Difference between revisions of "2010 UNCO Math Contest II Problems/Problem 11"

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== Solution ==
 
== Solution ==
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(a) <math>6</math> (b) <math>20</math> (c) <math>50</math> (d) <math>1\cdot n^2 + 2\cdot (n-1)^2+3\cdot (n-2)^2 + \cdots + n\cdot 1^2</math>
  
 
== See also ==
 
== See also ==

Latest revision as of 02:04, 13 January 2019

Problem

(a) The $3 \times 3$ square grid has $9$ dots equally spaced. How many squares (of all sizes) can you make using four of these dots as vertices? Two examples are shown.

[asy]  for (int x=0; x<3; ++x) {for (int y=0; y<3; ++y) {dot((x,y));dot((x+4,y));} } draw((0,0)--(1,0)--(1,1)--(0,1)--cycle,black); draw((4,1)--(5,0)--(6,1)--(5,2)--cycle,black); draw((3,-1)--(3,3),dashed);  [/asy]

(b) How many for a $4 \times 4$?

(c) How many for a $5 \times 5$?

(d) How many for an $(N+1) \times (N+1)$ grid of dots?

Solution

(a) $6$ (b) $20$ (c) $50$ (d) $1\cdot n^2 + 2\cdot (n-1)^2+3\cdot (n-2)^2 + \cdots + n\cdot 1^2$

See also

2010 UNCO Math Contest II (ProblemsAnswer KeyResources)
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