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19 problems match “systems of equations”. Clear

Two numbers with a percent relation ●●○○○ 20 ptsThe sum of two numbers is 4000. $15\%$ of one number equals $25\%$ of the other number. Find the two numbers.
Arithmetic then geometric progression integers ●●●○○ 50 ptsThree distinct integers $a$ , $b$ , $c$ satisfy the following conditions: - $abc = 17955$ ; - $a$ , $b$ , $c$ are three consecutive terms of an arithmetic sequence, in that order; - $3a+b$ , $3b+c$ , $3c+a$ are three consecutive terms of a geometric sequence, in that order. Find $a+b+c$ .
Sum of x and y in a system ●●●○○ 50 ptsGiven that $x+\sin y=2008$ and $x+2008 \cos y=2007$ , where $0 \leq y \leq \frac{\pi}{2}$ , find the value of $x+y$ . Give your answer as a decimal rounded to two decimal places.
Smallest modulus in a complex system ●●●○○ 50 ptsThe complex numbers $z$ and $w$ satisfy the system $\begin{align*} z + \frac{20i}{w} &= 5+i, \\ w+\frac{12i}{z} &= -4+10i. \end{align*}$ Find the smallest possible value of $|zw|^2$ .
Sum of four progression terms ●●●○○ 50 ptsIn an increasing sequence of four positive integers, the first three terms form an arithmetic progression, the last three terms form a geometric progression, and the first and fourth terms differ by $30$ . Find the sum of the four terms.
Rectangle area with vertices on a hyperbola ●●●○○ 50 ptsThe diagonals of a rectangle are both 10 and intersect at $(0,0)$ . All four vertices of the rectangle lie on the curve $y=\frac{12}{x}$ . Calculate the area of the rectangle.
Possible values of c in a polynomial identity ●●●○○ 50 ptsGiven integers $a$ , $b$ and $c$ such that the equation $(x+a)(x+b)+c(x-10)=(x-11)(x+1)$ holds for all $x$ , find all possible values of $c$ .
Two numbers from their means ●●●○○ 50 ptsTwo positive numbers have an arithmetic mean that exceeds their geometric mean by $2$ , and their geometric mean exceeds their harmonic mean by $1.6$ . Find the two numbers.
Maximum smallest ball in bowl Jia ●●●●○ 100 ptsPlace the 27 balls numbered $1$ to $27$ into three bowls, Jia, Yi and Bing, so that the average ball number in bowl Jia is $15$ , in bowl Yi is $3$ , and in bowl Bing is $18$ , and each bowl contains at least $4$ balls. Find the maximum possible value of the smallest ball number in bowl Jia.