MathDB

20

Part of 2015 AMC 12/AHSME

Problems(2)

Same Perimeter and Area, but Different Triangle

Source: 2015 AMC 12A #20

2/4/2015
Isosceles triangles TT and T′T' are not congruent but have the same area and the same perimeter. The sides of TT have lengths 55, 55, and 88, while those of T′T' have lengths aa, aa, and bb. Which of the following numbers is closest to bb?
<spanclass=′latex−bold′>(A)</span>3<spanclass=′latex−bold′>(B)</span>4<spanclass=′latex−bold′>(C)</span>5<spanclass=′latex−bold′>(D)</span>6<spanclass=′latex−bold′>(E)</span>8<span class='latex-bold'>(A) </span>3\qquad<span class='latex-bold'>(B) </span>4\qquad<span class='latex-bold'>(C) </span>5\qquad<span class='latex-bold'>(D) </span>6\qquad<span class='latex-bold'>(E) </span>8
geometryperimeterquadraticsarea of a triangleHeron's formulaalgebraquadratic formula
Modulo Ackermann

Source: 2015 amc 12b #20

2/26/2015
For every positive integer nn, let mod5⁡(n)\operatorname{mod_5}(n) be the remainder obtained when nn is divided by 55. Define a function f:{0,1,2,3,… }×{0,1,2,3,4}→{0,1,2,3,4}f : \{0, 1, 2, 3, \dots\} \times \{0, 1, 2, 3, 4\} \to \{0, 1, 2, 3, 4\} recursively as follows: f(i,j)={mod5⁡(j+1)if i=0 and 0≤j≤4f(i−1,1)if i≥1 and j=0, andf(i−1,f(i,j−1))if i≥1 and 1≤j≤4f(i, j) = \begin{cases} \operatorname{mod_5}(j+1) & \text{if }i=0\text{ and }0\leq j\leq 4 \\ f(i-1, 1) & \text{if }i\geq 1\text{ and }j=0 \text{, and}\\ f(i-1, f(i, j-1)) & \text{if }i\geq 1\text{ and }1\leq j\leq 4 \end{cases} What is f(2015,2)f(2015, 2)?
<spanclass=′latex−bold′>(A)</span>0<spanclass=′latex−bold′>(B)</span>1<spanclass=′latex−bold′>(C)</span>2<spanclass=′latex−bold′>(D)</span>3<spanclass=′latex−bold′>(E)</span>4<span class='latex-bold'>(A) </span>0 \qquad<span class='latex-bold'>(B) </span>1 \qquad<span class='latex-bold'>(C) </span>2 \qquad<span class='latex-bold'>(D) </span>3 \qquad<span class='latex-bold'>(E) </span>4
functioninductionalgebradomainratiogeometric sequenceAMC