Found problems: 744
In a rectangular array of points, with 5 rows and $N$ columns, the points are numbered consecutively from left to right beginning with the top row. Thus the top row is numbered 1 through $N,$ the second row is numbered $N+1$ through $2N,$ and so forth. Five points, $P_1, P_2, P_3, P_4,$ and $P_5,$ are selected so that each $P_i$ is in row $i.$ Let $x_i$ be the number associated with $P_i.$ Now renumber the array consecutively from top to bottom, beginning with the first column. Let $y_i$ be the number associated with $P_i$ after the renumbering. It is found that $x_1=y_2,$ $x_2=y_1,$ $x_3=y_4,$ $x_4=y_5,$ and $x_5=y_3.$ Find the smallest possible value of $N.$
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.
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.
The negative real numbers $x, y, z, t$ satisfy simultaneously equalities, $$x + y + z = t, \,\,\,\,\frac{1}{x}+ \frac{1}{y}+\frac{1}{z}= \frac{1}{t}, \\,\,\,\, x^3 + y^3 + z^3 = 1000^3$$ Compute $x + y + z + t$.
Find $x$ and $y$ ($a$ and $b$ parameters):
$$\begin{cases} \dfrac{x-y\sqrt{x^2-y^2}}{\sqrt{1-x^2+y^2}} = a\\ \\ \dfrac{y-x\sqrt{x^2-y^2}}{\sqrt{1-x^2+y^2}} = b\end{cases}$$
Determine all pairs of real numbers $(x,y)$ satisfying
\begin{align*} x^3+9x^2y&=10,\\
y^3+xy^2 &=2.
\end{align*}
Compute all ordered triples $(x, y, z)$ of real numbers satisfying the following system of equations:
$$xy + z = 40$$
$$xz + y = 51$$
$$x + y + z = 19.$$
Show that system of equations
$2ab=6(a+b)-13$
$a^2+b^2=4$
has not solutions in set of real numbers.
Find maximal value of the function $f(x)=8-3\sin^2 (3x)+6 \sin (6x)$
For any positive $a$ and $b$, find positive solutions of the system
$$\begin{cases} \dfrac{a^2}{x^2}- \dfrac{b^2}{y^2}=8(y^4-x^4)
\\ ax-by=x^4-y^4
\end{cases}$$
Find the triple of positive integers $(x,y,z)$ with $z$ least possible for which there are positive integers $a, b, c, d$ with the following properties:
(i) $x^y = a^b = c^d$ and $x > a > c$
(ii) $z = ab = cd$
(iii) $x + y = a + b$.
Solve the system $\begin{cases} x+y=a \\
x^5 +y^5 = b^5
\end{cases}$
Find all ordered triples $ (x,y,z)$ of real numbers which satisfy the following system of equations:
\[ \left\{\begin{array}{rcl} xy & \equal{} & z \minus{} x \minus{} y \\
xz & \equal{} & y \minus{} x \minus{} z \\
yz & \equal{} & x \minus{} y \minus{} z \end{array} \right.
\]
Let $n > 1$ be an odd positive integer. Assume that positive integers $x_1, x_2,..., x_n \ge 0$ satisfy:
$$\begin{cases} (x_2 - x_1)^2 + 2(x_2 +x_1) + 1 = n^2 \\
(x_3 -x_2)^2 + 2(x_3 +x_2) + 1 = n^2 \\
...\\
(x_1 - x_n)^2 + 2(x_1 + x_n)+ 1 = n^2 \end {cases}$$
Show that there exists $j, 1 \le j \le n$, such that $x_j = x_{j+1}$. Here $x_{n+1} = x_1$.
Solve in $N$:
$$\begin{cases} a^3=b^3+c^3+12a \\ a^2=5(b+c) \end{cases}$$
Find all real solutions of the system of equations
$$\begin{cases} x^2 + y^2 + u^2 + v^2 = 4 \\ xu + yv + xv + yu = 0 \\ xyu + yuv + uvx + vxy = - 2 \\ xyuv = -1 \end{cases}$$
Find all solutions $x_1, x_2, x_3, x_4, x_5$ of the system \[ x_5+x_2=yx_1 \] \[ x_1+x_3=yx_2 \] \[ x_2+x_4=yx_3 \] \[ x_3+x_5=yx_4 \] \[ x_4+x_1=yx_5 \] where $y$ is a parameter.
How many ordered pairs $(x,y)$ of real numbers satisfy the following system of equations?
\begin{align*}
x^2+3y&=9\\
(|x|+|y|-4)^2&=1\\
\end{align*}
$\textbf{(A)}\: 1\qquad\textbf{(B)} \: 2\qquad\textbf{(C)} \: 3\qquad\textbf{(D)} \: 5\qquad\textbf{(E)} \: 7$
Show that the only real solution to
\[\left\{ \begin{array}{l}
x(x+y)^2 = 9 \\
x(y^3 - x^3) = 7 \\
\end{array} \right.
\]
is $x = 1$, $y = 2$.
Solve the system of equations $$\begin{cases} \sin x = y \\ \sin y = x \end{cases}$$
Find all triples of reals $(a, b, c)$ such that
$$a - \frac{1}{b}=b - \frac{1}{c}=c - \frac{1}{a}.$$
Solve the system of equations
\[x^2 + y^2 + z^2 + t^2 = 50;\]
\[x^2 - y^2 + z^2 - t^2 = -24;\]
\[xy = zt;\]
\[x - y + z - t = 0.\]
Find all real solutions to the system of equations: $$\begin{cases} (x^2+xy+y^2)\sqrt{x^2+y^2} = 88 \\ (x^2-xy+y^2)\sqrt{x^2+y^2} = 40 \end{cases}$$
In $\triangle ABC$, $AC = BC$, and point $D$ is on $\overline{BC}$ so that $CD = 3 \cdot BD$. Let $E$ be the midpoint of $\overline{AD}$. Given that $CE = \sqrt{7}$ and $BE = 3$, the area of $\triangle ABC$ can be expressed in the form $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m+n$.
Find the total number of solutions to the following system of equations:
$ \{\begin{array}{l} a^2 + bc\equiv a \pmod{37} \\
b(a + d)\equiv b \pmod{37} \\
c(a + d)\equiv c \pmod{37} \\
bc + d^2\equiv d \pmod{37} \\
ad - bc\equiv 1 \pmod{37} \end{array}$