Difference between revisions of "2024 AIME II Problems/Problem 11"

Revision as of 09:31, 13 February 2024

Problem

Find the number of triples of nonnegative integers $(a,b,c)$ satisfying $a + b + c = 300$ and \begin{equation*} a^2b + a^2c + b^2a + b^2c + c^2a + c^2b = 6,000,000. \end{equation*}

solution 1

$ab(a+b)+bc(b+c)+ac(a+c)=300(ab+bc+ac)-3abc=6000000, 100(ab+bc+ac)-abc=2000000$

Note $(100-a)(100-b)(100-c)=1000000-10000(a+b+c)+100(ab+bc+ac)-abc=0$ . Thus, $a/b/c=100$ . There are $201$ cases for each but we need to subtract $2$ for $(100,100,100)$ . The answer is $\boxed{601}$

~Bluesoul

Solution 2

$a^2(b+c)+b^2(a+c)+c^2(a+b) = 6000000$ , thus $a^2(300-a)+b^2(300-b)+c^2(300-c) = 6000000$ . Complete the cube to get $-(a-100)^3-(b-100)^3+(c-100)^3 = 9000000-30000(a+b+c)$ , which so happens to be 0. Then we have $(a-100)^3+(b-100)^3+(c-100)^3 = 0$ . We can use Fermat's last theorem here to note that one of a, b, c has to be 100. We have 200+200+200+1 = 601.

Solution 3

We have \begin{align*} & a^2 b + a^2 c + b^2 a + b^2 c + c^2 a + c^2 b \\ & = ab \left( a + b \right) + bc \left( b + c \right) + ca \left( c + a \right) \\ & = ab \left( 300 - c \right) + bc \left( 300 - a \right) + ca \left( 300 - b \right) \\ & = 300 \left( ab + bc + ca \right) - 3 abc \\ & = -3 \left( \left( a - 100 \right) \left( b - 100 \right) \left( c - 100 \right) - 10^4 \left( a + b + c \right) + 10^6 \right) \\ & = -3 \left( \left( a - 100 \right) \left( b - 100 \right) \left( c - 100 \right) - 2 \cdot 10^6 \right) \\ & = 6 \cdot 10^6 . \end{align*} The first and the fifth equalities follow from the condition that $a+b+c = 300$ .

Therefore, \[ \left( a - 100 \right) \left( b - 100 \right) \left( c - 100 \right) = 0 . \]

Case 1: Exactly one out of $a - 100$ , $b - 100$ , $c - 100$ is equal to 0.

Step 1: We choose which term is equal to 0. The number ways is 3.

Step 2: For the other two terms that are not 0, we count the number of feasible solutions.

W.L.O.G, we assume we choose $a - 100 = 0$ in Step 1. In this step, we determine $b$ and $c$ .

Recall $a + b + c = 300$ . Thus, $b + c = 200$ . Because $b$ and $c$ are nonnegative integers and $b - 100 \neq 0$ and $c - 100 \neq 0$ , the number of solutions is 200.

Following from the rule of product, the number of solutions in this case is $3 \cdot 200 = 600$ .

Case 2: At least two out of $a - 100$ , $b - 100$ , $c - 100$ are equal to 0.

Because $a + b + c = 300$ , we must have $a = b = c = 100$ .

Therefore, the number of solutions in this case is 1.

Putting all cases together, the total number of solutions is $600 + 1 = \boxed{\textbf{(601) }}$ .

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

Solution 4

We will use Vieta's formulas to solve this problem. We assume $a + b + c = 300$ , $ab + bc + ca = m$ , and $abc = n$ . Thus $a$ , $b$ , $c$ are the three roots of a cubic polynomial $f(x)$ .

We note that $300m = (a + b + c)(ab + bc + ca)=\sum_{cyc} a^2b + 3abc = 6000000 + 3n$ , which simplifies to $100m - 2000000 = n$ .

Our polynomial $f(x)$ is therefore equal to $x^3 - 300x^2 + mx - (100m - 2000000)$ . Note that $f(100) = 0$ , and by polynomial division we obtain $f(x) = (x - 100)(x^2 - 200x - (m-20000))$ .

