2015 VMO
Finale
Competition
Countdown Round
Juniors + Seniors
Saturday, June 13th, 2015
New Westminster
Secondary School
Instructions
 The
Juniors Countdown round will take place first,
followed by the Seniors Countdown round.
 The brackets for both Juniors and Seniors are as
follows:
 The
rest of the instructions are delivered orally.
Please listen carefully!
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Computations
Evaluate 10 to one decimal
point.
Computations
Answer:3.2
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Algebra
2
2
If 𝑎 + 𝑏 = 5 and 𝑎 − 𝑏 = 2,
what is the positive value of 𝑎 +
𝑏? Express your answer in
simplest radical form.
Algebra
Answer: 6
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Geometry
Triangle 𝐴𝐵𝐶 is isosceles with
𝐴𝐵 = 𝐴𝐶 = 4 and angle 𝐵𝐴𝐶 =
120°. What’s the area of triangle
𝐴𝐵𝐶? Express your answer in
simplest radical form.
Geometry
Answer:4 3
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Number Theory
What is the value of:
lcm(20, 15, 48, 40)?
Number Theory
Answer:240
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Combinatorics
How many subsets of no more
than three elements are there
from the set {1, 2, 3, 4, 5, 6}?
Combinatorics
Answer:42 (subsets)
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Probability
When rolling three standard dice,
what’s the probability that the sum
of the numbers on the three dice is
equal to 5?
Probability
1
Answer:
36
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Number Theory
Two prime numbers sum to 151. What
is the product of these two prime
numbers?
Number Theory
Answer: 298
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Statistics
Determine the value of 𝜎 2 , where the
population has values {2, 0, 1, 5}. Express
your answer as a common fraction.
Note: 𝜎 2 is the variance.
Statistics
14
Answer:
3
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Computations
Calculate: 15.37 × 7.88 −
9.37 × 7.38 + 1.537 × 21.2 −
93.7 × 0.262 =
Computations
Answer:60
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Conics
What is the 𝑦-coordinate of the
focus of the parabola 𝑦 = 𝑥 2 ?
Express your answer as a common
fraction.
Conics
1
Answer:
4
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Complex Numbers
2+𝑖
1−𝑖
Express
as a complex number in
the form 𝑎 + 𝑏𝑖, where 𝑎 and 𝑏 are
common fractions.
Complex Numbers
1
Answer:
2
+
3
𝑖
2
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Trigonometry
Determine the smallest possible
positive value of 𝑥 for which
sin 2𝑥 = cos(2𝑥). Note that
0 < 𝑥 < 2𝜋.
Trigonometry
𝜋
Answer:
8
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Algebra
What is the average value of all
the terms in the arithmetic
sequence: 31, 62, 93, …, 2015?
Algebra
Answer: 1023
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Combinatorics
A permutation of the digits 1, 2, 3,
4, 5, 6, 7, 8, 9 forms a 9-digit
integer which is divisible by 2, 3,
4, 6, 8, and 9. How many such
permutations are there?
Combinatorics
Answer: 4320 permutations
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Algebra
When 213 is divided by 𝑛, the
remainder is 3. How many positive
integer values for 𝑛 are there?
Algebra
Answer: 13 values
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Geometry
A rectangular prism has three
different face areas: 15, 20, 25 in no
particular order. What’s the volume of
the rectangular prism? Express your
answer in simplest radical form.
Geometry
Answer:50 3
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Geometry
Trapezoid 𝐴𝐵𝐶𝐷 has 𝐴𝐵 = 𝐵𝐶 = 𝐶𝐷
and a circle is circumscribed about
𝐴𝐵𝐶𝐷 such that 𝐴𝐷 is the diameter of
the circle. Given that 𝐴𝐷 = 4, what’s
the area of the trapezoid? Express your
answer in simplest radical form.
Geometry
Answer:3 3
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Algebra
How many real solutions does
4
3
2
𝑥 +𝑥 +𝑥 +𝑥+1=0
have?
