Let there be given triples of integers $(r_j,s_j,t_j),~j=1,2,\ldots,N$, such that for each $j$, $r_j,t_j,s_j$ are not all even. Show that one can find integers $a,b,c$ such that $ar_j+bs_j+ct_j$ is odd for at least $\frac{4N}7$ of the indices $j$.
1. **Understanding the Problem:**
   We are given \( N \) triples of integers \((r_j, s_j, t_j)\) for \( j = 1, 2, \ldots, N \), where for each \( j \), not all of \( r_j, s_j, t_j \) are even. We need to show that there exist integers \( a, b, c \) such that \( ar_j + bs_j + ct_j \) is odd for at least \( \frac{4N}{7} \) of the indices \( j \).

2. **Setting Up the Problem:**
   Consider the set \(\mathbb{Z}_2^3 - \{(0,0,0)\}\), which consists of all non-zero 3-tuples of binary values (0 or 1). There are \( 2^3 - 1 = 7 \) such tuples.

3. **Counting Odd Sums:**
   For each \( j \), we need to count how many of the 7 possible non-zero tuples \((a, b, c)\) make \( ar_j + bs_j + ct_j \) odd. Since \( r_j, s_j, t_j \) are not all even, at least one of them is odd. This ensures that there is at least one combination of \( (a, b, c) \) that makes \( ar_j + bs_j + ct_j \) odd.

4. **Summing Over All \( j \):**
   We sum the number of times \( ar_j + bs_j + ct_j \) is odd over all \( j \) and all non-zero tuples \((a, b, c)\):
   \[
   \sum_{(a,b,c)\in\mathbb{Z}_2^3 - \{(0,0,0)\}} \left( \# \{ 1 \leq j \leq N : ar_j + bs_j + ct_j \text{ is odd} \} \right)
   \]
   This is equal to:
   \[
   \sum_{j=1}^N \left( \# \{ (a,b,c)\in\mathbb{Z}_2^3 - \{(0,0,0)\} : ar_j + bs_j + ct_j \text{ is odd} \} \right)
   \]

5. **Each \( j \) Contributes 4:**
   For each \( j \), there are exactly 4 non-zero tuples \((a, b, c)\) that make \( ar_j + bs_j + ct_j \) odd. This is because for any given \( j \), if \( r_j \) is odd, there is a unique \( a \) for which \( ar_j \) is odd, and similarly for \( s_j \) and \( t_j \). Thus, each \( j \) contributes 4 to the sum:
   \[
   \sum_{j=1}^N 4 = 4N
   \]

6. **Applying the Pigeonhole Principle:**
   By the pigeonhole principle, since there are 7 possible non-zero tuples \((a, b, c)\), there must be at least one tuple \((a, b, c)\) such that:
   \[
   \# \{ 1 \leq j \leq N : ar_j + bs_j + ct_j \text{ is odd} \} \geq \frac{4N}{7}
   \]

Conclusion:
\[
\boxed{\text{True}}
\]