅b⃗)+2x⃗⋅(a⃗×b⃗)F=(\vec x\cdot\vec a)(\vec x\cdot\vec b)+2\vec x\cdot(\vec a\times\vec b). Find 9max⁡F9\max F.
Not attempted
Your answer: —
Correct answer: 16.97
52
Let A=(0,0,0)A=(0,0,0), B=(6,0,0)B=(6,0,0), C=(0,8,0)C=(0,8,0). A point PP lies on the boundary of triangle ABCABC. Find three times the minimum of ∣PA⃗∣2+∣PB⃗∣2+∣PC⃗∣2|\vec{PA}|^2+|\vec{PB}|^2+|\vec{PC}|^2.
Not attempted
ABC
Your answer: —
Correct answer: 223.04
53
Let a⃗,b⃗\vec a,\vec b be perpendicular unit vectors. A unit vector x⃗\vec x satisfies x⃗⋅(a⃗+b⃗)=1\vec x\cdot(\vec a+\vec b)=1. Find the greatest value of 16(x⃗⋅a⃗)2(x⃗⋅b⃗)2∣x⃗×(a⃗+b⃗)∣216(\vec x\cdot\vec a)^2(\vec x\cdot\vec b)^2|\vec x\times(\vec a+\vec b)|^2.
Not attempted
Your answer: —
Correct answer: 1
54
Let a⃗\vec a be a unit vector. For each unit vector b⃗\vec b with a⃗⋅b⃗=1/2\vec a\cdot\vec b=1/2, let x⃗\vec x solve 2x⃗+a⃗×x⃗=b⃗2\vec x+\vec a\times\vec x=\vec b. Find 100 b⃗⋅x⃗100\,\vec b\cdot\vec x.
Not attempted
Your answer: —
Correct answer: 42.5
55
Vectors a⃗,b⃗\vec a,\vec b satisfy ∣a⃗∣2=5|\vec a|^2=5, ∣b⃗∣2=5|\vec b|^2=5, and a⃗⋅b⃗=3\vec a\cdot\vec b=3. A unit vector n⃗\vec n is perpendicular to a⃗−b⃗\vec a-\vec b and satisfies n⃗⋅(a⃗×b⃗)=2\vec n\cdot(\vec a\times\vec b)=2. Find the greatest possible value of 4[n⃗⋅(a⃗+b⃗)]24[\vec n\cdot(\vec a+\vec b)]^2.
Not attempted
Your answer: —
Correct answer: 48
56
Points P,QP,Q move on the ellipse x2/9+y2/4=1x^2/9+y^2/4=1 in the xyxy plane and obey OP⃗⋅OQ⃗=0\vec{OP}\cdot\vec{OQ}=0. Find the largest possible area of triangle OPQOPQ.
Not attempted
xyOOPQ
Your answer: —
Correct answer: 3
57
Let a⃗=(1,2,2)\vec a=(1,2,2) and b⃗=(2,−1,0)\vec b=(2,-1,0). Count the vectors x⃗\vec x satisfying x⃗×a⃗=b⃗\vec x\times\vec a=\vec b, ∣x⃗∣=2|\vec x|=2, and x⃗⋅a⃗≥0\vec x\cdot\vec a\ge0.
Not attempted
Your answer: —
Correct answer: 1
58
Let A=(0,0,0)A=(0,0,0), B=(4,0,0)B=(4,0,0), C=(4,3,0)C=(4,3,0), D=(0,3,0)D=(0,3,0). A point XX on diagonal ACAC satisfies AX:XC=1:2AX:XC=1:2. Define the oriented area vector of the closed walk A,B,X,D,C,AA,B,X,D,C,A as S⃗=12(a⃗×b⃗+b⃗×x⃗+x⃗×d⃗+d⃗×c⃗+c⃗×a⃗)\vec S=\frac12(\vec a\times\vec b+\vec b\times\vec x+\vec x\times\vec d+\vec d\times\vec c+\vec c\times\vec a). Find ∣S⃗∣2|\vec S|^2.
Not attempted
ABCDX
Your answer: —
Correct answer: 4
59
Let a⃗,b⃗,c⃗\vec a,\vec b,\vec c be vectors with ∣a⃗∣2+∣b⃗∣2+∣c⃗∣2=12|\vec a|^2+|\vec b|^2+|\vec c|^2=12 and a⃗+b⃗+c⃗=s⃗\vec a+\vec b+\vec c=\vec s, where ∣s⃗∣=3|\vec s|=3. Find the largest possible value of [a⃗ b⃗ c⃗]2[\vec a\ \vec b\ \vec c]^2.
Not attempted
Your answer: —
Correct answer: 60.75
60
Let OABCOABC be a tetrahedron of volume 1010. For real tt, points have positions p⃗=a⃗+tb⃗\vec p=\vec a+t\vec b, q⃗=b⃗+tc⃗\vec q=\vec b+t\vec c, r⃗=c⃗+ta⃗\vec r=\vec c+t\vec a. Find the sum of all distinct real tt for which the tetrahedron OPQROPQR has volume 2020.
