IPMAT Indore 2024 — LRDI

The Setup: Think of these companies as sorted C++ arrays. We need to maximize the final element of array A without throwing a logic error when it's compared to the very first element of array B. It's a strict inequality check, so we need to min-max the data to push Company A's ceiling as high as possible.

Step 1: Decode the median mechanics for Company B. Company B has $28$ employees (an even number). The median age ($30$) is the average of the two middle elements. If we use standard 1-based math indexing, that's the $14$th and $15$th employees:
$$\frac{b_{14} + b_{15}}{2} = 30$$

Step 2: Find the lowest possible starting age for Company B. To give array A the most room to scale up, we must push B's values as low as the rules allow. We can initialize the first $15$ elements in B to exactly $30$ without breaking the median requirement: $b_1 = b_2 = \dots = b_{14} = b_{15} = 30$.
Thus, the absolute minimum age for the youngest employee in B is $30$.

Step 3: Lock in Company A's max age. The constraint dictates that *every* employee in A must be strictly younger than *every* employee in B. In code terms, $a_{32} < b_1$. Since ages are strictly typed integers, if $b_1 = 30$, the absolute maximum allowed for $a_{32}$ is $29$.

Step 4: Verify this doesn't break Company A's own median constraint. Company A has $32$ employees with a median of $24$. This requires the average of $a_{16}$ and $a_{17}$ to be $24$. We can easily assign $a_{16} = 24$ and $a_{17} = 24$, which leaves plenty of capacity for elements $a_{18}$ through $a_{32}$ to cap out at $29$. The backend logic runs with zero lag, and the max age holds up perfectly.

Final Answer: 29

The Setup: This is a logical reasoning Data Interpretation problem based on an election matrix. We must deduce the exact vote distribution for parties A, B, and C across 5 constituencies using their total vote counts, win conditions, and numerical sequence constraints.

Step 1: Analyze Party C's winning constraints.
Every constituency has exactly $20$ voters and a clear winner (no ties for first place). To guarantee a win without tying, a party must secure a minimum of $8$ votes (since the remaining $12$ could be split $6-6$; a score of $7$ allows a $7-7-6$ tie).
Party C won *only* S2 and S3. Thus, $c_2 \ge 8$ and $c_3 \ge 8$.
Party C's total votes across all constituencies is $16$.
Therefore, C must have obtained exactly $8$ votes in S2, $8$ votes in S3, and $0$ votes in S1, S4, and S5.

Step 2: Analyze Party B's winning constraints.
Party A won only S1, and C won only S2 and S3. By elimination, Party B must be the winner of S4 and S5.
In S4 and S5, C has $0$ votes, meaning A and B split the $20$ votes entirely. For B to win clearly, B must secure more than half the votes: $b_4 \ge 11$ and $b_5 \ge 11$.

Step 3: Apply Party B's sequence constraints.
B's votes across S1 to S5 are distinct natural numbers in increasing order: $b_1 < b_2 < b_3 < b_4 < b_5$.
Since $b_4 \ge 11$, and $b_5$ must be strictly greater than $b_4$, $b_5 \ge 12$.
B's total votes equal $35$. To leave enough votes for the first three constituencies, we must minimize $b_4$ and $b_5$.
Let $b_4 = 11$ and $b_5 = 12$.
The remaining votes for the first three constituencies are: $b_1 + b_2 + b_3 = 35 - 11 - 12 = 12$.

Step 4: Determine Party B's exact sequence.
In S2 and S3, C wins with $8$ votes. Therefore, A and B must each have fewer than $8$ votes ($b_2 \le 7, b_3 \le 7, a_2 \le 7, a_3 \le 7$).
Since $a_i + b_i = 20 - 8 = 12$ in these constituencies, the only valid integer pairs for $(a_i, b_i)$ bounded by $7$ are $(7,5), (6,6),$ and $(5,7)$.
Thus, B's votes in S2 and S3 must be chosen from the set $\{5, 6, 7\}$.
Maintaining the strictly increasing sequence $b_2 < b_3$, we test combinations to satisfy $b_1 + b_2 + b_3 = 12$:
If $(b_2, b_3) = (5, 6)$, then $b_1 = 12 - 11 = 1$. (Valid natural number)
If $(b_2, b_3) = (5, 7)$, then $b_1 = 12 - 12 = 0$. (Invalid, natural numbers begin at $1$)
Therefore, B's exact vote sequence across S1-S5 is $1, 5, 6, 11, 12$.

