Found problems: 744
Determine all real solutions to the system of equations:
$$x_1 + x_2 +...+ x_{2004 }= 2004$$
$$x^4_1+ x^4_2+ ... + x^4_{2004} = x^3_1+x^3_2+... + x^3_{2004}$$
Solve the system of equation for $(x,y) \in \mathbb{R}$
$$\left\{\begin{matrix}
\sqrt{x^2+y^2}+\sqrt{(x-4)^2+(y-3)^2}=5\\
3x^2+4xy=24
\end{matrix}\right.$$ \\
Explain your answer
Let $d$ be any positive integer not equal to $2, 5$ or $13$. Show that one can find distinct $a,b$ in the set $\{2,5,13,d\}$ such that $ab-1$ is not a perfect square.
Find all real numbers $a, b, c$ that satisfy
$$ 2a - b =a^2b, \qquad 2b-c = b^2 c, \qquad 2c-a= c^2 a.$$
Prove that if the numbers $a, b, c, d$ satisfy the system of equations
$$\begin{cases} a^2+b^2=2cd \\ b^2+c^2=2da \\ c^2+d^2=2ab \end{cases}$$ then $a=b=c=d$.
Let $a,b,c$ be positive numbers with $\sqrt{a}+\sqrt{b}+\sqrt{c}= \frac{\sqrt{3}}{2}$
Prove that the system of equations
\[\sqrt{y-a}+\sqrt{z-a}=1\]
\[\sqrt{z-b}+\sqrt{x-b}=1\]
\[\sqrt{x-c}+\sqrt{y-c}=1\]
has exactly one solution $(x,y,z)$ in real numbers.
It was proposed by Poland. Have fun! :lol:
Let $a$ and $b$ satisfy the conditions $\begin{cases}
a^3 - 6a^2 + 15a = 9 \\
b^3 - 3b^2 + 6b = -1 \end{cases}$ .
The value of $(a - b)^{2014}$ is:
(A): $1$, (B): $2$, (C): $3$, (D): $4$, (E) None of the above.
Solve the system of equation for $(x,y) \in \mathbb{R}$
$$\left\{\begin{matrix}
\sqrt{x^2+y^2}+\sqrt{(x-4)^2+(y-3)^2}=5\\
3x^2+4xy=24
\end{matrix}\right.$$ \\
Explain your answer
Find all functions $f$ and $g$ defined from $\mathbb{R}_{>0}$ to $\mathbb{R}_{>0}$ such that for all $x, y > 0$ the two equations hold
$$ (f(x) + y - 1)(g(y) + x - 1) = {(x + y)}^2 $$
$$ (-f(x) + y)(g(y) + x) = (x + y + 1)(y - x - 1) $$
[i]Note: $\mathbb{R}_{>0}$ denotes the set of positive real numbers.[/i]
Determine all $4$-tuples $(a,b, c, d)$ of positive real numbers satisfying $a + b +c + d = 1$ and
$\max (\frac{a^2}{b},\frac{b^2}{a}) \cdot \max (\frac{c^2}{d},\frac{d^2}{c}) = (\min (a + b, c + d))^4$
Let $x\ne 1,y\ne 1$ and $x\ne y.$ Show that if \[\frac{yz-x^2}{1-x}=\frac{zx-y^2}{1-y},\] then \[\frac{yz-x^2}{1-x}=\frac{zx-y^2}{1-y}=x+y+z.\]
Find all sets of four real numbers $x_1, x_2, x_3, x_4$ such that the sum of any one and the product of the other three is equal to 2.
Let $n,k$ be given natural numbers. Determine all ordered n-tuples of non-negative real numbers $(x_1,x_2,...,x_n)$ that satisfy the system of equations
$$x_1^k+x_2^k+...+x_n^k=1$$
$$(1+x_1)(1+x_2)...(1+x_n)=2$$
Determine the value of the expression $x^2 + y^2 + z^2$,
if $x + y + z = 13$ , $xyz= 72$ and $\frac1x + \frac1y + \frac1z = \frac34$.
Find all positive real numbers $x, y, z$, such that
$$x - \frac{1}{y^2} = y - \frac{1}{z^2}= z - \frac{1}{x^2}$$
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 ?
Prove that there are infinitely many natural numbers $a,b,c,u$ and $v$ with greatest common divisor $1$ satisfying the system of equations: $a+b+c=u+v$ and $a^2+b^2+c^2=u^2+v^2$
Does there exist a finite set of real numbers such that their sum equals $2$, the sum of their squares equals $3$, the sum of their cubes equals $4$, ..., and the sum of their ninth powers equals $10$?
Let $m$ positive integers $a_1, \dots , a_m$ be given. Prove that there exist fewer than $2^m$ positive integers $b_1, \dots , b_n$ such that all sums of distinct $b_k$’s are distinct and all $a_i \ (i \leq m)$ occur among them.
The first four terms of an arithmetic sequence are $a, x, b, 2x$. The ratio of $a$ to $b$ is
$ \textbf{(A)}\ \frac{1}{4} \qquad\textbf{(B)}\ \frac{1}{3} \qquad\textbf{(C)}\ \frac{1}{2} \qquad\textbf{(D)}\ \frac{2}{3} \qquad\textbf{(E)}\ 2 $
Determine all triples (x, y, z) of non-negative real numbers that satisfy the following system of equations
$\begin{cases} x^2 - y = (z - 1)^2\\
y^2 - z = (x - 1)^2 \\
z^2 - x = (y -1)^2 \end{cases}$.
Find, with justification, all positive real numbers $a,b,c$ satisfying the system of equations:
$$\begin{cases} a\sqrt{b}=a+c \\ b\sqrt{c}=b+a \\ c\sqrt{a}=c+b \end{cases}$$
Solve in $R$ the system:
$$\begin{cases} \dfrac{xyz}{x + y + 1}= 1998000\\ \\
\dfrac{xyz}{y + z - 1}= 1998000 \\ \\
\dfrac{xyz}{z+x}= 1998000 \end{cases}$$
Determine, for any positive real number $a$, the number of solutions $(x,y)$ to the system of equations
$$\begin{cases} |x|+|y|= 1 \\ x^2 + y^2 = a \end{cases}$$
where $x$ and $y$ are real numbers.
Determine all triples of real numbers $(a,b,c)$ that satisfy simultaneously the equations:
$$\begin{cases} a(b^2 + c) = c(c + ab) \\ b(c^2 + a) = a(a + bc) \\ c(a^2 + b) = b(b + ca) \end{cases}$$