Not attempted
Your answer: —
Correct answer: -0.44
61
Noncoplanar vectors have [a⃗ b⃗ c⃗]=2[\vec a\ \vec b\ \vec c]=2. Set u⃗=(b⃗×c⃗)+(c⃗×a⃗)\vec u=(\vec b\times\vec c)+(\vec c\times\vec a), v⃗=(c⃗×a⃗)+(a⃗×b⃗)\vec v=(\vec c\times\vec a)+(\vec a\times\vec b), w⃗=(a⃗×b⃗)+(b⃗×c⃗)\vec w=(\vec a\times\vec b)+(\vec b\times\vec c). Find [u⃗×v⃗  v⃗×w⃗  w⃗×u⃗][\vec u\times\vec v\ \ \vec v\times\vec w\ \ \vec w\times\vec u].
Not attempted
Your answer: —
Correct answer: 64
62
Let a⃗\vec a be a unit vector and b⃗\vec b a vector with ∣b⃗∣2=10|\vec b|^2=10 and a⃗⋅b⃗=1\vec a\cdot\vec b=1. Set x⃗0=b⃗\vec x_0=\vec b, and for n=0,1,2n=0,1,2 let x⃗n+1+a⃗×x⃗n+1=x⃗n\vec x_{n+1}+\vec a\times\vec x_{n+1}=\vec x_n. Find 8∣x⃗3∣28|\vec x_3|^2.
Not attempted
Your answer: —
Correct answer: 17
63
Let a⃗=(0,0,1)\vec a=(0,0,1) and b⃗=(2,0,−2)\vec b=(2,0,-2). Count all real vectors x⃗\vec x satisfying x⃗×(x⃗×a⃗)+x⃗×a⃗=b⃗\vec x\times(\vec x\times\vec a)+\vec x\times\vec a=\vec b.
Not attempted
Your answer: —
Correct answer: 2
64
Let a⃗,b⃗\vec a,\vec b be unit vectors with a⃗⋅b⃗=1/2\vec a\cdot\vec b=1/2. A vector x⃗\vec x satisfies a⃗×(x⃗×a⃗)+2b⃗×(x⃗×b⃗)=a⃗+b⃗\vec a\times(\vec x\times\vec a)+2\vec b\times(\vec x\times\vec b)=\vec a+\vec b. Find 3∣x⃗∣23|\vec x|^2.
Not attempted
Your answer: —
Correct answer: 21
65
In a nondegenerate triangle ABCABC, put P=(1−s)A+sBP=(1-s)A+sB, Q=(1−t)B+tCQ=(1-t)B+tC, R=(1−u)C+uAR=(1-u)C+uA. Coefficients may be negative. Match each triple (s,t,u)(s,t,u) with area⁡(PQR)/area⁡(ABC)\operatorname{area}(PQR)/\operatorname{area}(ABC). List-IList-IIP) (2,3,4)1) 18Q) (2,−1,3)2) 2R) (1/2,2,−2)3) 7/2S) (1/3,1/2,3/4)4) 5/245) 6\begin{array}{ll}\text{List-I}&\text{List-II}\\\text{P)}\ (2,3,4)&\text{1)}\ 18\\\text{Q)}\ (2,-1,3)&\text{2)}\ 2\\\text{R)}\ (1/2,2,-2)&\text{3)}\ 7/2\\\text{S)}\ (1/3,1/2,3/4)&\text{4)}\ 5/24\\&\text{5)}\ 6\end{array} The schematic illustrates (s,t,u)=(1/2,2,−2)(s,t,u)=(1/2,2,-2) only.
