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Omni-MATH / For each positive digit D and positive integer k, we use the symbol D_((k)) to represent the positive integer having exactly k digits, each of which is equal to D. For example, …

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For each positive digit DD and positive integer kk, we use the symbol D(k)D_{(k)} to represent the positive integer having exactly kk digits, each of which is equal to DD. For example, 2(1)=22_{(1)}=2 and 3(4)=33333_{(4)}=3333. There are NN quadruples (P,Q,R,k)(P, Q, R, k) with P,QP, Q and RR positive digits, kk a positive integer with k≤2018k \leq 2018, and P(2k)−Q(k)=(R(k))2P_{(2k)}-Q_{(k)}=\left(R_{(k)}\right)^{2}. What is the sum of the digits of NN?
Plain-text mathematical notation (without MathML)
For each positive digit D and positive integer k, we use the symbol D_((k)) to represent the positive integer having exactly k digits, each of which is equal to D. For example, 2_((1))=2 and 3_((4))=3333. There are N quadruples (P,Q,R,k) with P,Q and R positive digits, k a positive integer with k≤2018, and P_((2k))−Q_((k))=(R_((k)))². What is the sum of the digits of N?
Original LaTeX notation
For each positive digit $D$ and positive integer $k$, we use the symbol $D_{(k)}$ to represent the positive integer having exactly $k$ digits, each of which is equal to $D$. For example, $2_{(1)}=2$ and $3_{(4)}=3333$. There are $N$ quadruples $(P, Q, R, k)$ with $P, Q$ and $R$ positive digits, $k$ a positive integer with $k \leq 2018$, and $P_{(2k)}-Q_{(k)}=\left(R_{(k)}\right)^{2}$. What is the sum of the digits of $N$?

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task_id
7a9d2552-7837-582b-9d9f-4f42cc443d8f
task_key
test--7a9d2552-7837-582b-9d9f-4f42cc443d8f
task_revision_id
1
config / split
/ test

Official source

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