AIME 2024 II · 第 1 题

Solution 1

Let $w,x,y,z$ denote the number of residents who own $1,2,3$ and $4$ of these items, respectively. We know $w+x+y+z=900$, since there are $900$ residents in total. This simplifies to

$w+z=229$, since we know $x=437$ and $y=234$.

Now, we set an equation of the total number of items. We know there are $195$ rings, $367$ clubs, $562$ spades, and $900$ candy hearts. Adding these up, there are $2024$ (wow! the year!) items in total. Thus, $w+2x+3y+4z=2024$ since we are not adding the number of items each group of people contributes, and this must be equal to the total number of items.

Plugging in $x$ and $y$ once more, we get $w+4z=448$. Solving $w+z=229$ and $w+4z=448$, we get $z=\boxed{073}$

-Westwoodmonster

Solution 2

We know that there are $195$ diamond rings, $367$ golf clubs, and $562$ garden spades, so we can calculate that there are $195+367+562=1124$ items, with the exclusion of candy hearts which is irrelevant to the question. There are $437$ people who owns $2$ items, which means $1$ item since candy hearts are irrelevant, and there are $234$ people who own $2$ items plus a bag of candy hearts, which means that the 234 people collectively own $234*2=468$ items. We can see that there are $1124-437-468=219$ items left, and since the question is asking us for the people who own $4$ items, which means $3$ items due to the irrelevance of candy hearts, we simply divide $219$ by $3$ and get $219/3=\boxed{073}$.

~Callisto531

Solution 3

Let $a$ be the number of people who have exactly one of these things and let $b$ be the number of people who have exactlty four of these objects. We have $a + 437 + 234 + d = 900,$ so $a + d = 229.$

Including those who have more than one object, we have

$$195 + 367 + 562 + 900 = a + 2\cdot 437 + 3\cdot 234 + 4d.$$ This is because we count those who own exactly $2$ objects twice, those who own $3$ thrice, and those who own $4$ four times. Solving gives $a + 4d = 448.$

Solving the system $a + 4d = 448, a + d = 229$ gives $3d = 219,$ so $d = \boxed{\textbf{(073)}}.$

-Benedict T (countmath1)

Video Solution

https://youtu.be/n60sIVSYWlc

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)