Prove: if an infinte arithmetic sequence ($ a_n\equal{}a_0\plus{}nd$) of positive real numbers contains two different powers of an integer $ a>1$, then the sequence contains an infinite geometric sequence ($ b_n\equal{}b_0q^n$) of real numbers.

As evidence that the correct answer does not mean the correctness of the proof, the teacher cited next example. Let's take the fraction $\frac{19}{95}$. After crossing out $9$ in the numerator and denominator (“reduction” by $9$), we get $\frac{1}{5}$ which is the correct answer. In the same way, a fraction $\frac{1999}{9995}$ can be “reduced” by three nines (cross out $999$ in the numerator and denominator). Is it possible that as a result of such a “reduction” we also get the correct answer, equal to $\frac13$ ? (We consider fractions of the form $\frac{1a}{a3}$. Here, with the letter $a$ we denote several numbers that follow in the same order in the numerator after $1$, and in the denominator before $3$. “Reduce” by $a$.)

Let $\omega$ be a primitive $7$th root of unity. Find $$\prod_{k=0}^6\left(1+\omega^k-\omega^{2k}\right).$$ (A complex number is a primitive root of unity if and only if it can be written in the form $e^{2k\pi i/n}$, where $k$ is relatively prime to $n$.)

Let $n\geq 2$ be an integer. Consider the solutions of the system $$\begin{cases} n=a+b-c \\ n=a^2+b^2-c^2 \end{cases}$$ where $a,b,c$ are integers. Show that there is at least one solution and that the solutions are finitely many.

Define $a_n$: the coefficient of then item $x$ in $(3-\sqrt{x})^n$, where $n$ is a positive integer. Then $\lim_{n\to\infty}\left(\frac{3^2}{a_2}+\frac{3^3}{a_3}+\cdots+\frac{3^n}{a_n}\right)=$________.

suppose that $G<S_n$ is a subgroup of permutations of $\{1,...,n\}$ with this property that for every $e\neq g\in G$ there exist exactly one $k\in \{1,...,n\}$ such that $g.k=k$. prove that there exist one $k\in \{1,...,n\}$ such that for every $g\in G$ we have $g.k=k$.(20 points)

Let $f(x) = ax^2+bx+c$ be a quadratic trinomial with $a$,$b$,$c$ reals such that any quadratic trinomial obtained by a permutation of $f$'s coefficients has an integer root (including $f$ itself). Show that $f(1)=0$.

None of the positive integers $k,m,n$ are divisible by $5$. Prove that at least one of the numbers $k^2-m^2,m^2-n^2,n^2-k^2$ is divisible by $5$.

The equation \[2^{333x-2}+2^{111x+2}=2^{222x+1}+1\] has three real roots. Given that their sum is $m/n$ where $m$ and $n$ are relatively prime positive integers, find $m+n$.

Let $ P(x,y)$ be a polynomial in two variables $ x,y$ such that $ P(x,y)\equal{}P(y,x)$ for every $ x,y$ (for example, the polynomial $ x^2\minus{}2xy\plus{}y^2$ satisfies this condition). Given that $ (x\minus{}y)$ is a factor of $ P(x,y)$, show that $ (x\minus{}y)^2$ is a factor of $ P(x,y)$.

The sequence of integers $a_1,a_2,a_3 ,.. $such that $a_1 = 1$, $a_2 = 2$ and for every natural $n \ge 1$ $$a_{n+2}=\begin{cases} 2001a_{n+1} - 1999a_n , \text{\,\,if\,\,the\,\,product\,\,} a_{n+1}a_n \text{is\,\,an\,\,even\,\,number} /\\ a_{n+1}-a_n , \text{\,\,if\,\,the\,\,product\,\,} a_{n+1}a_n \text{is\,\,an\,\,odd\,\,number} \end{cases}$$ Is there such a natural $m$ that $a_m= 2000$?

Evaluate the sum $$\sum_{n=0}^{\infty}\frac{5n+7}{6^n}$$

Given $m\in\mathbb{N}$. Find all functions $f:\mathbb{R^{+}}\rightarrow\mathbb{R^{+}}$ such that $$f(f(x)+y)-f(x)=\left( \frac{f(y)}{y}-1\right)x+f^m(y)$$ holds for all $x,y\in\mathbb{R^{+}}.$ ($f^m(x) =$ $f$ applies $m$ times.)

Peter wants to create a new multiplication table for the four numbers $1, 2, 3, 4$ in such a way that the product of two of them is also one of them. He wants also that $(a\cdot b)\cdot c = a\cdot (b\cdot c)$ holds and that $ab \ne ac$ and $ba \ne ca$ and $b \ne c$. Peter is successful in constructing the new table. In his new table, $1\cdot 3 = 2$ and $2\cdot 2 = 4$. What is the product $3\cdot 1$ according to Peter's table?

