IPMAT Indore 2025 — LRDI

The Setup: We are dealing with a logical puzzle disguised as a matrix grid. We have 5 teams of 15 members each, meaning there are 75 total members. Since the three roles (Biologists, Geologists, Explorers) have equal total numbers, we just divide 75 by 3. That means exactly 25 Biologists, 25 Geologists, and 25 Explorers across the entire board. Time to build the matrix and min-max the stats.

Step 1: The Explorer Distribution (The Meta)
The problem states that the number of Explorers in each team strictly decreases in the order A > B > C > D > E. We also know they must sum up to 25, and every single team must have at least 2 of *each* role.
Let's find the maximum number of Explorers team A can have. Team A already has 6 Geologists (given) and must have at least 2 Biologists.
$$Max\_Explorers_A = 15 - 6 - 2 = 7$$
If Team A has a maximum of 7 Explorers, we need 5 strictly decreasing integers starting from $\le 7$ that sum to 25. There is literally only one valid sequence in the entire universe that satisfies this:
$$7 + 6 + 5 + 4 + 3 = 25$$
Lock it in:
Team A = 7, Team B = 6, Team C = 5, Team D = 4, Team E = 3.

Step 2: The Biologist Constraints (Vibe Check)
Now let's look at the Biologists ($B$). We know the total is 25.
The prompt feeds us free intel: $B_C = 6$ and $B_D = 6$.
This means the remaining Biologists for teams A, B, and E are:
$$25 - 6 - 6 = 13$$
So, $B_A + B_B + B_E = 13$.

Now we apply the heavy constraint: "Every team except A has more biologists than explorers."
* For Team B: $B_B > E_B \rightarrow B_B > 6 \rightarrow B_B \ge 7$
* For Team E: $B_E > E_E \rightarrow B_E > 3 \rightarrow B_E \ge 4$
* For Team A (the exception): It just needs to meet the baseline minimum of $B_A \ge 2$.

Step 3: Solving the Biologist Equation
We have our minimum thresholds: $B_A \ge 2$, $B_B \ge 7$, and $B_E \ge 4$.
Let's add these bare minimums together:
$$2 + 7 + 4 = 13$$
Since the sum of the absolute minimums equals exactly the total number of remaining Biologists (13), there is zero wiggle room. The math is mathing perfectly. None of these values can be higher, or the sum would break 13.
Therefore, $B_A = 2$, $B_B = 7$, and $B_E = 4$.

Step 4: The Audit (Double Check Protocol)
Let's run a full matrix audit to ensure we aren't throwing the game. We'll fill in the Geologists ($G$) using the formula $G = 15 - B - E$ to make sure no team breaks the rules.
* Team A: E=7, B=2, G=6. (Sum = 15, all $\ge 2$. Works. $G=6$ matches the prompt.)
* Team B: E=6, B=7, G=2. (Sum = 15, all $\ge 2$, $B > E$. Works.)
* Team C: E=5, B=6, G=4. (Sum = 15, all $\ge 2$, $B > E$. Works.)
* Team D: E=4, B=6, G=5. (Sum = 15, all $\ge 2$, $B > E$. Works.)
* Team E: E=3, B=4, G=8. (Sum = 15, all $\ge 2$, $B > E$. Works.)
Now, let's verify total Geologists:
$$6 + 2 + 4 + 5 + 8 = 25$$
Perfectly balanced, as all things should be. The board is fully validated and legally solved.

Final Answer: 4

The Setup:
We are dealing with a round-robin tournament combinatorics problem. The base logic requires calculating the total number of matches in a perfect bracket, and then subtracting the matches that got canceled because a team got hit with the ban hammer. I have set up a stat box to extract the core data before we run the calculations.

Step 1: Extracting the Meta (Stat Box)
Let's map out exactly what is happening in this bracket before the suspension ruins the schedule.

Step 2: The Base Timeline
In a flawless, perfectly run round-robin tournament, every team plays every other team exactly once. The formula to find total unique handshakes (or matches) is $^nC_2$:
$$^8C_2 = \frac{8 \times 7}{2}$$
$$^8C_2 = 28$$
So, the organizers originally scheduled exactly 28 matches.

Step 3: Calculating the Dropped Matches
One team gets toxic and gets suspended.
Every single team in this tournament was scheduled to play 7 matches.
This suspended team managed to play exactly 3 matches before getting kicked.
To find the number of canceled matches, subtract what they played from what they were supposed to play:
$$7 - 3 = 4$$
Exactly 4 scheduled matches literally evaporated from the bracket.

Step 4: The Final Tally
Subtract the canceled matches from the total base schedule to find out how many matches actually happened.
$$28 - 4 = 24$$
There were 24 matches officially played that year.

Step 5: The Audit (Double Check Protocol)
Running it back using an alternative method to guarantee we aren't throwing.
Let's look at the tournament from the perspective of the 7 teams that *didn't* get banned.
If those 7 teams just played a private tournament among themselves, the total matches would be:
$$^7C_2 = \frac{7 \times 6}{2} = 21$$
Now, we just add the matches that the banned team played against 3 of these valid teams before getting kicked:
$$21 + 3 = 24$$
Both methods independently output exactly 24. The logic is completely locked in and validated. No cap.

