【全网首发】2018AIME II 2卷真题及答案-翰林国际教育

Triangle $ABC$ has side lengths $AB = 9$ , $BC =$ $5\sqrt{3}$ , and $AC = 12$ . Points $A = P_{0}, P_{1}, P_{2}, ... , P_{2450} = B$ are on segment $\overline{AB}$ with $P_{k}$ between $P_{k-1}$ and $P_{k+1}$ for $k = 1, 2, ..., 2449$ , and points $A = Q_{0}, Q_{1}, Q_{2}, ... , Q_{2450} = C$ are on segment $\overline{AC}$ with $Q_{k}$ between $Q_{k-1}$ and $Q_{k+1}$ for $k = 1, 2, ..., 2449$ . Furthermore, each segment $\overline{P_{k}Q_{k}}$ , $k = 1, 2, ..., 2449$ , is parallel to $\overline{BC}$ . The segments cut the triangle into $2450$ regions, consisting of $2449$ trapezoids and $1$ triangle. Each of the $2450$ regions has the same area. Find the number of segments $\overline{P_{k}Q_{k}}$ , $k = 1, 2, ..., 2450$ , that have rational length.

Problem8

A frog is positioned at the origin of the coordinate plane. From the point $(x, y)$ , the frog can jump to any of the points $(x + 1, y)$ , $(x + 2, y)$ , $(x, y + 1)$ , or $(x, y + 2)$ . Find the number of distinct sequences of jumps in which the frog begins at $(0, 0)$ and ends at $(4, 4)$ .

Problem9

Octagon $ABCDEFGH$ with side lengths $AB = CD = EF = GH = 10$ and $BC = DE = FG = HA = 11$ is formed by removing 6-8-10 triangles from the corners of a $23$ x $27$ rectangle with side $\overline{AH}$ on a short side of the rectangle, as shown. Let $J$ be the midpoint of $\overline{AH}$ , and partition the octagon into 7 triangles by drawing segments $\overline{JB}$ , $\overline{JC}$ , $\overline{JD}$ , $\overline{JE}$ , $\overline{JF}$ , and $\overline{JG}$ . Find the area of the convex polygon whose vertices are the centroids of these 7 triangles.

Problem 10

Find the number of functions $f(x)$ from $\{ 1,2,3,4,5 \}$ to $\{ 1,2,3,4,5 \}$ that satisfy $f(f(x)) = f(f(f(x)))$ for all $x$ in $\{ 1,2,3,4,5 \}$ .

Problem 11

Find the number of permutations of $1,2,3,4,5,6$ such that for each $k$ with $1\leq k\leq 5$ , at least one of the first $k$ terms of the permutation is greater than $k$ .

Problem 12

Let $ABCD$ be a convex quadrilateral with $AB=CD=10$ , $BC=14$ , and $AD=2\sqrt{65}$ . Assume that the diagonals of $ABCD$ intersect at point $P$ , and that the sum of the areas of $\triangle APB$ and $\triangle CPD$ equals the sum of the areas of $\triangle BPC$ and $\triangle APD$ . Find the area of quadrilateral $ABCD$ .

Problem 13

Misha rolls a standard, fair six-sided die until she rolls $1$ - $2$ - $3$ in that order on three consecutive rolls. The probability that she will roll the die an odd number of times is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .