Found problems: 85335
Let $\mathbb R$ be the set of real numbers. We denote by $\mathcal F$ the set of all functions $f\colon\mathbb R\to\mathbb R$ such that
$$f(x + f(y)) = f(x) + f(y)$$
for every $x,y\in\mathbb R$ Find all rational numbers $q$ such that for every function $f\in\mathcal F$, there exists some $z\in\mathbb R$ satisfying $f(z)=qz$.
Find the best possible $k,k'$ such that \[k<\frac{v}{v+w}+\frac{w}{w+x}+\frac{x}{x+y}+\frac{y}{y+z}+\frac{z}{z+v}<k'\]
for all positive reals $v,w,x,y,z$.
Find all triples $(x,y,z)$ of positive integers such that $x \leq y \leq z$ and
\[x^3(y^3+z^3)=2012(xyz+2).\]
Quadrilateral $XABY$ is inscribed in the semicircle $\omega$ with diameter $XY$. Segments $AY$ and $BX$ meet at $P$. Point $Z$ is the foot of the perpendicular from $P$ to line $XY$. Point $C$ lies on $\omega$ such that line $XC$ is perpendicular to line $AZ$. Let $Q$ be the intersection of segments $AY$ and $XC$. Prove that \[\dfrac{BY}{XP}+\dfrac{CY}{XQ}=\dfrac{AY}{AX}.\]
Let $O$ be the center of the inscribed circle of $\Delta ABC$ and point $D$ be the middle point of $AB$.
If $\angle AOD=90^\circ$, prove that $AB+BC=3AC$.
Let $n>1$ be an integer. Prove that there exist $m>n^n $ such that $\frac {n^m-m^n}{m+n} $ is a positive integer.
We say that a strictly increasing positive real sequence $a_1,a_2,\cdots $ is an [i]elf sequence[/i] if for any $c>0$ we can find an $N$ such that $a_n<cn$ for $n=N,N+1,\cdots$. Furthermore, we say that $a_n$ is a [i]hat[/i] if $a_{n-i}+a_{n+i}<2a_n$ for $\displaystyle 1\le i\le n-1$. Is it true that every elf sequence has infinitely many hats?
In a finite sequence of real numbers the sum of any seven successive terms is negative and the sum of any eleven successive terms is positive. Determine the maximum number of terms in the sequence.
The sequence $\left(a_{n}\right)_{n\in\mathbb{N}}$ is defined recursively as $a_{0}=a_{1}=1$, $a_{n+2}=5a_{n+1}-a_{n}-1$, $\forall n\in\mathbb{N}$
Prove that
$$a_{n}\mid a_{n+1}^{2}+a_{n+1}+1$$
for any $n\in\mathbb{N}$
a) Given a prime number $p$ and $2$ polynomials$$P(x)=a_nx^n+...+a_1x+a_0; Q(x)=b_mx^m+...+b_1x+b_0.$$
We know that the product $P(x)Q(x)$ is a polynomial whose coefficents are all divisible by $p.$
Prove that: at least $1$ in $2$ polynomials $P(x),Q(x)$ has all coefficents are all divisible by $p.$
b) Prove that the product of $2$ original polynomials is a original polynomial.
The numbers from $1$ to $100$ are placed without repetition in each cell of a \(10 \times 10\) board. An increasing path of length \(k\) on this board is a sequence of cells \(c_1, c_2, \ldots, c_k\) such that, for each \(i = 2, 3, \ldots, k\), the following properties are satisfied:
• The cells \(c_i\) and \(c_{i-1}\) share a side or a vertex;
• The number in \(c_i\) is greater than the number in \(c_{i-1}\).
What is the largest positive integer \(k\) for which we can always find an increasing path of length \(k\), regardless of how the numbers from 1 to 100 are arranged on the board?
How many ordered pairs of integers (x; y) satisfy
the equation: 2$x^2$ + $y^2$ + xy = 2(x + y)?
Are there positive integers $a$ and $b$ such that both $a^2 + 4b$ and $b^2 + 4a$ are perfect squares?
Find all two-digit numbers $x$ the sum of whose digits is the same as that of $2x$, $3x$, ... , $9x$.
In a plane we choose a cartesian system of coordinates. A point $A(x,y)$ in the plane is called an integer point if and only if both $x$ and $y$ are integers. An integer point $A$ is called invisible if on the segment $(OA)$ there is at least one integer point.