Final Answer: 24

The Setup:
We are literally back in the exact same lobby as Question 2. This is a follow-up question utilizing the same logical matrix we already decoded. Since we previously theory-crafted the exact distribution of all 75 members across the grid, we can just pull up our cached data and run a quick stat check to find out who has higher Geologist DPS than Biologist DPS. Free points, honestly.

Step 1: Retrieving the Master Matrix (From Memory)
Let's pull the fully solved grid we established earlier. Remember, every team has exactly 15 members, and each class (Explorer, Biologist, Geologist) totals exactly 25 across all teams.

Step 2: The Stat Check
The prompt asks for the number of teams where Geologists > Biologists ($G > B$). Let's run down the roster and vibe check every single team:
* Team A: $G=6$, $B=2$. ($6 > 2$) $\rightarrow$ Valid (W)
* Team B: $G=2$, $B=7$. ($2 < 7$) $\rightarrow$ Invalid (L)
* Team C: $G=4$, $B=6$. ($4 < 6$) $\rightarrow$ Invalid (L)
* Team D: $G=5$, $B=6$. ($5 < 6$) $\rightarrow$ Invalid (L)
* Team E: $G=8$, $B=4$. ($8 > 4$) $\rightarrow$ Valid (W)

Only Team A and Team E meet the specific criteria.

Step 3: The Audit (Double Check Protocol)
Let's do a quick sanity check to ensure our base matrix didn't glitch.
Are there exactly 25 of each role?
Explorers: $7+6+5+4+3 = 25$ (Check).
Biologists: $2+7+6+6+4 = 25$ (Check).
Geologists: $6+2+4+5+8 = 25$ (Check).
Do all teams have 15 members? Yes.
Does every team have $\ge 2$ in each role? Yes.
Do C and D have 6 Biologists? Yes.
Does A have 6 Geologists? Yes.
Does every team except A have $B > E$?
Team B ($7 > 6$), Team C ($6 > 5$), Team D ($6 > 4$), Team E ($4 > 3$). Yes.
Does $E$ strictly decrease from A to E? $7 > 6 > 5 > 4 > 3$. Yes.
The matrix is perfectly locked in. The final count of teams with $G > B$ is strictly 2.

Final Answer: 2

The Setup:
We are dropping back into the team arrangement lobby for round 3. Since we already theory-crafted and completely solved the master matrix in Question 2, we just need to query our Biologist data and run a basic median calculation. Free elo.

Step 1: Retrieving the Biologist Stats
Let's pull the exact number of Biologists ($B$) for each team from our previously validated master grid:
* Team A: $2$
* Team B: $7$
* Team C: $6$ (Given in prompt)
* Team D: $6$ (Given in prompt)
* Team E: $4$

Step 2: Sorting the Array
To find the median, we can't just pick the middle team. We have to sort the Biologist values in ascending order to find the true statistical middle (the 50th percentile).
Unsorted array: $\{2, 7, 6, 6, 4\}$
Sorted array: $\{2, 4, 6, 6, 7\}$

Step 3: Finding the Median
Since there are exactly 5 teams (an odd number), the median is literally just the dead-center value—the 3rd number in our sorted array.
Looking at our sorted list $\{2, 4, \mathbf{6}, 6, 7\}$, the middle value is 6.

Step 4: The Audit (Double Check Protocol)
Let's run a quick sanity check to make sure the data hasn't been corrupted.
Did they sum to 25? $2 + 4 + 6 + 6 + 7 = 25$. (Yes).
Did every team except A have more Biologists than Explorers?
Team B: $7 > 6$. Team C: $6 > 5$. Team D: $6 > 4$. Team E: $4 > 3$. (Yes).
Are we picking the 3rd index of a 5-item sorted list? Yes.
The math is flawless. The logic is strictly locked in.

Final Answer: 6

We don't need to calculate exact percentages - we can compare the fractions directly to find the largest one.

From the Apparel sales data, let's identify consecutive months where sales decreased:

$$\% \text{ Decline} = \frac{\text{Previous Month Sales} - \text{Current Month Sales}}{\text{Previous Month Sales}} \times 100\%$$

**February $\rightarrow$ March:**
$$\% \text{ Decline} = \frac{279 - 236}{279} \times 100\% = \frac{43}{279} \times 100\% = 15.41\%$$

**September $\rightarrow$ October:**
$$\% \text{ Decline} = \frac{288 - 222}{288} \times 100\% = \frac{66}{288} \times 100\% = 22.92\%$$

Comparing via fractions $\rightarrow$ March: $\frac{43}{279}$ and October: $\frac{66}{288}$
* October has a numerator ($66$) that is about $1.5$ times larger than March's numerator ($43$).
* October has a denominator ($288$) that is only about $1.03$ times larger than March's denominator ($279$).

Since the numerator grows much faster than the denominator, October's fraction $\frac{66}{288}$ is larger than March's fraction $\frac{43}{279}$.

October had the highest percentage decline in Apparel sales, dropping from 288 lakhs in September to 222 lakhs in October.

