Difference between revisions of "2015 UNM-PNM Statewide High School Mathematics Contest II Problems"
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− | UNM - PNM STATEWIDE MATHEMATICS CONTEST XLVII. February 7, | + | UNM - PNM STATEWIDE MATHEMATICS CONTEST XLVII. February 7, 2015. Second Round. Three Hours |
==Problem 1== | ==Problem 1== | ||
In the sequence | In the sequence | ||
− | <math>1, 2, 2, 4, 4, 4, 4, 8, 8, 8, 8, 8, 8, 8, 8, \cdots</math> | + | <math>1, 2, 2, 4, 4, 4, 4, 8, 8, 8, 8, 8, 8, 8, 8, \cdots </math> |
what number occupies position <math>2015</math>? | what number occupies position <math>2015</math>? | ||
− | [[UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 1|Solution]] | + | [[2015 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 1|Solution]] |
==Problem 2== | ==Problem 2== | ||
Line 16: | Line 16: | ||
configuration. | configuration. | ||
− | [[UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 2|Solution]] | + | [[2015 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 2|Solution]] |
==Problem 3== | ==Problem 3== | ||
Line 27: | Line 27: | ||
to the adjacent sides. You do not have to write a proof of this fact. | to the adjacent sides. You do not have to write a proof of this fact. | ||
− | [[UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 3|Solution]] | + | [[2015 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 3|Solution]] |
==Problem 4== | ==Problem 4== | ||
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There are <math>12</math> coins in a parking meter and we know that one of them is counterfeit. The | There are <math>12</math> coins in a parking meter and we know that one of them is counterfeit. The | ||
counterfeit coin is either heavier or lighter than the others. How can we find the fake coin | counterfeit coin is either heavier or lighter than the others. How can we find the fake coin | ||
− | and also if it is heavier or lighter in three weighings using a balance scale? Hint:<math>4=3+1</math>. | + | and also if it is heavier or lighter in three weighings using a balance scale? Hint: <math>4=3+1</math>. |
− | [[UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 4|Solution]] | + | [[2015 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 4|Solution]] |
==Problem 5== | ==Problem 5== | ||
Let <math>A</math> and <math>B</math> be two points in the plane. Describe the set <math>S</math> of all points in the plane such | Let <math>A</math> and <math>B</math> be two points in the plane. Describe the set <math>S</math> of all points in the plane such | ||
− | that for any point <math>P</math> in <math>S</math> we have <cmath>| | + | that for any point <math>P</math> in <math>S</math> we have <cmath>|PA| = 3|PB|</cmath>. |
− | [[UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 5|Solution]] | + | [[2015 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 5|Solution]] |
==Problem 6== | ==Problem 6== | ||
− | A faulty calculator displayed <math>\ | + | A faulty calculator displayed <math>\diamond 38 \diamond 1625</math> as an output of a calculation. We know that |
− | two of the digits of this number are missing and these are replaced with the symbol | + | two of the digits of this number are missing and these are replaced with the symbol <math>\diamond</math>. |
− | Furthermore, we know that < | + | Furthermore, we know that <math>9</math> and <math>11</math> divide the computed output. What are the missing |
digits and the complete output of our calculation? | digits and the complete output of our calculation? | ||
− | [[UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 6|Solution]] | + | [[2015 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 6|Solution]] |
==Problem 7== | ==Problem 7== | ||
− | Let < | + | Let <math>A</math> be the average of the three numbers <math>\sin{2\alpha}, \sin{2\beta} , \sin(2\gamma)</math> where <math>\alpha + \beta + \gamma = \pi.</math> |
− | Express the product P = sin | + | Express the product <math>P = sin(\alpha) sin(\beta) sin(\gamma)</math> in terms of <math>A</math>. |
− | [[UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 7|Solution]] | + | [[2015 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 7|Solution]] |
==Problem 8== | ==Problem 8== | ||
− | Suppose we draw circles of radius < | + | Suppose we draw circles of radius <math>r</math> with centers at every point in the plane with integer |
− | coordinates. What is the smallest < | + | coordinates. What is the smallest <math>r</math> such that every line with slope <math>2/7</math> has a point in |
common with at least one of these circles? | common with at least one of these circles? | ||
− | [[UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 8|Solution]] | + | [[2015 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 8|Solution]] |
==Problem 9== | ==Problem 9== | ||
Line 74: | Line 74: | ||
all three lie in a semicircle? | all three lie in a semicircle? | ||
− | [[UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 9|Solution]] | + | [2015 [UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 9|Solution]] |
==Problem 10== | ==Problem 10== | ||
− | Solve aa | + | Solve <math>\underbrace{\overline{aa\ldots a}}_{\text{2k}} - \underbrace{\overline{bb\ldots b}}_{\text{k}}=(\underbrace{\overline{cc\ldots c}}_{\text{k}})^2</math> |
− | + | , where <math>(\underbrace{\overline{cc\ldots c}}_{\text{k}}</math> | |
− | + | denotes a number with <math>k</math> digits each one | |
− | + | equal to <math>c</math>. | |
− | k | ||
− | |||
− | + | [[2015 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 10|Solution]] | |
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− | [[UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 10|Solution]] | ||
==See Also == | ==See Also == |
Latest revision as of 03:48, 12 January 2019
UNM - PNM STATEWIDE MATHEMATICS CONTEST XLVII. February 7, 2015. Second Round. Three Hours
Contents
Problem 1
In the sequence
what number occupies position
?
Problem 2
Show that if is a set of finitely many non-collinear points in the plane (i.e., not all of
the points are on the same line), then there is a line which contains exactly two of the
points of
. Is the claim true if
has infinitely many points? Hint: Use an extremal
configuration.
Problem 3
Show that the bisect of an angle in a triangle divides the opposite side in segments whose
lengths have the same ratio as the ratio of the adjacent sides,
in the picture below. NOTE: The same is true for the bisector of an exterior angle of a
triangle, i.e., it divides the opposite side externally into segments that are proportional
to the adjacent sides. You do not have to write a proof of this fact.
Problem 4
There are coins in a parking meter and we know that one of them is counterfeit. The
counterfeit coin is either heavier or lighter than the others. How can we find the fake coin
and also if it is heavier or lighter in three weighings using a balance scale? Hint:
.
Problem 5
Let and
be two points in the plane. Describe the set
of all points in the plane such
that for any point
in
we have
.
Problem 6
A faulty calculator displayed as an output of a calculation. We know that
two of the digits of this number are missing and these are replaced with the symbol
.
Furthermore, we know that
and
divide the computed output. What are the missing
digits and the complete output of our calculation?
Problem 7
Let be the average of the three numbers
where
Express the product
in terms of
.
Problem 8
Suppose we draw circles of radius with centers at every point in the plane with integer
coordinates. What is the smallest
such that every line with slope
has a point in
common with at least one of these circles?
Problem 9
What is the probability of picking at random three points on a circle of radius one so that all three lie in a semicircle?
[2015 [UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 9|Solution]]
Problem 10
Solve
, where
denotes a number with
digits each one
equal to
.