Let $f(x) = ax^7+bx^3+cx-5$, where $a,b$ and $c$ are constants. If $f(-7) = 7$, the $f(7)$ equals $\textbf {(A) } -17 \qquad \textbf {(B) } -7 \qquad \textbf {(C) } 14 \qquad \textbf {(D) } 21\qquad \textbf {(E) } \text{not uniquely determined}$

The function $f : (0,1) \to R$ is defined by $f(x) = x$ if $x$ is irrational, $f(x) = \frac{p+1}{q}$ if $x =\frac{p}{q}$ , where $(p,q) = 1$. Find the maximum value of $f$ on the interval $(7/8,8/9)$.

Consider a $ 7\times 7$ numbers table $ a_{ij} \equal{} (i^2 \plus{} j)(i \plus{} j^2), 1\le i,j\le 7.$ When we add arbitrarily each term of an arithmetical progression consisting of $ 7$ integers to corresponding to term of certain row (or column) in turn, call it an operation. Determine whether such that each row of numbers table is an arithmetical progression, after a finite number of operations.

Prove that $\cos{\frac{\pi}{7}}-\cos{\frac{2\pi}{7}}+\cos{\frac{3\pi}{7}}=\frac{1}{2}$

How many base ten integers of the form 1010101...101 are prime?

A twelve foot tree casts a five foot shadow. How long is Henry’s shadow (at the same time of day) if he is five and a half feet tall?

For the sequence of real numbers $a_1,a_2,\dots ,a_k$ we say it is [i]invested[/i] on the interval $[b,c]$ if there exists numbers $x_0,x_1,\dots ,x_k$ in the interval $[b,c]$ such that $|x_i-x_{i-1}|=a_i$ for $i=1,2,3,\dots k$ . A sequence is [i]normed[/i] if all its members are not greater than $1$ . For a given natural $n$ , prove : a)Every [i]normed[/i] sequence of length $2n+1$ is [i]invested[/i] in the interval $\left[ 0, 2-\frac{1}{2^n} \right ]$. b) there exists [i]normed[/i] sequence of length $4n+3$ wich is not [i]invested[/i] on $\left[ 0, 2-\frac{1}{2^n} \right ]$.

A positive real number sequence $a_1, a_2, a_3,\dots $ and a positive integer \(s\) is given. Let $f_n(0) = \frac{a_n+\dots+a_1}{n}$ and for each $0<k<n$ \[f_n(k)=\frac{a_n+\dots+a_{k+1}}{n-k}-\frac{a_k+\dots+a_1}{k}\] Then for every integer $n\geq s,$ the condition \[a_{n+1}=\max_{0\leq k<n}(f_n(k))\] is satisfied. Prove that this sequence must be eventually constant.

For each pair of integers \( j, k \geq 2 \), define the function \( f_{jk} : \mathbb{R} \to \mathbb{R} \) given by \[ f_{jk}(x) = 1 - (1 - x^j)^k. \] (a) Prove that for any integers \( j, k \geq 2 \), there exists a unique real number \( p_{jk} \in (0, 1) \) such that \( f_{jk}(p_{jk}) = p_{jk} \). Furthermore, defining \( \lambda_{jk} := f'_{jk}(p_{jk}) \), prove that \( \lambda_{jk} > 1 \). (b) Prove that \( p^j_{jk} = 1 - p_{kj} \) for any integers \( j, k \geq 2 \). (c) Prove that \( \lambda_{jk} = \lambda_{kj} \) for any integers \( j, k \geq 2 \).

If $ a_2 \neq 0$ and $ r$ and $ s$ are the roots of $ a_0 \plus{} a_1x \plus{} a_2x^2 \equal{} 0$, then the equality $ a_0 \plus{} a_1x \plus{} a_2x^2 \equal{} a_0\left (1 \minus{} \frac {x}{r} \right ) \left (1 \minus{} \frac {x}{s} \right )$ holds: $ \textbf{(A)}\ \text{for all values of }x, a_0\neq 0$ $ \textbf{(B)}\ \text{for all values of }x$ $ \textbf{(C)}\ \text{only when }x \equal{} 0$ $ \textbf{(D)}\ \text{only when }x \equal{} r \text{ or }x \equal{} s$ $ \textbf{(E)}\ \text{only when }x \equal{} r \text{ or }x \equal{} s, a_0 \neq 0$

Let $ P(x) \equal{} x^3 \plus{} mx \plus{} n$ be an integer polynomial satisfying that if $ P(x) \minus{} P(y)$ is divisible by 107, then $ x \minus{} y$ is divisible by 107 as well, where $ x$ and $ y$ are integers. Prove that 107 divides $ m$.