Algebra
Answer:0 solutions
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Algebra
A boat takes 3 hours to travel
downstream from Port A to Port B,
and 4 hours to travel upstream from
Port B to Port A. The speed of the
river is 3km/hr. What’s the distance
between Port A and Port B, in km?
Algebra
Answer:72 km
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Number Theory
When the following fractions are listed
from the smallest to the largest, which
fraction is in the middle?
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89
Number Theory
Answer:
6
19
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Number Theory
2015
2048
When
is converted to decimal
form, what is the length of the
decimal part of the number (the part
after the decimal point)?
Number Theory
Answer:11 digits
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Geometry
The Ancient Chinese used a very ingenious
method to determine the volume of a body of
water. First a 10m rod is placed into the water, and
the amount by which the rod exceeds the water
level is recorded to be 2m. The rod is then tilted
diagonally to the side until it touches the midpoint
of an edge of the body of water, at which point the
entire rod is just precisely submerged. Assuming
that the body of water has a cross sectional area of
a square, determine the volume of contained in the
body of water, in 𝑚3 .
Geometry
Answer:1152 𝑚
3
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Combinatorics
The combination of letters “VMO” has
the following properties:
(i)
It contains three distinct capital
letters from the English alphabet
(ii) All three letters contain one
vertical line of symmetry, when
capitalized
How many permutations of letters,
including “VMO”, satisfy the above two
properties?
Combinatorics
Answer:990 permutations
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Combinatorics
Two distinct letters of the English
alphabet are randomly chosen. What’s
the probability that these two letters
are consecutive in the alphabet?
Combinatorics
1
Answer:
13
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Computations
What is the smallest positive
𝑛
integer 𝑛 such that 𝑖=1 𝑖 >
2015?
Computations
Answer:63
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Computations
It turns out that
0.5 × 0.8 × 0.04 × 1.25 × 0.2 × 0.025 = 10𝑎
where 𝑎 is an integer. What is the value of
𝑎?
Computations
Answer:−4
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Statistics
At the 2014 International Mathematical Olympiad (IMO), there
were 6 problems worth 7 points each.
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The fourth quartile of scorers had an average score of 4
points.
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The third quartile of scorers had an average score of 10
points.
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The second quartile of scorers (Bronze medalists) had an
average score of 19 points.
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The 76th to 95th percentile scorers (Silver medalists) had an
average score of 25 points.
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The 96th to 100th percentile scorers (Gold medalists) had an
average score of 35 points.
What is the average score of all of the contestants at the 2014
IMO?
Statistics
Answer:15 points
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Number Theory
Earth orbits the Sun once every Earth
1
year; Mercury orbits the Sun once every
4
of an Earth year; the Moon orbits the
1
Earth once every of an Earth year.
13
Suppose Earth, Moon, and Mercury are
right now collinear with the Sun’s center.
What’s the least number of years it will
take until the Earth, Moon, and Mercury
are collinear with the Sun’s center again?
Number Theory
Answer:1 year
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Algebra
The ship Yamato II has been struck by a
small iceberg. When the leak was
discovered, 800 tons of water had
already gushed into the ship. Two water
pumps are then immediately installed
and begin pumping water out at 18 tons
and 14 tons per minute, respectively. 50
minutes later, all of the water has been
pumped out. How much water gushes
into the Yamato II per minute, in tons?
Algebra
Answer:16 tons
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Algebra
There are two baskets of fruits. The larger basket
weighs 4kg more than triple the weight of the
smaller basket. If 5kg of fruit is transferred from
the smaller basket to the larger basket, then the
larger basket is exactly six times the weight of
the smaller basket. What is the sum of the
weights of the two baskets of fruits, in kg?
Algebra
Answer:56kg
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Number Theory
A rectangular sheet of paper has
dimensions 112cm by 80cm. I want
to cut the sheet up into a number of
congruent squares with integer side
lengths, with no parts of the sheet
left over. I want to maximize the
area of each individual square. How
many square can I cut?
Number Theory
Answer:35 squares
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Geometry
A cylinder of diameter 10 is inscribed
in a cone of base radius 20 and height
20, with one of its circular bases on the
base of the cone. What is the total
surface area of the cylinder? Express
your answer in terms of 𝜋.