Final Answer: October

The Setup: We need to identify which product categories had the value of products returned (as a percentage of sales) increase for three consecutive months. The concept here is to track each category's return percentage month by month and look for patterns where the percentage consistently rises over a three-month span (meaning the current month's return percentage is strictly higher than the previous month's for three transitions).

Step 1: Analyze the Apparel category.
For Apparel, we see continuous increases from January to February to March:
* January: 13%
* February: 16%
* March: 20%
This gives us three consecutive months of increasing percentages ($13\% \rightarrow 16\% \rightarrow 20\%$).

Step 2: Analyze the Footwear category.
For Footwear, we see continuous increases from October to November to December:
* October: 8%
* November: 9%
* December: 10%
This gives us three consecutive months of increasing percentages ($8\% \rightarrow 9\% \rightarrow 10\%$).

Step 3: Analyze the Electronics category.
Scanning the Electronics return percentages, we do not find any sequence where the values strictly increase for three consecutive months.

Step 4: Conclusion.
The key insight is that we are looking for a pattern where each month's return percentage is strictly greater than the previous month's percentage for three consecutive months. Both the Apparel and Footwear categories satisfy this condition.

Final Answer: Both Apparel and Footwear

$$\text{Net Sales} = \text{Sales} - \text{Value of Products Returned}$$

The value of products returned is given as a percentage of sales, so we need to convert this to actual rupees first.

For May in the Footwear category:
* $\text{Sales} = 69 \text{ lakhs}$
* $\text{Return percentage} = 6\% \text{ of sales}$
* $\text{Value of products returned} = \frac{6}{100} \times 69 = 4.14 \text{ lakhs}$
* $\text{Net sales for May} = 69 - 4.14 = 64.86 \text{ lakhs}$

For June in the Footwear category:
* $\text{Sales} = 111 \text{ lakhs}$
* $\text{Return percentage} = 5\% \text{ of sales}$
* $\text{Value of products returned} = \frac{5}{100} \times 111 = 5.55 \text{ lakhs}$
* $\text{Net sales for June} = 111 - 5.55 = 105.45 \text{ lakhs}$

To find the percentage increase from May to June:
$$\text{\% Increase} = \frac{\text{June net sales} - \text{May net sales}}{\text{May net sales}} \times 100\%$$

$$= \frac{105.45 - 64.86}{64.86} \times 100\%$$

$$= \frac{40.59}{64.86} \times 100\%$$

$$= 62.58\%$$

Final Answer: 62.58 percent

The Setup: The contribution of Apparel to Total Sales is calculated as the ratio of Apparel sales to the total sales of all categories for that specific month, converted to a percentage.
$$\text{Apparel contribution} = \frac{\text{Apparel sales}}{\text{Total sales of all categories}} \times 100\%$$

Step 1: Calculate the total sales for each given month.
We find the total sales by adding the sales from all three categories (Apparel + Footwear + Electronics):
* January: $262 + 104 + 289 = 655$
* April: $258 + 58 + 325 = 641$
* August: $252 + 60 + 336 = 648$
* December: $221 + 86 + 268 = 575$

Step 2: Calculate the exact contribution percentage for each month.
* January: $\frac{262}{655} \times 100\% = 40.00\%$
* April: $\frac{258}{641} \times 100\% \approx 40.25\%$
* August: $\frac{252}{648} \times 100\% \approx 38.89\%$
* December: $\frac{221}{575} \times 100\% \approx 38.43\%$

Step 3: Compare and Conclude.
Comparing the percentages, April has the highest contribution ($40.25\%$).

*Conceptual insight:* Even though January had higher absolute Apparel sales ($262$ vs $258$), April had proportionally lower sales from other categories, making Apparel's share of the total pie larger.

Final Answer: April

The Setup: The value of products returned is given as a percentage of sales, so we need to calculate the actual value in lakhs of rupees using the formula:
$$\text{Value of products returned} = \frac{\text{Return percentage}}{100} \times \text{Sales}$$

Step 1: Calculate the exact return value for each given month.
Extracting the Footwear sales and return percentage data from the table:
* September: $\frac{9}{100} \times 118 = 10.62 \text{ lakhs}$
* July: $\frac{9}{100} \times 119 = 10.71 \text{ lakhs}$
* June: $\frac{5}{100} \times 111 = 5.55 \text{ lakhs}$
* March: $\frac{7}{100} \times 121 = 8.47 \text{ lakhs}$

Step 2: Compare and Conclude.
Comparing the calculated values, July has the highest absolute value of products returned ($10.71 \text{ lakhs}$).

*Conceptual insight:* The absolute value of returns depends on both the return percentage and the total sales volume. July combines a high return percentage ($9\%$) with substantial sales ($119 \text{ lakhs}$), creating the largest absolute return amount.
* September $\rightarrow$ percentage is the same ($9\%$) but the sales are lower ($118$).
* June $\rightarrow$ sales are around the same but the percentage is almost half ($5\%$).
* March $\rightarrow$ percentage is $2\%$ lower ($7\%$) and sales are only $2 \text{ lakhs}$ more than July, which makes it definitively lower.

Final Answer: July