Study the given information carefully and answer the questions given below.

Seven corporate seminars will be scheduled on the 13th, 5th, 23rd, 17th, 29th, 7th, and 2nd across six consecutive months: January, February, March, April, May, and June of a leap year. Exactly two seminars will be held in one of these months, while the other months will have one seminar each. The Marketing seminar is scheduled for an even-numbered date in a month that has exactly 30 days. The Finance seminar is scheduled two months prior to the Marketing seminar. The Operations seminar will take place on the 5th, in the month immediately following the one in which the HR seminar is held. The IT seminar is scheduled for March, but its month is not immediately before the month of the HR seminar. The sum of the dates for the two seminars held in the same month is 25. The Sales seminar is scheduled for the 13th but cannot be held in May. Neither the HR seminar nor the Legal seminar is scheduled for the 7th.

Based on the details provided, answer the following question.

In which of the following months will the Operations seminar be held?

Correct Answer: June

Let's analyze the given information step-by-step to determine the complete schedule for all seven corporate seminars.

Step 1: Understand the basic constraints

1. Dates: 2nd, 5th, 7th, 13th, 17th, 23rd, 29th.

2. Months (6 consecutive months in a leap year):

• January (31 days)

• February (29 days - leap year)

• March (31 days)

• April (30 days)

• May (31 days)

• June (30 days)

3. Distribution: Exactly two seminars will be held in one month, and one seminar in each of the remaining five months.

Step 2: Determine the Marketing and Finance Seminars

• The Marketing seminar is scheduled for an even-numbered date in a month with exactly 30 days.

• The only even-numbered date available is the 2nd.

• The 30-day months are April and June.

• The Finance seminar is scheduled two months prior to Marketing. If Marketing were in April, Finance would be in February. If Marketing were in June, Finance would be in April.

Step 3: Analyze the IT, HR, and Operations Seminars

• IT is scheduled for March.

• IT's month (March) is NOT immediately before HR's month, which means HR cannot be in April. Therefore, HR must be in January, February, May, or June.

• The Operations seminar takes place on the 5th, in the month immediately following the HR seminar (HR month + 1 = Operations month).

Step 4: Check the month with two seminars and date sum

• The sum of the two dates in the single month with two seminars is 25.

• Looking at the available dates (2, 5, 7, 13, 17, 23, 29):

$2+23=25$

• No other combination of two dates from the set sums to 25. Thus, the two dates in that month are the 2nd and the 23rd.

• Since Marketing is on the 2nd (in April or June), the other seminar in that same month must be on the 23rd.

Step 5: Deduce the exact placement of seminars

• Sales seminar: Scheduled for the 13th, but cannot be held in May.

• HR and Legal seminars: Neither is scheduled for the 7th.

• Now let's evaluate the position of HR and Operations:

- HR cannot be in March (IT is in March).

- HR cannot be in April (since March is not immediately before HR).

- If HR is in May, then Operations is on June 5th.

- If HR is in January, Operations is in February. But then April/June configurations need to fit Marketing, Finance, Sales, Legal, IT, etc.

- Testing HR in May and Operations in June:

1. HR = May

2. Operations = June 5th

3. If Operations is June 5th, then Marketing must be June 2nd (the month with 2 seminars is June, containing June 2nd and June 23rd).

4. Since Marketing is in June, Finance is two months prior, which is April.

5. Finance in April means April has one seminar. The dates in June are 2nd (Marketing) and 23rd (another seminar).

6. IT is in March.

7. That leaves January, February, March, April, May, June with complete distribution!

• Let's assign the remaining dates to ensure everything fits perfectly:

- June has two seminars: Marketing (2nd) and another on the 23rd (Legal seminar). Operations is on June 5th? Wait, June would have 3 seminars if Operations was also in June! Thus, Operations is in June on the 5th, and Marketing is in April (2nd), meaning the double-seminar month is April (2nd and 23rd)!

- Let's re-verify Marketing in April:

1. Marketing = April 2nd (April has 2 seminars: 2nd and 23rd).

2. Finance = February (2 months prior to April).

3. IT = March.

4. HR = May, so Operations = June 5th.

5. Sales = 13th (not May). It fits in January 13th.

6. The 23rd of April must be Legal (since HR/Legal cannot be 7th, and HR is in May).

7. The remaining date 7th goes to HR on May 7th? Wait, neither HR nor Legal is on the 7th. So 7th goes to Finance (February 7th) or IT (March 7th)!

8. This completely satisfies all constraints!

Conclusion:

The Operations seminar is held in June (on the 5th of June).