Found problems: 744
In a sports meeting a total of $m$ medals were awarded over $n$ days. On the first day one medal and $\frac{1}{7}$ of the remaining medals were awarded. On the second day two medals and $\frac{1}{7}$ of the remaining medals were awarded, and so on. On the last day, the remaining $n$ medals were awarded. How many medals did the meeting last, and what was the total number of medals ?
Find all pairs of real numbers $(x, y)$, that satisfy the system of equations: $$\left\{\begin{matrix} 6(1-x)^2=\dfrac{1}{y} \\ \\6(1-y)^2=\dfrac{1}{x}.\end{matrix}\right.$$
Suppose $x$ and $y$ are real numbers such that \[x^2+xy+y^2=2\qquad\text{and}\qquad x^2-y^2=\sqrt5.\] The sum of all possible distinct values of $|x|$ can be written in the form $\textstyle\sum_{i=1}^n\sqrt{a_i}$, where each of the $a_i$ is a rational number. If $\textstyle\sum_{i=1}^na_i=\frac mn$ where $m$ and $n$ are positive realtively prime integers, what is $100m+n$?
[i] Proposed by David Altizio [/i]
Solve the following system (over the real numbers):
\[\begin{cases}5x+5y+5xy-2xy^2-2x^2y=20 &\\
3x+3y+3xy+xy^2+x^2y=23&\end{cases}\]
Solve the system of equations:
$
\begin{matrix}
|x+y| + |1-x| = 6 \\ |x+y+1| + |1-y| = 4.
\end{matrix}
$
Let real numbers $x_1, x_2, \cdots , x_n$ satisfy $0 < x_1 < x_2 < \cdots< x_n < 1$ and set $x_0 = 0, x_{n+1} = 1$. Suppose that these numbers satisfy the following system of equations:
\[\sum_{j=0, j \neq i}^{n+1} \frac{1}{x_i-x_j}=0 \quad \text{where } i = 1, 2, . . ., n.\]
Prove that $x_{n+1-i} = 1- x_i$ for $i = 1, 2, . . . , n.$
The reals $a, b$ satisfy $$\begin{cases} a^3 - 3a^2 + 5a - 17 = 0 \\ b^3 - 3b^2 + 5b + 11 = 0 .\end{cases}$$ Find $a+b$.
Determine all triples of real numbers $(x, y, z)$ which satisfy the following system of equations:
\[ \begin{cases} x+y+z=0 \\ x^3+y^3+z^3 = 90 \\ x^5+y^5+z^5=2850. \end{cases} \]
Solve : $y(x+y)^2=9 $ ; $y(x^3-y^3)=7$
Find the solutions of positive integers for the system $xy + x + y = 71$ and $x^2y + xy^2 = 880$.
Determine all integers $a, b, c$ satisfying identities:
$a + b + c = 15$
$(a - 3)^3 + (b - 5)^3 + (c -7)^3 = 540$
Let $x,y\in\mathbb{R}$ be such that $x = y(3-y)^2$ and $y = x(3-x)^2$. Find all possible values of $x+y$.
Solve the system of equations:
$\{\begin{array}{l}(1+\frac{12}{3x+y}).\sqrt{x}=2\\(1-\frac{12}{3x+y}).\sqrt{y}=6\end{array}$
Let $a,b,c$ be positive numbers. Prove that a triangle with sides $a,b,c$ exists if and only if the system of equations
$$\begin{cases}\dfrac{y}{z}+\dfrac{z}{y}=\dfrac{a}{x} \\ \\ \dfrac{z}{x}+\dfrac{x}{z}=\dfrac{b}{y} \\ \\ \dfrac{x}{y}+\dfrac{y}{x}=\dfrac{c}{z}\end{cases}$$ has a real solution.
Find all solutions of the following system of $n$ equations in $n$ variables:
\[\begin{array}{c}\ x_1|x_1| - (x_1 - a)|x_1 - a| = x_2|x_2|,x_2|x_2| - (x_2 - a)|x_2 - a| = x_3|x_3|,\ \vdots \ x_n|x_n| - (x_n - a)|x_n - a| = x_1|x_1|\end{array}\]
where $a$ is a given number.
Suppose $a, b, c$ are real numbers such that $abc \ne 0$.
Determine $x, y, z$ in terms of $a, b, c$ such that $bz + cy = a, cx + az = b, ay + bx = c$.
Prove also that $\frac{1 - x^2}{a^2} = \frac{1 - y^2}{b^2} = \frac{1 - z^2}{c^2}$.
Solve the system of equations: \[ x+y+z=a \] \[ x^2+y^2+z^2=b^2 \] \[ xy=z^2 \] where $a$ and $b$ are constants. Give the conditions that $a$ and $b$ must satisfy so that $x,y,z$ are distinct positive numbers.
Determine all positive integers $n$ for which there are positive real numbers $x,y$ and $z$ such that $\sqrt x +\sqrt y +\sqrt z=1$ and $\sqrt{x+n} +\sqrt{y+n} +\sqrt{z+n}$ is an integer.
Find the number of different positive integer triples $(x, y,z)$ satisfying the equations
$x^2 + y -z = 100$ and $x + y^2 - z = 124$:
Determine all numbers $x, y$ and $z$ satisfying the system of equations
$$\begin{cases} x^2 + yz = 1 \\ y^2 - xz = 0 \\ z^2 + xy = 1\end{cases}$$
Let $n$ be a positive integer that is not a perfect cube. Define real numbers $a,b,c$ by
\[a=\root3\of n\kern1.5pt,\qquad b={1\over a-[a]}\kern1pt,\qquad c={1\over b-[b]}\kern1.5pt,\]
where $[x]$ denotes the integer part of $x$. Prove that there are infinitely many such integers $n$ with the property that there exist integers $r,s,t$, not all zero, such that $ra+sb+tc=0$.
Prove that if the real numbers $ a, b, c, d $ satisfy the equations
$$
\; a^2 + b^2 = 1,\; c^2 + d^2 = 1, \; ac + bd = -\frac{1}{2},$$
then
$$a^2 + ac + c^2 = b^2 + bd + d^2.$$
Let $a,b,c,d x,y,z, w$ be real numbers such that $$\begin{matrix}
ax -by-c z-dw =0\\
b x +a y -d z +cw=0\\
c x+ d y +a z -b w=0\\
dx-c y+bz+aw=0
\end{matrix}$$
prove that $$a=b=c=d=0, \ \ or \ \ x=y=z=w=0$$
Find all real numbers $(x,y)$ satisfying the following: $$x+\frac{3x-y}{x^2+y^2}=3$$ $$y-\frac{x+3y}{x^2+y^2}=0$$
Let $n \ne 0$ be a natural number and integers $x_1, x_2, ...., x_n, y_1, y_2, ...., y_n$ with the properties:
a) $x_1 + x_2 + .... + x_n = y_1 + y_2 + .... + y_n = 0,$
b) $x_1 ^ 2 + y_1 ^ 2 = x_2 ^ 2 + y_2 ^ 2 = .... = x_n ^ 2 + y_n ^ 2$.
Show that $n$ is even.