Found problems: 744
In a certain city, age is reckoned in terms of real numbers rather than integers. Every two citizens $x$ and $x'$ either know each other or do not know each other. Moreover, if they do not, then there exists a chain of citizens $x = x_0, x_1, \ldots, x_n = x'$ for some integer $n \geq 2$ such that $ x_{i-1}$ and $x_i$ know each other. In a census, all male citizens declare their ages, and there is at least one male citizen. Each female citizen provides only the information that her age is the average of the ages of all the citizens she knows. Prove that this is enough to determine uniquely the ages of all the female citizens.
There are $n$ children around a round table. Erika is the oldest among them and she has $n$ candies, while no other child has any candy. Erika decided to distribute the candies according to the following rules. In every round, she chooses a child with at least two candies and the chosen child sends a candy to each of his/her two neighbors. (So in the first round Erika must choose herself). For which $n \ge 3$ is it possible to end the distribution after a finite number of rounds with every child having exactly one candy?
Show that there are infinitely many positive integers $c$, such that the following equations both have solutions in positive integers: $(x^2 - c)(y^2 -c) = z^2 -c$ and $(x^2 + c)(y^2 - c) = z^2 - c$.
Suppose $a$, $b$, and $c$ are relatively prime integers such that \[\frac{a}{b+c} = 2\qquad\text{and}\qquad \frac{b}{a+c} = 3.\] What is $|c|$?
Given $k_1,k_2,...,k_n\in R^+$, find all the naturals $n$ such that
$$k_1+k_2+...+k_n=2n-3$$
$$\frac{1}{k_1}+\frac{1}{k_2}+...+\frac{1}{k_n}=3$$
[i](Zhuge Liang)[/i]
The first four terms of an arithmetic sequence are $ p,9,3p\minus{}q,$ and $ 3p\plus{}q$. What is the $ 2010^{\text{th}}$ term of the sequence?
$ \textbf{(A)}\ 8041\qquad \textbf{(B)}\ 8043\qquad \textbf{(C)}\ 8045\qquad \textbf{(D)}\ 8047\qquad \textbf{(E)}\ 8049$
Given $n > 2$ nonnegative distinct integers $a_1,...,a_n$, find all nonnegative integers $y$ and $x_1,...,x_n$ satisfying $gcd(x_1,...,x_n) = 1$ and
$$\begin{cases} a_1x_1 + a_2x_2 +...+ a_nx_n = yx_1 \\
a_2x_1 + a_3x_2 +...+ a_1x_n = yx_2 \\
... \\
a_nx_1 + a_1x_2 +...+ a_{n-1}x_n = yx_n \end{cases}$$
Find all nonnegative solutions $(x,y,z)$ to the system
$$\begin{cases} \sqrt{x+y}+\sqrt{z} = 7 \\
\sqrt{x+z}+\sqrt{y} = 7 \\
\sqrt{y+z}+\sqrt{x} = 5 \end{cases}$$
Find all triples $(a, b, c)$ of positive integers for which $$\begin{cases} a + bc=2010 \\ b + ca = 250\end{cases}$$
Solve:
$$ \left\{ \begin{matrix} x+y=\sqrt{4z -1} \\ y+z=\sqrt{4x -1} \\ z+x=\sqrt{4y -1}\end{matrix}\right. . $$
Find all triples $(x, y, z)$ of real numbers satisfying:
$x + y - z = -1$ , $x^2 - y^2 + z^2 = 1$ and $- x^3 + y^3 + z^3 = -1$
Positive integers $ a$, $ b$, and $ c$ are chosen so that $ a<b<c$, and the system of equations
\[ 2x\plus{}y\equal{}2003\text{ and }y\equal{}|x\minus{}a|\plus{}|x\minus{}b|\plus{}|x\minus{}c|
\]has exactly one solution. What is the minimum value of $ c$?
$ \textbf{(A)}\ 668 \qquad
\textbf{(B)}\ 669 \qquad
\textbf{(C)}\ 1002 \qquad
\textbf{(D)}\ 2003 \qquad
\textbf{(E)}\ 2004$
Determine all real solutions $(x,y)$ of the following system of equations:
\begin{align*}
x&=3x^2y-y^3,\\
y &= x^3-3xy^2
\end{align*}
Solve the system of equations
$$\begin{cases} \dfrac{x+y}{xy}+\dfrac{xy}{x+y}= a+ \dfrac{1}{a}\\ \\\dfrac{x-y}{xy}+\dfrac{xy}{x-y}= c+ \dfrac{1}{c}\end{cases}$$
Determine all triples $(x, y, z)$ of positive real numbers which satisfies the following system of equations
\[2x^3=2y(x^2+1)-(z^2+1), \] \[ 2y^4=3z(y^2+1)-2(x^2+1), \] \[ 2z^5=4x(z^2+1)-3(y^2+1).\]
Determine all tuples of positive integers $(x, y, z, t)$ such that:
$$ xyz = t!$$
$$ (x+1)(y+1)(z+1) = (t+1)!$$
holds simultaneously.
In a magic square, the sum of the three entries in any row, column, or diagonal is the same value. The figure shows four of the entries of a magic square. Find $x.$
[asy]
size(100);defaultpen(linewidth(0.7));
int i;
for(i=0; i<4; i=i+1) {
draw((0,2*i)--(6,2*i)^^(2*i,0)--(2*i,6));
}
label("$x$", (1,5));
label("$1$", (1,3));
label("$19$", (3,5));
label("$96$", (5,5));[/asy]
Determine all real solutions to the system of equations:
$$x^2 - y = z^2$$
$$y^2 - z = x^2$$
$$z^2 - x = y^2$$
Given a positive integer number $a$ and two real numbers $A$ and $B$, find a necessary and sufficient condition on $A$ and $B$ for the following system of equations to have integer solution:
\[ \left\{\begin{array}{cc} x^2+y^2+z^2=(Ba)^2\\ x^2(Ax^2+By^2)+y^2(Ay^2+Bz^2)+z^2(Az^2+Bx^2)=\dfrac{1}{4}(2A+B)(Ba)^4\end{array}\right. \]
Determine all real solutions of the following system of equations:
$$x+y^2=y^3$$
$$y+x^2=x^3$$
Find, with proof, all integers $n$ such that there is a solution in nonnegative real numbers $(x,y,z)$ to the system of equations
\[2x^2+3y^2+6z^2=n\text{ and }3x+4y+5z=23.\]
In a Martian civilization, all logarithms whose bases are not specified are assumed to be base $b$, for some fixed $b \geq 2$. A Martian student writes down
\begin{align*}3 \log(\sqrt{x}\log x) &= 56\\\log_{\log (x)}(x) &= 54
\end{align*}
and finds that this system of equations has a single real number solution $x > 1$. Find $b$.
Find all triples $(a,b,c)$ of real numbers such that the following system holds:
$$\begin{cases} a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \\a^2+b^2+c^2=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\end{cases}$$
[i]Proposed by Dorlir Ahmeti, Albania[/i]
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 $B$?
Let the real parameter $p$ be such that the system $\begin{cases} p(x^2 - y^2) = (p^2- 1)xy \\ |x - 1|+ |y| = 1 \end{cases}$ has at least three different real solutions. Find $p$ and solve the system for that $p$.