We now notice that the solutions to the quadratic equation above are $x = 100 \pm \frac{\sqrt{200^2 - 4(m - 20000)}}{2} = 100 \pm \sqrt{90000 - 4m}$ , and that by changing the value of $m$ we can let the roots of the equation be any pair of two integers which sum to $200$ . Thus any triple in the form $(100, 100 - x, 100 + x)$ where $x$ is an integer between $0$ and $100$ satisfies the conditions.

Now to count the possible solutions, we note that when $x \ne 100$ , the three roots are distinct; thus there are $3! = 6$ ways to order the three roots. As we can choose $x$ from $0$ to $99$ , there are $100 \cdot 3! = 600$ triples in this case. When $x = 100$ , all three roots are equal to $100$ , and there is only one triple in this case.

In total, there are thus $\boxed{601}$ distinct triples.

~GaloisTorrent <3

Solution 5

Let's define $a=100+x$ , $b=100+y$ , $c=100+z$ . Then we have $x+y+z=0$ and $6000000 = \sum a^2(b+c)$

$= \sum (100+x)^2(200-x) = \sum (10000+200x+x^2)(200-x) = \sum (20000 - 10000 x + x(40000-x^2))$

$= \sum (20000 + 30000 x -x^3) = 6000000 - \sum x^3$ , so we get $x^3 + y^3 + z^3 = 0$ . Then from $x+y+z = 0$ , we can find $0 = x^3+y^3+z^3 = x^3+y^3-(x+y)^3 = 3xyz$ , which means that one of $a$ , $b$ , $c$ must be 0. There are 201 solutions for each of $a=0$ , $b=0$ and $c=0$ , and subtract the overcounting of 2 for solution $(200, 200, 200)$ , the final result is $201 \times 3 - 2 = \boxed{601}$ .

Dan Li

dan

Video Solution

https://youtu.be/YMYe9chPLdY

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

@@ Line 83: / Line 83: @@
 ==Solution 5==
-Let's define <math>a=100+x</math>, <math>b=100+y</math>, <math>c=100+z</math>. Then we have <math>x+y+z=0</math> and
+Let's define <math>a=100+x</math>, <math>b=100+y</math>, <math>c=100+z</math>. Then we have <math>x+y+z=0</math> and <math>6000000 = \sum a^2(b+c) </math>
-<math>6000000 = \sum a^2(b+c) = \sum (100+x)^2(200-x) = \sum (10000+200x+x^2)(200-x) </math>
-<math>= \sum (20000 - 10000 x + x(40000-x^2)) = \sum (20000 + 30000 x -x^3) = 6000000 - \sum x^3</math>, so we get <math>x^3 + y^3 + z^3 = 0</math>. Then from <math>x+y+z = 0</math>, we can find <math>0 = x^3+y^3+z^3 = x^3+y^3-(x+y)^3 = 3xyz</math>, which means that one of <math>a</math>, <math>b</math>,<math>c</math> must be 0. There are 201 solutions for each of <math>a=0</math>, <math>b=0</math> and <math>c=0</math>, and subtract the overcounting of 2 for solution <math>(200, 200, 200)</math>, the final result is <math>201 \times 3 - 2 = \boxed{601}</math>.
+<math>= \sum (100+x)^2(200-x) = \sum (10000+200x+x^2)(200-x) = \sum (20000 - 10000 x + x(40000-x^2)) </math>
+<math>= \sum (20000 + 30000 x -x^3) = 6000000 - \sum x^3</math>, so we get <math>x^3 + y^3 + z^3 = 0</math>. Then from <math>x+y+z = 0</math>, we can find <math>0 = x^3+y^3+z^3 = x^3+y^3-(x+y)^3 = 3xyz</math>, which means that one of <math>a</math>, <math>b</math>,<math>c</math> must be 0. There are 201 solutions for each of <math>a=0</math>, <math>b=0</math> and <math>c=0</math>, and subtract the overcounting of 2 for solution <math>(200, 200, 200)</math>, the final result is <math>201 \times 3 - 2 = \boxed{601}</math>.
 Dan Li

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