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.

How many Zentars did Lian spend in the two countries he visited?

The correct answer is 13000.

To solve this step-by-step, we first need to establish the exchange rates for the currencies of the four countries relative to Zentars. Let the exchange rate of 1 Aurel be $a$ Zentars, 1 Brin be $b$ Zentars, and 1 Crown be $c$ Zentars. The problem explicitly gives us the exchange rate for Crowns:

Next, we can determine which countries the three travelers (Jano, Kira, and Lian) visited. There are four countries in total, but they departed from Zerathania. The problem states that exactly three travelers visited exactly two countries each, resulting in a total of 6 country visits. Since each visited country received exactly two travelers, exactly 3 countries were visited. Based on the citizen observations, these three countries are Aurevia, Brelosia, and Cyrenia. We can map the specific visits as follows:

1. The Aurevia citizen noted Jano and Kira visited Aurevia.

2. The Brelosia citizen noted Kira and Lian visited Brelosia.

3. Because each traveler visited exactly two countries, Lian must have also visited Cyrenia. Since Cyrenia must have two visitors, Jano (who needs one more country) is the second visitor to Cyrenia.

Now we analyze the spending amounts in local currencies. The possible spends are 1000, 2000, or 3000. For the 6 total visits, there were exactly two spends of 1000 and exactly one spend of 3000. Therefore, the remaining three spends must all be 2000. The distribution of these local spends is: 1000, 1000, 2000, 2000, 2000, 3000.

From the Brelosia citizen's observation, Lian spent 3000 in Brelosia, and Kira spent 2000 in Brelosia. This accounts for our single 3000 spend. Because each traveler must have a different spend in the two countries they visit, Lian's spend in Cyrenia must be either 1000 or 2000. Similarly, Kira's spend in Aurevia must be either 1000 or 3000. Since the only 3000 spend was already used by Lian, Kira must have spent 1000 in Aurevia.

This leaves three spends remaining to be assigned: 1000, 2000, and 2000. These belong to Jano in Aurevia, Jano in Cyrenia, and Lian in Cyrenia. Because Jano's spends in his two countries must be different, Jano must have spent 1000 in one country and 2000 in the other. This leaves the remaining 2000 spend for Lian in Cyrenia. Thus, Lian's local currency spends were 3000 Brins in Brelosia and 2000 Crowns in Cyrenia.

Next, let's look at the Travel Costs to deduce the exchange rates and missing Flight Costs. Kira's total Travel Cost consists of her Flight Cost (which we will denote as $F_K$) plus her local spends converted to Zentars.

The citizens state Travel Costs converted entirely into their local currencies. The Brelosia citizen observes Kira's cost as 4000 Brins. Converting this back to Zentars by multiplying by $b$:

The Aurevia citizen observes Kira's cost as 8000 Aurels. Converting this to Zentars by multiplying by $a$:

Substituting the first equation into the second equation:

We know the unknown Flight Costs are 5000 and 6000 Zentars. If $F_K$ were 5000, $a$ would be a fraction. Assuming $F_K$ is 6000 Zentars gives us a clean exchange rate of $a$ = 2. By substituting these back into our earlier equation:

So 1 Aurel is worth 2 Zentars, and 1 Brin is worth 4 Zentars. Finally, we can calculate Lian's exact spending in Zentars. Lian spent 3000 Brins in Brelosia and 2000 Crowns in Cyrenia. Multiplying these local amounts by their respective exchange rates into Zentars:

To find the total amount of Zentars Lian spent across both countries, we simply add these two values together:

Therefore, Lian spent a total of 13000 Zentars.