Prove that for each positive integer $n$ there exists a square of side $n$ in which all the interior integer points are invisible.
Let $x_i \in \{0, 1\} (i = 1, 2, \cdots, n)$. If the function $f = f(x_1, x_2, \cdots, x_n)$ only equals $0$ or $1$, then define $f$ as an "$n$-variable Boolean function" and denote
$$D_n (f) = \{ (x_1, x_2, \cdots, x_n) | f(x_1, x_2, \cdots, x_n) = 0 \}$$.
$(1)$ Determine the number of $n$-variable Boolean functions;
$(2)$ Let $g$ be a $10$-variable Boolean function satisfying
$$g(x_1, x_2, \cdots, x_{10}) \equiv 1 + x_1 + x_1 x_2 + x_1 x_2 x_3 + \cdots + x_1 x_2\cdots x_{10} \pmod{2}$$
Evaluate the size of the set $D_{10} (g)$ and $\sum\limits_{(x_1, x_2, \cdots, x_{10}) \in D_{10} (g)} (x_1 + x_2 + x_3 + \cdots + x_{10})$.
The expression $\sin2^\circ\sin4^\circ\sin6^\circ\cdots\sin90^\circ$ is equal to $p\sqrt{5}/2^{50}$, where $p$ is an integer. Find $p$.
Find the sum of all such natural numbers from $1$ to $500$ that are not divisible by $5$ or $7$.
What time was it $2011$ minutes after midnight on January 1, 2011?
$\textbf{(A)} \text{January 1 at 9:31PM}$
$\textbf{(B)} \text{January 1 at 11:51PM}$
$\textbf{(C)} \text{January 2 at 3:11AM}$
$\textbf{(D)} \text{January 2 at 9:31AM}$
$\textbf{(E)} \text{January 2 at 6:01PM}$
Let $d = a^{1999} + b^{1999} + c^{1999}$ , where $a, b$ and $c$ are integers such that $a + b + c = 0$.
(a) May it happen that $d = 2$?
(b) May it happen that $d$ is prime?
(V Senderov)
Starting with two points A and B, some circles and points are constructed as shown in
the figure:[list][*]the circle with centre A through B
[*]the circle with centre B through A
[*]the circle with centre C through A
[*]the circle with centre D through B
[*]the circle with centre E through A
[*]the circle with centre F through A
[*]the circle with centre G through A[/list]
[i][size=75](I think the wording is not very rigorous, you should assume intersections from the drawing)[/size][/i]
Show that $M$ is the midpoint of $AB$.
[img]https://cdn.artofproblemsolving.com/attachments/d/4/2352ab21cc19549f0381e88ddde9dce4299c2e.png[/img]
Solve the system of equations
$$\begin{cases} \dfrac{x+y}{1+xy}=\dfrac{1-2y}{2-y} \\ \dfrac{x-y}{1-xy}=\dfrac{1-3x}{3-x} \end{cases}$$
A group of $25$ CCA students decide they want to go to Disneyland, which is $105$ miles away. To save some time, they rent a bus with capacity $10$ people which can travel up to $60$ miles per hour. On the other hand, a student will run up to $9$ miles per hour. However, because a complicated plan of getting on and off the bus may be confusing to some students, a student may only board the bus once. What is the least number of minutes it will take for all students to reach Disneyland?
Note: both the bus and students may travel backwards.
[i]2017 CCA Math Bonanza Team Round #8[/i]
Given natural number $a>1$ and different odd prime numbers $p_1,p_2,...,p_n$ with
$a^{p_1}\equiv 1$ (mod $p_2$), $a^{p_2}\equiv 1$ (mod $p_3$), ..., $a^{p_n}\equiv 1$(mod $p_1$).
Prove that
a) $(a-1)\vdots p_i$ for some $i=1,..,n$
b) Can $(a-1)$ be divisible by $p_i $for exactly one $i$ of $i=1,...,n$?
I. Bliznets
(a) Show that for every $n\in\mathbb{N}$ there is exactly one $x\in\mathbb{R}^+$ so that $x^n+x^{n+1}=1$. Call this $x_n$.
(b) Find $\lim\limits_{n\rightarrow+\infty}x_n$.