Practice (97)

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Under the new AMC 10, 12 scoring method, 6 points are given for each correct answer, 2.5 points are given for each unanswered question, and no points are given for an incorrect answer. Some of the possible scores between 0 and 150 can be obtained in only one way, for example, the only way to obtain a score of 146.5 is to have 24 correct answers and one unanswered question. Some scores can be obtained in exactly two ways; for example, a score of 104.5 can be obtained with 17 correct answers, 1 unanswered question, and 7 incorrect, and also with 12 correct answers and 13 unanswered questions. There are three scores that can be obtained in exactly three ways. What is their sum?

Solve the following question in integers $$x^6 + 3x^3 +1 = y^4$$

Solve in positive integers the equation $x^2y + y^2z +z^2x = 3xyz$

Solve the following equation in positive integers: $$x^2 +3x^2y^2 = 30y^2 + 517$$

Solve in positive integers the equation $x^3 - y^3 = xy + 61$.

Solve the equation in integers $(x^2-y^2)^2 = 1+16y$.

Show the equation $x^2 + y^2-8z^3 = 6$ has no integer solution.

Find all the integer solutions to the equation $xy - 10(x+ y)= 1$.


Solve in integers the equation $x^2 - xy +2x -3 y = 0$


Solve the equation in integers $x^2 +4xy + 5y^2 + 2x + 4y -7 =0$

Solve the equation in integers $x^2 - 2xy -3y^2 +3x-5y-6=0$


Show that the equation $x^4 + y^4 + z^4 = 2x^2y^2 + 2y^2 z^2 + 2z^2x^2 +24$ has no integer solution.

Let $n$ be a positive integer. If the equation $x + 2y + 2z = n$ has exactly $28$ positive integer solutions, find the value of $n$.


Let $x$, $y$, and $z$ be three positive integers, If - $7x^2 - 3y^2 + 4z^2 = 8$ - $16x^2 - 7y^2 + 9z^2=-3$ Find the value of $x^2 + y^2 + z^2$

Solve the equation in integers $(x+y)^x = y^x + 1413$

How many integer solutions does the equation $(x+1)(y+1)=25$ have?


Alice, Bob, and Charlie are visiting Princeton and decide to go to the Princeton U-Store to buy some tiger plushies. They each buy at least one plushie at price p. $A$ day later, the U-Store decides to give a discount on plushies and sell them at $p'$ with $0 < p' < p$. Alice, Bob, and Charlie go back to the U-Store and buy some more plushies with each buying at least one again. At the end of that day, Alice has 12 plushies, Bob has 40, and Charlie has 52 but they all spent the same amount of money: \$42. How many plushies did Alice buy on the first day?

Real numbers $x, y, z$ satisfy the following equality: $$4(x + y + z) = x^2 + y^2 + z^2$$ Let $M$ be the maximum of $xy + yz + zx$, and let $m$ be the minimum of $xy + yz + zx$. Find $M + 10m$.

$x, y, z$ are positive real numbers that satisfy $x^3+2y^3+6z^3 = 1$. Let $k$ be the maximum possible value of $2x + y + 3z$. Let $n$ be the smallest positive integer such that $k^n$ is an integer. Find the value of $k^n + n$.

Find $\textit{any}$ quadruple of positive integers $(a, b, c, d)$ satisfying $a^3+b^4+c^5=d^{11}$ and $abc<10^5$.

The sum of the three different positive unit fractions is $\frac{6}{7}$. What is the least number that could be the sum of the denominators of these fractions?

The sum of $n$ consecutive positive integers is 100. What is the greatest possible value of $n$?

The sides of a right triangle all have lengths that are whole numbers. The sum of the length of one leg and the hypotenuse is 49. Find the sum of all the possible lengths of the other leg. (A) 7 (B) 49 (C) 63 (D) 71 (E) 96

Find the sum of all positive integer $x$ such that $3\times 2^x = n^2-1$ for some positive integer $n$.

Find the number of pairs of integer solution $(x, y)$ that satisfies the equation $$(x-y + 2)(x-y-2) =-(x-2)(y-2)$$