Found problems: 53
2023 LMT Fall, 3C
Determine the least integer $n$ such that for any set of $n$ lines in the 2D plane, there exists either a subset of $1001$ lines that are all parallel, or a subset of $1001$ lines that are pairwise nonparallel.
[i]Proposed by Samuel Wang[/i]
[hide=Solution][i]Solution.[/i] $\boxed{1000001}$
Since being parallel is a transitive property, we note that in order for this to not exist, there must exist at most $1001$ groups of lines, all pairwise intersecting, with each group containing at most $1001$ lines. Thus, $n = 1000^2 + 1 =
\boxed{1000001}$.[/hide]
2024 LMT Fall, B4
Let $S$, $K$, $I$, $B$, $D$, $Y$ be distinct integers from $0$ to $9,$ inclusive. Given that they follow this equation:
$$\begin{array}{rrrrr}
& S & K & I & B \\
- & I & D & I & D \\
\hline
& & & D & Y
\end{array}$$find the maximum value of $\overline{SKIBIDI}$.
2024 LMT Fall, A1
In Genshin Impact, $PRIMOGEM'$ is the octagon in the diagram below. Let $A$ be the intersection of $PO$ and $IE$. Suppose $PR=RI=IM=MO=OG=GE=EM'=M'P$, $AP=AI=AO=AE=4$, and $AR=AM=AG=AM'=\sqrt{2}$. Find the area of $PRIMOGEM'$.
[asy]
size(5cm);
pair P = (0, 4), R = (1, 1), I = (4, 0), M = (1, -1), O = (0, -4), G = (-1, -1), E = (-4, 0), MM = (-1, 1), origin = (0, 0);
draw(P--R--I--M--O--G--E--MM--P);
draw(origin--P);
draw(origin--I);
draw(origin--O);
draw(origin--E);
draw(R--G);
draw(MM--M);
label("$P$", P, N);
label("$R$", R, NE);
label("$I$", I, SE);
label("$M$", M, SE);
label("$G$", G, SW);
label("$E$", E, W);
label("$M'$", MM, NW);
label("$O$", O, S);
[/asy]