Found problems: 744
Let a system of equations:
$\left\{\begin{matrix}x-ay=yz\\y-az=zx\\z-ax=xy\end{matrix}\right.$
a)Find (x,y,z) if a=0
b)Prove that: the system have 5 distinct roots $\forall$a>1,a$\in\mathbb{R}.$
Find the largest real number $a$ such that \[\left\{ \begin{array}{l}
x - 4y = 1 \\
ax + 3y = 1\\
\end{array} \right.
\] has an integer solution.
Given $n^2$ numbers $x_{i,j}$ ($i,j=1,2,...,n$) satisfying the system of $n^3$ equations $$x_{i,j}+x_{j,k}+x_{k,i}=0 \,\,\, (i,j,k = 1,...,n)$$Prove that there exist such numbers $a_1,a_2,...,a_n$, that $x_{i,j}=a_i-a_j$ for all $i,j=1,...n$.
If $$\frac{x_1}{x_1+1} = \frac{x_2}{x_2+3} = \frac{x_3}{x_3+5} = ...= \frac{x_{1006}}{x_{1006}+2011}$$ and $x_1+x_2+...+x_{1006} = 503^2$, determine the value of $x_{1006}$.
Solve the system of equations $$
\left\{\begin{array}{l}
x \log x+y \log y+z \log x=0\\ \\
\dfrac{\log x}{x}+\dfrac{\log y}{y}+\dfrac{\log z}{z}=0
\end{array} \right.
$$
The players on a basketball team made some three-point shots, some two-point shots, and some one-point free throws. They scored as many points with two-point shots as with three-point shots. Their number of successful free throws was one more than their number of successful two-point shots. The team's total score was 61 points. How many free throws did they make?
$\textbf{(A)}\,13 \qquad\textbf{(B)}\,14 \qquad\textbf{(C)}\,15 \qquad\textbf{(D)}\,16 \qquad\textbf{(E)}\,17$
Show that there are two real solutions to:
$$\begin{cases} x + y^2 + z^4 = 0 \\
y + z^2 + x^4 = 0 \\
z + x^2 + y^5 = 0\end {cases}$$
When $ 15$ is appended to a list of integers, the mean is increased by $ 2$. When $ 1$ is appended to the enlarged list, the mean of the enlarged list is decreased by $ 1$. How many integers were in the original list?
$ \textbf{(A)}\ 4 \qquad
\textbf{(B)}\ 5 \qquad
\textbf{(C)}\ 6 \qquad
\textbf{(D)}\ 7 \qquad
\textbf{(E)}\ 8$
Let $a,b,c$ be real numbers with $a$ non-zero. It is known that the real numbers $x_1,x_2,\ldots,x_n$ satisfy the $n$ equations:
\[ ax_1^2+bx_1+c = x_{2} \]\[ ax_2^2+bx_2 +c = x_3\]\[ \ldots \quad \ldots \quad \ldots \quad \ldots\]\[ ax_n^2+bx_n+c = x_1 \] Prove that the system has [b]zero[/b], [u]one[/u] or [i]more than one[/i] real solutions if $(b-1)^2-4ac$ is [b]negative[/b], equal to [u]zero[/u] or [i]positive[/i] respectively.
Find all triples $(x,y, z)$ of real (but not necessarily positive) numbers satisfying
$3(x^2 + y^2 + z^2) = 1$ , $x^2y^2 + y^2z^2 + z^2x^2 = xyz(x + y + z)^3$.
Find all triplets of real numbers $(x,y,z)$ that are solutions to the system of equations
$x^2+y^2+25z^2=6xz+8yz$
$ 3x^2+2y^2+z^2=240$
The number of solutions, in real numbers $a$, $b$, and $c$, to the system of equations $$a+bc=1,$$$$b+ac=1,$$$$c+ab=1,$$ is
$\text{(A) }3\qquad\text{(B) }4\qquad\text{(C) }5\qquad\text{(D) more than }5\text{, but finitely many}\qquad\text{(E) infinitely many}$
For an acute triangle $ABC$ and a point $X$ satisfying $\angle{ABX}+\angle{ACX}=\angle{CBX}+\angle{BCX}$.