Not attempted
ABCPQR
A
P-1, Q-2, R-3, S-4
B
P-2, Q-3, R-4, S-1
C
P-3, Q-4, R-1, S-2
D
P-4, Q-1, R-2, S-3
66
A unit vector has direction cosines l,m,nl,m,n satisfying l+m+n=1l+m+n=1. Match each expression with its greatest possible value. List-IList-IIP) l1) 1Q) l+m2) 4/3R) l−m3) 2/3S) l+2m+3n4) 2+2/35) 3\begin{array}{ll}\text{List-I}&\text{List-II}\\\text{P)}\ l&\text{1)}\ 1\\\text{Q)}\ l+m&\text{2)}\ 4/3\\\text{R)}\ l-m&\text{3)}\ 2/\sqrt3\\\text{S)}\ l+2m+3n&\text{4)}\ 2+2/\sqrt3\\&\text{5)}\ \sqrt3\end{array}
Not attempted
A
P-2, Q-3, R-4, S-1
B
P-1, Q-2, R-3, S-4
C
P-3, Q-4, R-1, S-2
D
P-4, Q-1, R-2, S-3
67
Let A=(0,0,0)A=(0,0,0), B=(2,0,0)B=(2,0,0), C=(0,2,0)C=(0,2,0). For each weight triple (a,b,c)(a,b,c) in List-I, minimize a∣XA⃗∣2+b∣XB⃗∣2+c∣XC⃗∣2a|\vec{XA}|^2+b|\vec{XB}|^2+c|\vec{XC}|^2 over points XX on the entire line x+y=1x+y=1, z=0z=0. Match with the minimum. List-IList-IIP) (1,1,1)1) 11/2Q) (2,1,1)2) 6R) (1,2,1)3) 15/2S) (1,1,3)4) 89/105) 8\begin{array}{ll}\text{List-I}&\text{List-II}\\\text{P)}\ (1,1,1)&\text{1)}\ 11/2\\\text{Q)}\ (2,1,1)&\text{2)}\ 6\\\text{R)}\ (1,2,1)&\text{3)}\ 15/2\\\text{S)}\ (1,1,3)&\text{4)}\ 89/10\\&\text{5)}\ 8\end{array}
Not attempted
A
P-2, Q-3, R-4, S-1
B
P-3, Q-4, R-1, S-2
C
P-1, Q-2, R-3, S-4
D
P-4, Q-1, R-2, S-3
68
Let a⃗=(0,0,1)\vec a=(0,0,1) and b⃗=(1,0,0)\vec b=(1,0,0). For each tt in List-I, minimize ∣x⃗∣2|\vec x|^2 subject to x⃗×a⃗=tb⃗\vec x\times\vec a=t\vec b and x⃗⋅(a⃗+b⃗)=1\vec x\cdot(\vec a+\vec b)=1. Match with this minimum. List-IList-IIP) 01) 1Q) 12) 2R) −13) 5S) 24) 45) 0\begin{array}{ll}\text{List-I}&\text{List-II}\\\text{P)}\ 0&\text{1)}\ 1\\\text{Q)}\ 1&\text{2)}\ 2\\\text{R)}\ -1&\text{3)}\ 5\\\text{S)}\ 2&\text{4)}\ 4\\&\text{5)}\ 0\end{array}
Not attempted
A
P-2, Q-2, R-3, S-1
B
P-2, Q-3, R-1, S-2
C
P-3, Q-1, R-2, S-2
D
P-1, Q-2, R-2, S-3
69
Let a⃗,b⃗,c⃗\vec a,\vec b,\vec c be unit vectors with equal pairwise dot product tt. Match the listed value of tt with the squared volume of their parallelepiped. If no configuration exists, use the corresponding List-II item. List-IList-IIP) −11) No configurationQ) −1/22) 0R) 03) 1S) 1/24) 1/25) 1/4\begin{array}{ll}\text{List-I}&\text{List-II}\\\text{P)}\ -1&\text{1)}\ \text{No configuration}\\\text{Q)}\ -1/2&\text{2)}\ 0\\\text{R)}\ 0&\text{3)}\ 1\\\text{S)}\ 1/2&\text{4)}\ 1/2\\&\text{5)}\ 1/4\end{array}
Not attempted
A
P-1, Q-2, R-3, S-4
B
P-2, Q-3, R-4, S-1
C
P-3, Q-4, R-1, S-2
D
P-4, Q-1, R-2, S-3
70
For a unit vector a⃗\vec a and a vector b⃗\vec b with ∣b⃗∣2=5|\vec b|^2=5 and a⃗⋅b⃗=1\vec a\cdot\vec b=1, let x⃗\vec x solve kx⃗+a⃗×x⃗=b⃗k\vec x+\vec a\times\vec x=\vec b. Match each kk with ∣x⃗∣2|\vec x|^2, or with the no-solution item. List-IList-IIP) 01) No solutionQ) 12) 3R) 23) 21/20S) −24) 55) 4/5\begin{array}{ll}\text{List-I}&\text{List-II}\\\text{P)}\ 0&\text{1)}\ \text{No solution}\\\text{Q)}\ 1&\text{2)}\ 3\\\text{R)}\ 2&\text{3)}\ 21/20\\\text{S)}\ -2&\text{4)}\ 5\\&\text{5)}\ 4/5\end{array}
Not attempted
A
P-2, Q-3, R-3, S-1
B
P-1, Q-2, R-3, S-3
C
P-3, Q-3, R-1, S-2
D
P-3, Q-1, R-2, S-3
The axial and perpendicular equations have different scaling factors.
Dotting with a⃗\vec a gives k(a⃗⋅x⃗)=1k(\vec a\cdot\vec x)=1, so k=0k=0 is inconsistent. For nonzero kk the axial squared norm is 1/k21/k^2. The transverse right side has squared norm 44, while the transverse map multiplies squared norms by k2+1k^2+1. Thus ∣x⃗∣2=1/k2+4/(k2+1)|\vec x|^2=1/k^2+4/(k^2+1). This gives 33 at k=1k=1 and 21/2021/20 at k=2,−2k=2,-2. A negative kk reverses the axial component but leaves its squared norm unchanged. The fifth listed expression counts only the transverse part for ∣k∣=2|k|=2.