Found problems: 744
Find all real solutions $(x,y,z)$ of the system of equations
$$\begin{cases} x^3 = 2y-1 \\y^3 = 2z-1\\ z^3 = 2x-1\end{cases}$$
Find the smallest positive real $t$ such that
\[\left\{ \begin{array}{l}
x_1 + x_3 = 2t x_2 \\
x_2 + x_4 = 2t x_3 \\
x_3 + x_5=2t x_4 \\
\end{array} \right.
\]
has a solution $x_1$, $x_2$, $x_3$, $x_4$, $x_5$ in non-negative reals, not all zero.
The formula which expresses the relationship between $x$ and $y$ as shown in the accompanying table is:
\[ \begin{tabular}[t]{|c|c|c|c|c|c|}\hline
x&0&1&2&3&4\\\hline
y&100&90&70&40&0\\\hline
\end{tabular}\]
$\textbf{(A)}\ y=100-10x \qquad
\textbf{(B)}\ y=100-5x^2 \qquad
\textbf{(C)}\ y=100-5x-5x^2 \qquad\\
\textbf{(D)}\ y=20-x-x^2 \qquad
\textbf{(E)}\ \text{None of these}$
Given $a$ and $b$ distinct positive integers, show that the system of equations
$x y +zw = a$
$xz + yw = b$
has only finitely many solutions in integers $x, y, z,w$.
Find all quadruples of real numbers $(a,b,c,d)$ satisfying the system of equations
\[\begin{cases}(b+c+d)^{2010}=3a\\ (a+c+d)^{2010}=3b\\ (a+b+d)^{2010}=3c\\ (a+b+c)^{2010}=3d\end{cases}\]
Determine all the triples $(a, b, c)$ of positive real numbers such that the system
\[ax + by -cz = 0,\]\[a \sqrt{1-x^2}+b \sqrt{1-y^2}-c \sqrt{1-z^2}=0,\]
is compatible in the set of real numbers, and then find all its real solutions.
Real numbers $x$ and $y$ satisfy the system of equations
$$x^3 + 3x^2 = -3y - 1$$
$$y^3 + 3y^2 = -3x - 1.$$
What is the greatest possible value of $x$?
Find all integers $n \geq 3$ for which there exist real numbers $a_1, a_2, \dots a_{n + 2}$ satisfying $a_{n + 1} = a_1$, $a_{n + 2} = a_2$ and
$$a_ia_{i + 1} + 1 = a_{i + 2},$$
for $i = 1, 2, \dots, n$.
[i]Proposed by Patrik Bak, Slovakia[/i]
Determine all pairs $ (a,b)$ of positive integers, each integer having two decimal digits, such that $ 100a\plus{}b$ and $ 201a\plus{}b$ are both perfect squares.
Solve the system $\begin{cases} 3xyz -x^3 - y^3-z^3 = b^3 \\
x + y+ z = 2b \\
x^2 + y^2-z^2 = b^2
\end{cases}$ in $C$
Let $P$ be a cubic polynomial with $P(0) = k, P(1) = 2k,$ and $P(-1) = 3k$. What is $P(2) + P(-2)$?
$ \textbf{(A) }0 \qquad\textbf{(B) }k \qquad\textbf{(C) }6k \qquad\textbf{(D) }7k\qquad\textbf{(E) }14k\qquad $
Let $x_1,x_2, ..., x_n$ be non-negative real numbers. Solve the system of equations:
$$x_k+x_{k+1}=x^2_{k+2}\,\,,\,\,\, (k =1,2,...,n),$$
where $x_{n+1} = x_1$, $x_{n+2} = x_2$.
Let $a, b, c$ be positive real numbers and $p, q, r$ complex numbers. Let $S$ be the set of all solutions $(x, y, z)$ in $\mathbb C$ of the system of simultaneous equations
\[ax + by + cz = p,\]\[ax2 + by2 + cz2 = q,\]\[ax3 + bx3 + cx3 = r.\]
Prove that $S$ has at most six elements.