Fi nd the minimum length of $AX$ if $AB=13$, $BC=14$, and $CA=15$.
Given real numbers $(a_i)$ and $(b_i)$ (for $i=1,2,3,4$) such that $a_1 b _2 \ne a_2 b_1 .$ Consider the set of all solutions $(x_1 ,x_2 ,x_3 , x_4)$ of the simultaneous equations
$$ a_1 x_1 +a _2 x_2 +a_3 x_3 +a_4 x_4 =0 \;\; \text{and}\;\; b_1 x_1 +b_2 x_2 +b_3 x_3 +b_4 x_4 =0 $$
for which no $x_i$ is zero. Each such solution generates a $4$-tuple of plus and minus signs (by considering the sign of $x_i$).
[list=a]
[*] Determine, with proof, the maximum number of distinct $4$-tuples possible.
[*] Investigate necessary and sufficient conditions on $(a_i)$ and $(b_i)$ such that the above maximum of distinct $4$-tuples is attained.
Find all ordered pairs $(x,y)$ of real numbers that satisfy the following system of equations:
$$\begin{cases}
y(x+y)^2=2\\
8y(x^3-y^3) = 13.
\end{cases}$$
Find all real pairs $(x,y)$ that solve the system of equations \begin{align} x^5 &= 21x^3+y^3
\\ y^5 &= x^3+21y^3. \end{align}
Find all the pairs of real numbers $(x,y)$ that are solutions of the system:
$(x^{2}+y^{2})^{2}-xy(x+y)^{2}=19 $
$| x - y | = 1$
Integers $a$, $b$, and $c$ satisfy $ab + c = 100$, $bc + a = 87$, and $ca + b = 60$. What is $ab + bc + ca$?
$
\textbf{(A) }212 \qquad
\textbf{(B) }247 \qquad
\textbf{(C) }258 \qquad
\textbf{(D) }276 \qquad
\textbf{(E) }284 \qquad
$
Solve the system of equations:
$\left\{ \begin{aligned} x^4+y^2-xy^3-\frac{9}{8}x = 0 \\ y^4+x^2-yx^3-\frac{9}{8}y=0 \end{aligned} \right.$
Let $a$ be a real number and $A = \{(x, y) \in R \times R | \, x + y = a\}$, $B = \{(x,y) \in R \times R | \, x^3 + y^3 < a\}$ . Find all values of $a$ such that $A \cap B = \emptyset$ .
A positive integer $n$ and a real number $a$ are given. Find all $n$-tuples $(x_1, ... ,x_n)$ of real numbers that satisfy the system of equations $$\sum_{i=1}^{n} x_i^k= a^k \,\,\,\, for \,\,\,\, k = 1,2, ... ,n$$
For what real numbers $p$ has the system of equations
$$\begin{cases} x_1^4+\dfrac{1}{x_1^2}=px_2 \\ \\ x_2^4+\dfrac{1}{x_2^2}=px_3 \\ ... \\ x_{2004}^4+\dfrac{1}{x_{2004}^2}=px_{2005} \\ \\ x_{2005}^4+\dfrac{1}{x_{2005}^2}=px_{1}\end{cases}$$
just one solution $(x_1,x_2,...,x_{2005})$, where $x_1,x_2,...,x_{2005}$ are real numbers?
Find all triplets of real numbers $(x, y, z)$ that satisfies the system of equations
$x^5 = 2y^3 + y - 2$
$y^5 = 2z^3 + z - 2$
$z^5 = 2x^3 + x - 2$
Three real numbers are given. Fractional part of the product of every two of them is $ 1\over 2$. Prove that these numbers are irrational.
[i]Proposed by A. Golovanov[/i]
Let n be a positive integer. Find all complex numbers $x_{1}$, $x_{2}$, ..., $x_{n}$ satisfying the following system of equations:
$x_{1}+2x_{2}+...+nx_{n}=0$,
$x_{1}^{2}+2x_{2}^{2}+...+nx_{n}^{2}=0$,
...
$x_{1}^{n}+2x_{2}^{n}+...+nx_{n}^{n}=0$.