Geometry
Answer:200𝜋
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Geometry
Right triangle 𝐴𝐵𝐶 has 𝐴𝐶 = 3 and
𝐵𝐶 = 4. A circle is constructed
with 𝐵𝐶 as its diameter. Other than
point 𝐵, the circle intersects 𝐴𝐵 at
another point. Call this point 𝐷.
What’s the length of 𝐵𝐷? Express
your answer as a common fraction.
Geometry
Answer:
16
5
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Algebra
Alex, Brittany, Carl, Danielle, and
Erwin compete in a very difficult math
competition. Brittany, Carl, Danielle,
and Erwin received scores of 91, 82, 79,
and 78 respectively, out of 100. Alex’s
score is 6 points higher than the average
of all five competitors’ scores. What is
Alex’s score?
Algebra
Answer:90
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Algebra
It takes Hose A 10 hours by itself to fill
a pool, and it takes Hose B 8 hours by
itself to fill a pool. Given that Hose A
pumps out 3 tons more water per hour
than Hose B, what is the capacity of the
pool, in tons?
Algebra
Answer:120 tons
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Computations
Calculate
4.8×7.5×8.1
2.4×2.5×2.7
Computations
Answer:18
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Number Theory
On a highway there are 25 electric poles,
of which the distance between any two
consecutive poles is 45m. Now, the city
wants to increase the distance between
any two consecutive poles to 60m. What
is the maximum number of electric poles
that need not be moved?
Number Theory
Answer:7 poles
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Number Theory
Two numbers 𝑎, 𝑏, satisfy gcd(𝑎, 𝑏) =
6 and lcm(𝑎, 𝑏) = 144. Other than the
obvious pair (6, 144), what is the other
possible pair for (𝑎, 𝑏), where 𝑎 < 𝑏?
Number Theory
Answer:(18, 48)
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Geometry
In a circle of radius 1 + 2, 4 mutually
tangent and congruent circles are
inscribed. Each of the 4 circles iis also
internally tangent to the circle of radius
1 + 2. What is the area outside the 4
smaller circles but inside the larger
circle. Express your answer in the form
𝑎𝜋 𝑏 − 𝑐𝜋.
Geometry
Answer:2𝜋 2 − 𝜋
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Geometry
Water is poured into a sphere of radius
2 until the water level is 1 unit deep at
its deepest point. What is the area of
the water surface in the sphere?
Express your answer in terms of 𝜋.
Geometry
Answer: 3𝜋
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Combinatorics
Suppose the UN Universal Declaration
of Human Rights has 100 articles.
Grace memorized 75 consecutive
articles. Hazel memorized 60
consecutive articles. Ivy memorized 52
consecutive articles. What is the
minimum number of articles that all
three have memorized?
Combinatorics
Answer:12 articles
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Combinatorics
How many rectangles have all their vertices
as lattice points either on or within the
rectangle bounded by (0, 0), (0, 4), (4, 4), and
(4, 0) such that all of the edges of the
rectangle are either parallel to the 𝑥-axis or 𝑦axis?
Note: a lattice point is a point with integer
coordinates.
Combinatorics
Answer:100 rectangles
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Algebra
A bamboo stick is inserted into a
container of water, soaking 40cm of the
bottom portion of the bamboo stick. The
same bamboo stick is then inverted and
inserted into the container of water again.
Now, the portion of the bamboo stick that
is soaked is 13cm longer than half of the
total length of the bamboo stick. How
long is the bamboo stick, in cm?
Algebra
Answer:134cm
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Algebra
The square of an integer is equal to 12
multiplied by the integer plus 108.
What is the sum of all possible
integers which satisfy this property?
Algebra
Answer:12
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Combinatorics
In a multiple choice exam, there
were 50 questions. For every correct
answer, 4 points were awarded. For
every blank, 1 point was awarded.
For every incorrect answer, no
points were awarded. How many
possible scores, out of 200, can be
achieved?
Combinatorics
Answer:197 scores
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Combinatorics
What is the minimum number of
distinct positive integers that need to
be randomly chosen from the set of
natural numbers such that there must
exists a pair of positive integers
from those chosen whose difference
is divisible by 2015?
