There are 2023 dice on a table. For one dollar, choose one die and turn any one of its four side faces to the top. What is the least sum that always suffices to make all 2023 top numbers equal? Faces are numbered 1 through 6, with opposite numbers summing to 7.
For a nonnegative integer $n$, define $Q(x)=\lfloor x\rfloor+\lfloor x/2\rfloor+\cdots+\lfloor x/10^n\rfloor$. Find $Q(10^n)-Q(10^n-1)$. Here $\lfloor x\rfloor$ is the greatest integer not exceeding $x$.
Let $I$ be the incenter of triangle $ABC$, and let $N=BI\cap AC$. The tangent at $A$ to the circumcircle of $AIN$ meets the tangent at $C$ to the circumcircle of $CIN$ at $D$. Prove that $AC\perp DI$.
Let $a_1,a_2,\ldots$ and $b_1,b_2,\ldots$ be infinite strictly increasing arithmetic progressions of positive numbers. Suppose $a_k/b_k$ is an integer for every $k$. Can this ratio depend on $k$?
Every distance between two of five given points is greater than 2. Must some pair be at distance greater than 3 if the points lie (a) in a plane; (b) in three-dimensional space?
