Found problems: 744
Solve the system of equations:
$ y^2\equal{}(x\plus{}8)(x^2\plus{}2),$
$ y^2\minus{}(8\plus{}4x)y\plus{}(16\plus{}16x\minus{}5x^2)\equal{}0.$
Do there exist real numbers $a,b,c$ such that the system of equations
\begin{align*}
x+y+z&=a\\
x^2+y^2+z^2&=b\\
x^4+y^4+z^4&=c
\end{align*}
has infinitely many real solutions $(x,y,z)$?
Find all triples $(x,y,z)$ of positive integers satisfying the system of equations
\[\begin{cases} x^2=2(y+z)\\ x^6=y^6+z^6+31(y^2+z^2)\end{cases}\]
Let $a,b,c$ be coprime nonzero integers. Prove that for any coprime integers $u,v,w$ with $au+bv+cw = 0$ there exist integers $m,n, p$ such that $$\begin{cases} a = nw- pv \\ b = pu-mw \\ c = mv-nu \end{cases}$$
We consider the following system
with $q=2p$:
\[\begin{matrix} a_{11}x_{1}+\ldots+a_{1q}x_{q}=0,\\ a_{21}x_{1}+\ldots+a_{2q}x_{q}=0,\\ \ldots ,\\ a_{p1}x_{1}+\ldots+a_{pq}x_{q}=0,\\ \end{matrix}\]
in which every coefficient is an element from the set $\{-1,0,1\}$$.$ Prove that there exists a solution $x_{1}, \ldots,x_{q}$ for the system with the properties:
[b]a.)[/b] all $x_{j}, j=1,\ldots,q$ are integers$;$
[b]b.)[/b] there exists at least one j for which $x_{j} \neq 0;$
[b]c.)[/b] $|x_{j}| \leq q$ for any $j=1, \ldots ,q.$
Let $a$, $b$, and $c$ be complex numbers satisfying the system of equations \begin{align*}\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}&=9,\\\dfrac{a^2}{b+c}+\dfrac{b^2}{c+a}+\dfrac{c^2}{a+b}&=32,\\\dfrac{a^3}{b+c}+\dfrac{b^3}{c+a}+\dfrac{c^3}{a+b}&=122.\end{align*} Find $abc$.
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 $i = 1, 2, . . . , n.$
Determine all integers $n\ge1$ for which there exists $n$ real numbers $x_1,\ldots,x_n$ in the closed interval $[-4,2]$ such that the following three conditions are fulfilled:
- the sum of these real numbers is at least $n$.
- the sum of their squares is at most $4n$.
- the sum of their fourth powers is at least $34n$.
[i](Proposed by Gerhard Woeginger, Austria)[/i]
Determine all ordered quadruples of real numbers $(x_1, x_2, x_3, x_4)$ for which the following system of equations exists, is fulfilled:
$$x_1x_2 + x_1x_3 + x_2x_3 + x_4 = 2$$
$$x_1x_2 + x_1x_4 + x_2x_4 + x_3 = 2$$
$$x_1x_3 + x_1x_4 + x_3x_4 + x_2 = 2$$
$$x_2x_3 + x_2x_4 + x_3x_4 + x_1 = 2$$
Find all integer solutions to $m^{m+n} = n^{12}, n^{m+n} = m^3$.
Find four consecutive terms $a, b, c, d$ of an arithmetic progression and four consecutive terms $a_1, b_1, c_1, d_1$ of a geometric progression such that $$\begin{cases}a + a_1 = 27 \\\ b + b_1 = 27 \\ c + c_1 = 39 \\ d + d_1 = 87\end{cases}$$.
Determine if there are triplets ($x,y,z)$ of real numbers such that
$$\begin{cases} x+y+z=7 \\ xy+yz+zx=11\end{cases}$$
If the answer is affirmative, find the minimum and maximum values of $z$ in such a triplet.