Combinatorics
Answer:2016 integers
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Combinatorics
A bag contains 5 blue marbles and 7 red
marbles. Two marbles are drawn without
replacement. Given that the first marble
was not blue, what’s the probability that
the two marbles are the same color?
Combinatorics
7
Answer:
22
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Algebra
Spiders have 8 legs; dragonflies have 6
legs and two pairs of wings; moths have
6 legs and one pair of wings. There are
18 bugs in my bug collection, with a
total of 118 legs, and 20 pairs of wings.
How many of the legs belong to
spiders?
Algebra
Answer:40 legs
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Geometry
Triangle 𝐴𝐵𝐶 has 𝐴𝐶 = 5, 𝐵𝐶 = 12,
and 𝐴𝐵 = 13. Point 𝐶 is folded so that
edge 𝐴𝐶 coincides with edge 𝐴𝐵. Once
the folding is complete, what is the area
of the portion of the triangle that is only
one layer of paper thick? Express your
answer as a common fraction.
Geometry
Answer:
40
3
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Geometry
A solid whose edges are all
straight lines and whose faces are
all polygons has 25 faces, 60
edges, and how many vertices?
Geometry
Answer: 37 vertices
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Algebra
A train takes 35 seconds to go through
a 540m tunnel, or 53 seconds to go
through a 846m tunnel. What is the
length of the train, in m?
Note: the timer starts when the front
of the train enters the tunnel, and stops
when the rear of the train leaves the
tunnel.
Algebra
Answer: 55m
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Number Theory
There are 90 candies in a pile. Mr. Factor is
distributing these candy to the kids in his
class. It turns out that when every kid
receives the same number of candies, there
are no leftovers. What is the number of
possible class sizes that Mr. Factor could
have?
Number Theory
Answer: 12 class sizes
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Number Theory
The product of three positive
integers is 84. The sum of two of
these integers is equal to the third.
What is the sum of these three
integers?
Number Theory
Answer: 14
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Algebra
A three digit positive integer 𝑎𝑏𝑐 has units
digit 5. If a new three digit number is
formed with the digits 𝑐𝑎𝑏, in that order,
then the resulting number is 108 smaller
than the original number. What is the
original number?
Algebra
Answer:675
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Algebra
Train A and train B travel at 23m/s and 16m/s
respectively. When the two trains are lined up
head to head and begin travelling
simultaneously, then it takes train A 30 seconds
to fully surpass train B. When the two trains are
lined up tail to tail and begin travelling
simultaneously, then it takes train A 26 seconds
to fully surpass train B. What’s the combined
length of the two trains, in m?
Algebra
Answer:336m
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Combinatorics
How many distinct right angle
triangles with integer legs lengths have
area 48?
Note: only the two legs must be
integers, the hypotenuse need not be an
integer.
Combinatorics
Answer:12 triangles
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Number Theory
The fourteen digit number
1𝑎2𝑎3𝑎4𝑎5𝑎6𝑎7𝑎 is divisible
by 11. What is the value of the
digit 𝑎?
Number Theory
Answer:4
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Number Theory
What is the largest prime that divides
215215?
Number Theory
Answer: 43
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Geometry
A regular polygon with 𝑛 sides
has all of its internal diagonals
drawn. It turns out that there are
exactly 49 distinct points inside
the polygon where two or more
diagonals intersect. What is the
value of 𝑛?
Geometry
Answer:8
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Geometry
A 3 by 4 by 6 rectangular prism is on the floor
with one of its 3 by 6 faces touching the floor.
The rectangular prism is then rotated using
one of the length 6 edges touching the floor as
a hinge. After the rotation, one of the 4 by 6
faces is touching the floor. In terms of 𝜋, what
is the distance travelled by a point on the edge
opposing the edge which acted as the rotating
hinge?
Geometry
5𝜋
Answer:
2
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Geometry
If the side lengths of a square is
each increased by 5cm, then the
resulting square has an area
2
which is 95cm more than the
area of the original square.
