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Difference between revisions of "2015 UNM-PNM Statewide High School Mathematics Contest II Problems"

(Created page with "UNM - PNM STATEWIDE MATHEMATICS CONTEST XLVII. February 7, 2014. Second Round. Three Hours ==Problem 1== In the sequence <math>1, 2, 2, 4, 4, 4, 4, 8, 8, 8, 8, 8, 8, 8, 8, \...")
 
m
Line 4: Line 4:
  
 
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>?
  
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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]]
 
[[UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 4|Solution]]
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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>|P A| = 3|P B|</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]]
 
[[UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 5|Solution]]
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==Problem 6==
 
==Problem 6==
  
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A faulty calculator displayed <math>\circle 38 \circle 1625</math> as an output of a calculation. We know that
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A faulty calculator displayed <math>\diamond 38 \diamond 1625</math> as an output of a calculation. We know that
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two of the digits of this number are missing and these are replaced with the symbol %\circle<math>.
+
two of the digits of this number are missing and these are replaced with the symbol <math>\diamond</math>.
−
Furthermore, we know that </math>9<math> and </math>11<math> divide the computed output. What are the missing
+
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?
  
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==Problem 7==
 
==Problem 7==
  
−
Let </math>A<math> be the average of the three numbers </math>sin 2α, sin 2β<math> and </math>sin 2γ<math> where </math>α + β + γ = π<math>.
+
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 α sin β sin γ in terms of </math>A<math>.
+
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]]
 
[[UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 7|Solution]]
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==Problem 8==
 
==Problem 8==
  
−
Suppose we draw circles of radius </math>r<math> with centers at every point in the plane with integer
+
Suppose we draw circles of radius <math>r</math> with centers at every point in the plane with integer
−
coordinates. What is the smallest </math>r<math> such that every line with slope </math>2/7$ has a point in
+
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?
  

Revision as of 03:13, 12 January 2019

UNM - PNM STATEWIDE MATHEMATICS CONTEST XLVII. February 7, 2014. Second Round. Three Hours

Problem 1

In the sequence $1, 2, 2, 4, 4, 4, 4, 8, 8, 8, 8, 8, 8, 8, 8, \cdots$ what number occupies position $2015$?

Solution

Problem 2

Show that if $S$ 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 $S$. Is the claim true if $S$ has infinitely many points? Hint: Use an extremal configuration.

Solution

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, \[AN/NB = CA/CB\] 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.

Solution

Problem 4

There are $12$ 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: $4=3+1$.

Solution

Problem 5

Let $A$ and $B$ be two points in the plane. Describe the set $S$ of all points in the plane such that for any point $P$ in $S$ we have \[|PA| = 3|PB|\].

Solution

Problem 6

A faulty calculator displayed $\diamond 38 \diamond 1625$ 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 $\diamond$. Furthermore, we know that $9$ and $11$ divide the computed output. What are the missing digits and the complete output of our calculation?

Solution

Problem 7

Let $A$ be the average of the three numbers $\sin{2\alpha}, \sin{2\beta} , \sin(2\gamma)$ where $\alpha + \beta + \gamma = \pi.$ Express the product $P = sin(\alpha) sin(\beta) sin(\gamma)$ in terms of $A$.

Solution

Problem 8

Suppose we draw circles of radius $r$ with centers at every point in the plane with integer coordinates. What is the smallest $r$ such that every line with slope $2/7$ has a point in common with at least one of these circles?


Solution

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?

Solution

Problem 10

Solve aa...a | {z } 2k − bb...b | {z } k =

cc...c | {z } k !2 , where cc...c | {z } k denotes a number with k digits each one equal to c

Solution

See Also

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