Found problems: 85335
Determine all integers $n$ for which $\frac{4n-2}{n+5}$ is the square of a rational number.
$ ABCD$ is a terahedron: $ AD\plus{}BD\equal{}AC\plus{}BC,$ $ BD\plus{}CD\equal{}BA\plus{}CA,$ $ CD\plus{}AD\equal{}CB\plus{}AB,$ $ M,N,P$ are the mid points of $ BC,CA,AB.$ $ OA\equal{}OB\equal{}OC\equal{}OD.$ Prove that $ \angle MOP \equal{} \angle NOP \equal{}\angle NOM.$
An infinite increasing arithmetical progression consists of positive integers and contains a perfect cube. Prove that this progression also contains a term which is a perfect cube but not a perfect square.
We call a rectangle of the size $1 \times 2$ a domino. Rectangle of the $2 \times 3$ removing two opposite (under center of rectangle) corners we call tetramino. These figures can be rotated.
It requires to tile rectangle of size $2008 \times 2010$ by using dominoes and tetraminoes. What is the minimal number of dominoes should be used?
Let $I$ be a nonempty open subinterval of the set of positive real numbers. For which even $n \in \mathbb{N}$ are there injective function $f: I \to \mathbb{R}$ and positive function $p: I \to \mathbb{R}$, such that for all $x_1 , \ldots , x_n \in I$,
\[ f \left( \frac{1}{2} \left( \frac{x_1+\cdots+x_n}{n}+\sqrt[n]{x_1 \cdots x_n} \right) \right)=\frac{p(x_1)f(x_1)+\cdots+p(x_n)f(x_n)}{p(x_1)+\cdots+p(x_n)} \]
holds?
Prove that $\lfloor{\sqrt{9n+7}}\rfloor=\lfloor{\sqrt{n}+\sqrt{n+1}+\sqrt{n+2}}\rfloor$ for all postive integer $n$.
How many divisors of $12!$ are perfect squares?
[i]2015 CCA Math Bonanza Lightning Round #4.1[/i]
Solve the system of equations $$\begin{cases} \sin x = y \\ \sin y = x \end{cases}$$
$ \frac{\sqrt{10+\sqrt{1}}+\sqrt{10+\sqrt{2}}+...+\sqrt{10+\sqrt{99}}}{\sqrt{10-\sqrt{1}}+\sqrt{10-\sqrt{2}}+...+\sqrt{10-\sqrt{99}}}=? $
Let $(F_n)_{n\geq 1} $ be the Fibonacci sequence $F_1 = F_2 = 1, F_{n+2} = F_{n+1} + F_n (n \geq 1),$ and $P(x)$ the polynomial of degree $990$ satisfying
\[ P(k) = F_k, \qquad \text{ for } k = 992, . . . , 1982.\]
Prove that $P(1983) = F_{1983} - 1.$
Call a triple $(a, b, c)$ of positive integers a [b]nice[/b] triple if $a, b, c$ forms a non-decreasing arithmetic progression, $gcd(b, a) = gcd(b, c) = 1$ and the product $abc$ is a perfect square. Prove that given a nice triple, there exists some other nice triple having at least one element common with the given triple.
Given $a,x\in\mathbb{R}$ and $x\geq 0$,$a\geq 0$ . Also $\sin(\sqrt{x+a})=\sin(\sqrt{x})$ . What can you say about $a$??? Justify your answer.
The continuous function $f(x)$ satisfies $c^2f(x + y) = f(x)f(y)$ for all real numbers $x$ and $y,$ where $c > 0$ is a constant. If $f(1) = c$, find $f(x)$ (with proof).
Show that $n!=a^{n-1}+b^{n-1}+c^{n-1}$ has only finitely many solutions in positive integers.
[i]Proposed by Dorlir Ahmeti, Albania[/i]
What number should we put in place of the question mark such that the following statement becomes true?
$$11001_? = 54001_{10}$$
A number written in the subscript means which base the number is in.
Suppose that the set $\{1,2,\cdots, 1998\}$ has been partitioned into disjoint pairs $\{a_i,b_i\}$ ($1\leq i\leq 999$) so that for all $i$, $|a_i-b_i|$ equals $1$ or $6$. Prove that the sum \[ |a_1-b_1|+|a_2-b_2|+\cdots +|a_{999}-b_{999}| \] ends in the digit $9$.
Consider an infinite sequence of positive integers in which each positive integer occurs exactly once. Let $\{a_n\}, n\ge 1$ be such a sequence. We call it [i]consistent [/i] if, for an arbitrary natural $k$ and every natural $n ,m$ such that $a_n <a_m$, the inequality $a_{kn} <a _{km}$ also holds. For example, the sequence $a_n = n$ is consistent .
a) Prove that there are consistent sequences other than $a_n = n$.
b) Are there consistent sequences for which $a_n \ne n, n\ge 2$ ?
c) Are there consistent sequences for which $a n \ne n, n\ge 1$ ?
Let $m, n, k$ be positive integers with $m \ge n$ and $1 + 2 + ... + n = mk$. Prove that the numbers $1, 2, ... , n$ can be divided into $k$ groups in such a way that the sum of the numbers in each group equals $m$.
Is it possible to build a $ 4\times 4\times4$ cube from blocks of the following shape consisting of $ 4$ unit cubes?
If $a, b, c, d$ are the solutions of the equation $x^4-kx-15=0$, find the equation whose solutions are $\frac{a+b+c}{d^2}, \frac{a+b+d}{c^2}, \frac{a+c+d}{b^2}, \frac{b+c+d}{a^2}$.
Given a set $S_n = \{1, 2, 3, \ldots, n\}$, we define a [i]preference list[/i] to be an ordered subset of $S_n$. Let $P_n$ be the number of preference lists of $S_n$. Show that for positive integers $n > m$, $P_n - P_m$ is divisible by $n - m$.
[i]Note: the empty set and $S_n$ are subsets of $S_n$.[/i]
Let $NS$ and $EW$ be two perpendicular diameters of a circle $\mathcal{C}$. A line $\ell$ touches $\mathcal{C}$ at point $S$. Let $A$ and $B$ be two points on $\mathcal{C}$, symmetric with respect to the diameter $EW$. Denote the intersection points of $\ell$ with the lines $NA$ and $NB$ by $A'$ and $B'$, respectively. Show that $|SA'|\cdot |SB'|=|SN|^2$.
Find the highest point (largest possible $y$-coordinate) on the parabola \[y=-2x^2+28x+418.\]
In a word of more than $10$ letters, any two consecutive letters are different. Prove that one can change places of two consecutive letters so that the resulting word is not [i]periodic[/i], that is, cannot be divided into equal subwords.
Let $ ABC$ be an isosceles triangle with $ AB \equal{} AC$. $ M$ is the midpoint of $ BC$ and $ O$ is the point on the line $ AM$ such that $ OB$ is perpendicular to $ AB$. $ Q$ is an arbitrary point on $ BC$ different from $ B$ and $ C$. $ E$ lies on the line $ AB$ and $ F$ lies on the line $ AC$ such that $ E, Q, F$ are distinct and collinear. Prove that $ OQ$ is perpendicular to $ EF$ if and only if $ QE \equal{} QF$.