Step 5: Calculate Party A's vote distribution.
Using the formula $a_i = 20 - b_i - c_i$:
S1: $a_1 = 20 - 1 - 0 = 19$ (A wins)
S2: $a_2 = 20 - 5 - 8 = 7$ (C wins)
S3: $a_3 = 20 - 6 - 8 = 6$ (C wins)
S4: $a_4 = 20 - 11 - 0 = 9$ (B wins)
S5: $a_5 = 20 - 12 - 0 = 8$ (B wins)
Checking the total: $19 + 7 + 6 + 9 + 8 = 49$, which perfectly matches A's given total.

Step 6: Answer the specific prompt.
We must find the constituency where B got fewer votes than both A and C.
In S2, B has $5$ votes, A has $7$ votes, and C has $8$ votes. $5 < 7$ and $5 < 8$.

Final Answer: S2

The Setup: We must identify the specific number of votes obtained by Party B in constituency S2, using the comprehensive election matrix derived from the logical constraints.

Step 1: Reference the derived election matrix.
As established through the total vote counts and sequence constraints:
Party C's votes across S1-S5: $0, 8, 8, 0, 0$
Party B's votes across S1-S5: $1, 5, 6, 11, 12$
Party A's votes across S1-S5: $19, 7, 6, 9, 8$

Step 2: Isolate the requested data point.
We look at Party B's vote sequence ($b_1, b_2, b_3, b_4, b_5$) and identify the value for S2 ($b_2$).
$b_2 = 5$.

Final Answer: 5

The Setup: We must identify the specific number of votes obtained by Party A in constituency S5, utilizing the completed election matrix.

Step 1: Reference the derived election matrix.
The calculated vote distribution for Party A across the five constituencies (S1 through S5) is $19, 7, 6, 9,$ and $8$.

Step 2: Isolate the requested data point.
We evaluate Party A's sequence ($a_1, a_2, a_3, a_4, a_5$) and extract the specific value corresponding to S5 ($a_5$).
$a_5 = 8$.

Final Answer: 8

The Setup: We need to compare Party A's vote counts across all five constituencies to determine which constituency yielded their lowest performance.

Step 1: Retrieve Party A's vote distribution.
From our derived election matrix, the votes obtained by Party A in constituencies S1, S2, S3, S4, and S5 are respectively:
$19, 7, 6, 9, 8$

Step 2: Identify the minimum value.
Comparing the integers in the set $\{19, 7, 6, 9, 8\}$, the lowest number is $6$.

Step 3: Map the minimum value back to its constituency.
The vote count of $6$ corresponds to constituency S3.

Final Answer: S3

The Setup: We must model a hypothetical political alliance between Party A and Party C, combining their votes in each constituency to see how many total seats the new alliance would win against Party B.

Step 1: Calculate the alliance's combined votes per constituency.
We sum the individual votes of A and C for each constituency ($a_i + c_i$):
S1: $19 + 0 = 19$
S2: $7 + 8 = 15$
S3: $6 + 8 = 14$
S4: $9 + 0 = 9$
S5: $8 + 0 = 8$

Step 2: Compare the alliance's votes against Party B's votes.
To win a seat, the alliance's combined votes must exceed Party B's votes in that constituency.
S1: Alliance ($19$) vs Party B ($1$) $\Rightarrow$ Alliance wins.
S2: Alliance ($15$) vs Party B ($5$) $\Rightarrow$ Alliance wins.
S3: Alliance ($14$) vs Party B ($6$) $\Rightarrow$ Alliance wins.
S4: Alliance ($9$) vs Party B ($11$) $\Rightarrow$ Party B wins.
S5: Alliance ($8$) vs Party B ($12$) $\Rightarrow$ Party B wins.

Step 3: Total the seats won by the alliance.
The alliance successfully wins constituencies S1, S2, and S3, totaling $3$ seats.

Final Answer: 3