What is the area of the original
2
square, in cm ?
Geometry
Answer:49cm
2
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Number Theory
The Fibonacci sequence 1, 1, 2, 3, 5, 8, 13,
21, … is constructed beginning with 1, 1,
and beginning with the third term, every
term is equal to the sum of the previous
two terms. How many of the first 1000
terms of the Fibonacci sequence are even?
Number Theory
Answer: 333 terms
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Algebra
A book has 380 pages. How many
digits must be printed so that the
book has a complete set of page
numbers? For example, if a book has
10 pages, 11 digits must be printed.
Algebra
Answer:1032 digits
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Algebra
Kevin writes out the first 1000
natural numbers. How many times
did write the digit “1”?
Algebra
Answer:301
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Geometry
A convex nondegenerate quadrilateral
has perpendicular diagonals of length 6
and 11. What is the area of the
quadrilateral?
Geometry
Answer:33
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Geometry
In convex quadrilateral ABCD, M is
the midpoint of AB and N is the
midpoint of CD. If quadrilateral
ABCD has area 2016, what is the
area of quadrilateral DMBN?
Geometry
Answer:1008
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Number Theory
1
11
If is converted into decimal form,
what is the value of the 2015th digit
after the decimal point?
Number Theory
Answer:0
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Combinatorics
How many ways are there to give out
10 candies to 3 children such that one
child must have at least 5 candies?
Note: not every child needs to have a
piece of candy.
Combinatorics
Answer: 84 ways
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Combinatorics
How many of the first 100 natural
numbers are neither multiples of 3 nor
5?
Combinatorics
Answer: 53 numbers
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Algebra
2
Let 𝑓 𝑥 = 𝑥 − 2. What is the
minimum possible value of 𝑓(𝑓 𝑥 )
as 𝑥 ranges over the domain of
𝑓 𝑓 𝑥 ?
Algebra
Answer: −2
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Computations
Evaluate:
10
3
𝑛
𝑛=1
Computations
Answer:55
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Computations
Determine the largest integer
10
smaller than 𝑖=1 log 𝑖
Computations
Answer: 6
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Computations
The base 10 number 2015 in base 2
is 11111011111, which is a
palindrome that reads the same
forwards and backwards. What’s the
smallest base 10 number larger than
2015 whose base 2 representation is
also a palindrome?
Computations
Answer:2047
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Algebra
What is the minimum value of the
𝑥 2 +9
𝑥
function 𝑓 𝑥 =
as 𝑥 ranges
over the positive real numbers?
Algebra
Answer:6
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Geometry
What’s the volume of a
tetrahedron with edge length 1?
Express your answer in simplest
radical form.
Geometry
2
Answer:
12
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Geometry
How many 2D nets are there for a
cube? A net is a single 2D shape that
can be folded along straight edges to
form an appropriate 3D shape.
Geometry
Answer:11 nets
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Number Theory
When consecutive terms from the
Fibonacci sequence are divided by
2, the remainders are: 0, 1, 1, 0, 1,
1…and repeats every three terms.
When consecutive terms from the
Fibonacci sequence are divided by
3, the remainders repeat after how
many terms?
Number Theory
Answer: 8 terms
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Algebra
A cube has side length 2015 and volume
𝑉. When one dimension of the cube is
increased by 1, another dimension
decreased by 1, and the third dimension
unchanged, the new volume of the
rectangular prism formed is 𝑊. What’s the
value of 𝑉 − 𝑊?
Algebra
Answer:2015
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Geometry
The solid described by
2
2
2
𝑥 + 𝑦 + 𝑧 = 3 in the
𝑥 − 𝑦 − 𝑧 plane has volume 𝑉.
What is the value of 𝑉, in simplest
radical form?
Geometry
Answer:4𝜋 3
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Combinatorics
In Tic-Tac-Toe, how many unique
drawn positions are there? A drawn
position is one in which all 9 squares
of the board are filled by 5 X’s and 4
O’s, and neither player has a three in a
row. Two positions are called unique if
one cannot be rotated to get the other.
Combinatorics
Answer:3 positions