Determine all real solutions $(a_1,...,a_n)$ of the following system of equations:
$$\begin{cases}a_3 = a_2 + a_1\\
a_4 = a_3 + a_2\\
...\\
a_n = a_{n-1} + a_{n-2}\\
a_1= a_n +a_{n-1} \\
a_2 = a_1 + a_n \end{cases}$$
Does the system of equation
\begin{align*}
\begin{cases}
x_1 + x_2 &= y_1 + y_2 + y_3 + y_4 \\
x_1^2 + x_2^2 &= y_1^2 + y_2^2 + y_3^2 + y_4^2 \\
x_1^3 + x_2^3 &= y_1^3 + y_2^3 + y_3^3 + y_4^3
\end{cases}
\end{align*}
admit a solution in integers such that the absolute value of each of these integers is greater than $2020$?
Suppose $x,y$ and $z$ are integers that satisfy the system of equations \[x^2y+y^2z+z^2x=2186\] \[xy^2+yz^2+zx^2=2188.\] Evaluate $x^2+y^2+z^2.$
Carl and Bob can demolish a building in 6 days, Anne and Bob can do it in $3$, Anne and Carl in $5$. How many days does it take all of them working together if Carl gets injured at the end of the first day and can't come back?
Solve the following system of equations in real numbers:
\[ \begin{cases} \sqrt{x^2-2x+6}\cdot \log_{3}(6-y) =x \\ \sqrt{y^2-2y+6}\cdot \log_{3}(6-z)=y \\ \sqrt{z^2-2z+6}\cdot\log_{3}(6-x)=z \end{cases}. \]
How many real solutions does the following system have ?$\begin{cases} x+y=2 \\
xy - z^2 = 1 \end{cases}$
$C$ is the circle center $(0,1)$, radius $1$. $P$ is the parabola $y = ax^2$. They meet at $(0, 0)$. For what values of $a$ do they meet at another point or points?
Solve the system of equations $ \begin{cases} x^3 \plus{} x(y \minus{} z)^2 \equal{} 2\\ y^3 \plus{} y(z \minus{} x)^2 \equal{} 30\\ z^3 \plus{} z(x \minus{} y)^2 \equal{} 16\end{cases}$.
Find the polynomial $P(x)$ with real coefficients such that $P(2)=12$ and $P(x^2)=x^2(x^2+1)P(x)$ for each $x\in\mathbb{R}$.
Find all triplets of real numbers $(x, y, z)$ that satisfies the system of equations
$x^5 = 2y^3 + y - 2$
$y^5 = 2z^3 + z - 2$
$z^5 = 2x^3 + x - 2$
The vertices of a $ 3 \minus{} 4 \minus{} 5$ right triangle are the centers of three mutually externally tangent circles, as shown. What is the sum of the areas of the three circles?
[asy]unitsize(5mm);
defaultpen(fontsize(10pt)+linewidth(.8pt));
pair B=(0,0), C=(5,0);
pair A=intersectionpoints(Circle(B,3),Circle(C,4))[0];
draw(A--B--C--cycle);
draw(Circle(C,3));
draw(Circle(A,1));
draw(Circle(B,2));
label("$A$",A,N);
label("$B$",B,W);
label("$C$",C,E);
label("3",midpoint(B--A),NW);
label("4",midpoint(A--C),NE);
label("5",midpoint(B--C),S);[/asy]$ \textbf{(A) } 12\pi\qquad \textbf{(B) } \frac {25\pi}{2}\qquad \textbf{(C) } 13\pi\qquad \textbf{(D) } \frac {27\pi}{2}\qquad \textbf{(E) } 14\pi$
A system of equations
$$\begin{cases} x^2 \,\, ? \,\, z^2 = -8 \\ y^2 \,\, ? \,\, z^2 = 7 \end{cases}$$
is written on a piece of paper, but unfortunately two of the symbols are a little blurred. However, it is known that the system has at least one solution, and that each of the two question marks stands for either $+$ or $-$. What are the two symbols?
Let $n$ be a positive integer. Solve the system of equations \begin{align*}x_{1}+2x_{2}+\cdots+nx_{n}&= \frac{n(n+1)}{2}\\ x_{1}+x_{2}^{2}+\cdots+x_{n}^{n}&= n\end{align*} for $n$-tuples $(x_{1},x_{2},\ldots,x_{n})$ of nonnegative real numbers.