Let $a$ be a real number, such that $a\ne 0, a\ne 1, a\ne -1$ and $m,n,p,q$ be natural numbers .
Prove that if $a^m+a^n=a^p+a^q$ and $a^{3m}+a^{3n}=a^{3p}+a^{3q}$ , then $m \cdot n = p \cdot q$.
Solve the system of equations:
$
\begin{matrix}
|x+y| + |1-x| = 6 \\ |x+y+1| + |1-y| = 4.
\end{matrix}
$
Solve the system of equations
$$\begin{cases} \sqrt{\dfrac{y^2+x}{4x}}+\dfrac{y}{\sqrt{y^2+x}}=\dfrac{y^2}{4}\sqrt{\dfrac{4x}{y^2+x}} \\ \sqrt{x}+ \sqrt{x-y-1}=(y+1)(\sqrt{x}- \sqrt{x-y-1}) \end{cases}$$
Find the smallest odd prime $p$, such that there exist coprime positive integers $k$ and $\ell$ which satisfy
\[4k-3\ell=12\quad \text{ and }\quad \ell^2+\ell k +k^2\equiv 3\text{ }(\text{mod }p)\]
Let $a,b, c$ be real numbers with $a+b+c = 2018$.
Suppose $x, y$, and $z$ are the distinct positive real numbers which are satisfied $a = x^2 - yz - 2018, b = y^2 - zx - 2018$ , and $c = z^2 - xy - 2018$.
Compute the value of the following expression $P = \frac{\sqrt{a^3 + b^3 + c^3 - 3abc}}{x^3 + y^3 + z^3 - 3xyz}$
Real numbers $a, b, c$ are not equal $0$ and are solution of the system:
$\begin{cases} a^2 + a = b^2 \\ b^2 + b = c^2 \\ c^2 +c = a^2 \end{cases}$
Prove that $(a - b)(b - c)(c - a) = 1$.
Let $k > 1$ be a positive integer and $n \ge 2019$ be an odd positive integer. The non-zero rational numbers $x_1, x_2,..., x_n$ are not all equal, and satisfy the following chain of equalities:
$$x_1 +\frac{k}{x_2}= x_2 +\frac{k}{x_3}= x_3 +\frac{k}{x_4}= ... = x_{n-1} +\frac{k}{x_n}= x_n +\frac{k}{x_1}.$$
What is the smallest possible value of $k$?
Solve the system of equations:
$
\begin{matrix}
x^2 + x - 1 = y \\
y^2 + y - 1 = z \\
z^2 + z - 1 = x.
\end{matrix}
$
In a mathematical contest, three problems, $A,B,C$ were posed. Among the participants ther were 25 students who solved at least one problem each. Of all the contestants who did not solve problem $A$, the number who solved $B$ was twice the number who solved $C$. The number of students who solved only problem $A$ was one more than the number of students who solved $A$ and at least one other problem. Of all students who solved just one problem, half did not solve problem $A$. How many students solved only problem $B$?
Solve the system of equations
\[\tan x_1 +\cot x_1=3 \tan x_2,\]\[\tan x_2 +\cot x_2=3 \tan x_3,\]\[\vdots\]\[\tan x_n +\cot x_n=3 \tan x_1\]
Find all the triplets of real numbers $(x , y , z)$ such that :
$y=\frac{x^3+12x}{3x^2+4}$ , $z=\frac{y^3+12y}{3y^2+4}$ , $x=\frac{z^3+12z}{3z^2+4}$
Find all the quaterns $(x,y,z,w)$ of real numbers (not necessarily distinct) that solve the following system of equations:
$$x+y=z^2+w^2+6zw$$
$$x+z=y^2+w^2+6yw$$
$$x+w=y^2+z^2+6yz$$
$$y+z=x^2+w^2+6xw$$
$$y+w=x^2+z^2+6xz$$
$$z+w=x^2+y^2+6xy$$