Difference between revisions of "2017 UNM-PNM Statewide High School Mathematics Contest II Problems"
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==Problem 4== | ==Problem 4== | ||
+ | Find a second-degree polynomial with integer coefficients, <math>p(x) = ax^2 + bx + c</math>, such that <math>p(1),p(3),p(5)</math>, and <math>p(7)</math> are perfect squares, but <math>p(2)</math> is not. | ||
[[2017 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 4|Solution]] | [[2017 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 4|Solution]] | ||
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==Problem 5== | ==Problem 5== | ||
+ | Find all real triples <math>(x,y,z)</math> which are solutions to the system: | ||
+ | |||
+ | <math>x^3 + x^2y + x^2z = 40</math> | ||
+ | |||
+ | <math>y^3 + y^2x + y^2z = 90</math> | ||
+ | |||
+ | <math>z^3 + z^2x + z^2y = 250</math> | ||
[[2017 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 5|Solution]] | [[2017 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 5|Solution]] | ||
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==Problem 6== | ==Problem 6== | ||
+ | There are <math>12</math> stacks of <math>12</math> coins. Each of the coins in <math>11</math> of the <math>12</math> stacks weighs <math>10</math> grams each. Suppose the coins in the remaining stack each weigh <math>9.9</math> grams. You are given one time access to a precise digital scale. Devise a plan to weigh some coins in precisely one weighing to determine which pile has the lighter coins. | ||
[[2017 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 6|Solution]] | [[2017 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 6|Solution]] | ||
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==Problem 7== | ==Problem 7== | ||
+ | Find a formula for | ||
+ | <math>sum_{k=0}^{</math>\lceil \frac{n}{4} \rceil } binom{n}{4k}<math> for any natural number </math>n$. | ||
[[2017 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 7|Solution]] | [[2017 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 7|Solution]] |
Revision as of 03:01, 15 January 2019
UNM - PNM STATEWIDE MATHEMATICS CONTEST XLIX. February 4, 2017. Second Round. Three Hours
Contents
Problem 1
What are the last two digits of ?
Problem 2
Suppose , and all denote distinct digits from to . If , what are , and ?
Problem 3
Let and .
(a) Determine and .
(b) Denote . Determine all the functions in the set or for some a whole number.
Problem 4
Find a second-degree polynomial with integer coefficients, , such that , and are perfect squares, but is not.
Problem 5
Find all real triples which are solutions to the system:
Problem 6
There are stacks of coins. Each of the coins in of the stacks weighs grams each. Suppose the coins in the remaining stack each weigh grams. You are given one time access to a precise digital scale. Devise a plan to weigh some coins in precisely one weighing to determine which pile has the lighter coins.
Problem 7
Find a formula for $sum_{k=0}^{$ (Error compiling LaTeX. Unknown error_msg)\lceil \frac{n}{4} \rceil } binom{n}{4k}n$.