MathDB

9

Part of 2022 AIME Problems

Problems(2)

RBBYGGYROPPO GANGNAM STYLE

Source: AIME I #9

2/9/2022
Ellina has twelve blocks, two each of red (R),\left({\bf R}\right), blue (B),\left({\bf B}\right), yellow (Y),\left({\bf Y}\right), green (G),\left({\bf G}\right), orange (O),\left({\bf O}\right), and purple (P).\left({\bf P}\right). Call an arrangement of blocks even if there is an even number of blocks between each pair of blocks of the same color. For example, the arrangement R B B Y G G Y R O P P O {\text {\bf R B B Y G G Y R O P P O}}is even. Ellina arranges her blocks in a row in random order. The probability that her arrangement is even is mn\frac mn, where mm and nn are relatively prime positive integers. Find m+nm+n.
AIMEAIME I
When MAA can't decide between geo and combo

Source: 2022 AIME II Problem 9

2/17/2022
Let ℓA\ell_A and ℓB\ell_B be two distinct parallel lines. For positive integers mm and nn, distinct points A1,A2,A3,…,AmA_1, A_2, \allowbreak A_3, \allowbreak \ldots, \allowbreak A_m lie on ℓA\ell_A, and distinct points B1,B2,B3,…,BnB_1, B_2, B_3, \ldots, B_n lie on ℓB\ell_B. Additionally, when segments AiBj‾\overline{A_iB_j} are drawn for all i=1,2,3,…,mi=1,2,3,\ldots, m and j=1,2,3,…,nj=1,\allowbreak 2,\allowbreak 3, \ldots, \allowbreak n, no point strictly between ℓA\ell_A and ℓB\ell_B lies on more than two of the segments. Find the number of bounded regions into which this figure divides the plane when m=7m=7 and n=5n=5. The figure shows that there are 8 regions when m=3m=3 and n=2n=2. [asy] import geometry; size(10cm); draw((-2,0)--(13,0)); draw((0,4)--(10,4)); label("ℓA\ell_A",(-2,0),W); label("ℓB\ell_B",(0,4),W); point A1=(0,0),A2=(5,0),A3=(11,0),B1=(2,4),B2=(8,4),I1=extension(B1,A2,A1,B2),I2=extension(B1,A3,A1,B2),I3=extension(B1,A3,A2,B2); draw(B1--A1--B2); draw(B1--A2--B2); draw(B1--A3--B2); label("A1A_1",A1,S); label("A2A_2",A2,S); label("A3A_3",A3,S); label("B1B_1",B1,N); label("B2B_2",B2,N); label("1",centroid(A1,B1,I1)); label("2",centroid(B1,I1,I3)); label("3",centroid(B1,B2,I3)); label("4",centroid(A1,A2,I1)); label("5",(A2+I1+I2+I3)/4); label("6",centroid(B2,I2,I3)); label("7",centroid(A2,A3,I2)); label("8",centroid(A3,B2,I2)); dot(A1); dot(A2); dot(A3); dot(B1); dot(B2); [/asy]
AMCAIMEAIME II