Aurevia, Brelosia, Cyrenia and Zerathania are four countries with their currencies being Aurels, Brins, Crowns, and Zentars, respectively. The currencies have different exchange values. Crown’s currency exchange rate with Zentars = 0.5, i.e., 1 Crown is worth 0.5 Zentars.

Three travelers, Jano, Kira, and Lian set out from Zerathania visiting exactly two of the countries. Each country is visited by exactly two travelers. Each traveler has a unique Flight Cost, which represents the total cost of airfare in traveling to both the countries and back to Zerathania. The Flight Cost of Jano was 4000 Zentars, while that of the other two travelers were 5000 and 6000 Zentars, not necessarily in that order.

When visiting a country, a traveler spent either 1000, 2000 or 3000 in the country’s local currency. Each traveler had different spends (in the country’s local currency) in the two countries he/she visited. Across all the visits, there were exactly two spends of 1000 and exactly one spend of 3000 (in the country’s local currency).

The total ”Travel Cost” for a traveler is the sum of his/her Flight Cost and the money spent in the countries visited.

The citizens of the four countries with knowledge of these travels made a few observations, with spends measured in their respective local currencies:

i. Aurevia citizen: Jano and Kira visited our country, and their Travel Costs were 3500 and 8000, respectively.
ii. Brelosia citizen: Kira and Lian visited our country, spending 2000 and 3000, respectively. Kira’s Travel Cost was 4000.
iii. Cyrenia citizen: Lian visited our country and her Travel Cost was 36000.

What was Jano’s total spend in the two countries he visited, in Aurels?

The correct answer is 1500.

Here is the step-by-step breakdown of the solution:

1. Setup and Notation

Let the countries be:

• Aurevia (A) with currency Aurels
• Brelosia (B) with currency Brins
• Cyrenia (C) with currency Crowns
• Zerathania (Z) with currency Zentars

Let the exchange rates (value of one unit of local currency in Zentars) be $e_A$, $e_B$, and $e_C$ respectively. We are given the exchange rate for Crowns:

$e_C=0.5\text{Zentars}$

Three travelers—Jano (J), Kira (K), and Lian (L)—set out from Zerathania (Z), and each visits exactly two of the other countries. Since each country is visited by exactly two travelers, the travel matrix is:

• Aurevia (A): Visited by Jano and Kira
• Brelosia (B): Visited by Kira and Lian
• Cyrenia (C): Visited by Lian and Jano

Each traveler spends either 1000, 2000, or 3000 in the visited country's local currency. Across all 6 visits, the distribution of spends is:

• Exactly two spends of 1000
• Exactly three spends of 2000
• Exactly one spend of 3000

The Flight Costs in Zentars are:

• Jano: 4000 Zentars
• Kira and Lian: One is 5000 Zentars, the other is 6000 Zentars

2. Analyzing the Observations

Observation (ii) - Brelosia:

Kira and Lian visited Brelosia, spending 2000 and 3000 in Brins, respectively. Kira’s Travel Cost was 4000 Brins.

Thus:

• Kira's spend in Brelosia ($S_{KB}$) = 2000 Brins
• Lian's spend in Brelosia ($S_{LB}$) = 3000 Brins (this is the unique 3000 spend)
• Kira's Travel Cost = 4000 Brins

Observation (i) - Aurevia:

Jano and Kira visited Aurevia, and their Travel Costs were 3500 and 8000 Aurels, respectively.

Since Kira's Travel Cost is 4000 Brins or 8000 Aurels, we can relate the exchange rates:

$4000e_B=8000e_A\Rightarrow e_B=2e_A$

Kira's Travel Cost in Zentars is:

$T{C}_K=F{C}_K+S_{KA}{e}_A+S_{KB}{e}_B$

$8000e_A=F{C}_K+S_{KA}{e}_A+2000(2e_A)$

$(4000-S_{KA})e_A=F{C}_K$

Each traveler has different spends in the two countries visited. Since Kira spent 2000 in Brelosia and cannot spend 3000 (already taken by Lian), Kira's spend in Aurevia ($S_{KA}$) must be 1000 Aurels. Substituting this in:

$3000e_A=F{C}_K$

Since $F{C}_K$ must be either 5000 or 6000 Zentars:

• If $F{C}_K=6000$, then $e_A=2$ and $e_B=4$. This means Lian's Flight Cost ($F{C}_L$) is 5000 Zentars.

3. Determining Jano's Spend

From Observation (i), Jano's Travel Cost was 3500 Aurels. Since $e_A=2$, Jano's Travel Cost in Zentars is:

$3500\times 2=7000\text{Zentars}$

Jano's Flight Cost is 4000 Zentars. Therefore, Jano's total spend in the two visited countries (Aurevia and Cyrenia) in Zentars is:

$7000-4000=3000\text{Zentars}$

To convert Jano's total spend from Zentars to Aurels, we divide the spend in Zentars by the exchange rate of Aurels ($e_A=2$):

$\frac{3000}{2}=1500\text{Aurels}$