Found problems: 85335
Let $a_1, a_2, a_3, a_4, a_5$ be positive integers such that $a_1, a_2, a_3$ and $a_3, a_4, a_5$ are both geometric sequences and $a_1, a_3, a_5$ is an arithmetic sequence. If $a_3 = 1575$, find all possible values of $\vert a_4 - a_2 \vert$.
The number $ \left (2+2^{96} \right )!$ has $2^{93}$ trailing zeroes when expressed in base $B$.
[b]
a)[/b] Find the minimum possible $B$.
[b]b)[/b] Find the maximum possible $B$.
[b]c)[/b] Find the total number of possible $B$.
[i]Proposed by Lewis Chen[/i]
Find all functions $f:\mathbb{Q}\rightarrow\mathbb{Q}$ such that\[f(x)+f(t)=f(y)+f(z)\]for all rational numbers $x<y<z<t$ that form an arithmetic progression. ($\mathbb{Q}$ is the set of all rational numbers.)
Find number of terms when we expand $(a+b+c)^{99}$ (in the general case).
Nick is taking a $10$ question test where each answer is either true or false with equal probability. Nick forgot to study, so he guesses randomly on each of the $10$ problems. What is the probability that Nick answers exactly half of the questions correctly?
Let $\alpha, \beta$ be real numbers such that $\sin\alpha\sin\beta=\frac{1}{3}$. Prove that the set of possible values of $\cos \alpha \cos \beta$ is the interval $\left[-\frac{2}{3}, \frac{2}{3}\right]$.
A strictly increasing sequence $\{x_i\}_{i=1}^{\infty}$ of positive integers is said to be [i]large[/i] if, for every real number $L$, there exists an integer $n$ such that $\frac{1}{x_1} + \frac{1}{x_2} + \cdots + \frac{1}{x_n} > L$. Do there exist large sequences $\{a_i\}_{i=1}^\infty$ and $\{b_i\}_{i=1}^{\infty}$ such that the sequence $\{a_i+b_i\}_{i=1}^{\infty}$ is not large?
[i]Proposed by Lewis Chen[/i]
For every natural number $n$ we define $f(n)$ by the following rule: $f(1) = 1$ and for $n>1$ then $f(n) = 1 + a_1 \cdot p_1 + \ldots + a_k \cdot p_k$, where $n = p_1^{a_1} \cdots p_k^{a_k}$ is the canonical prime factorisation of $n$ ($p_1, \ldots, p_k$ are distinct primes and $a_1, \ldots, a_k$ are positive integers). For every positive integer $s$, let $f_s(n) = f(f(\ldots f(n))\ldots)$, where on the right hand side there are exactly $s$ symbols $f$. Show that for every given natural number $a$, there is a natural number $s_0$ such that for all $s > s_0$, the sum $f_s(a) + f_{s-1}(a)$ does not depend on $s$.
Given that $a$ is a real solution to the polynomial equation $$nx^n-x^{n-1}-x^{n-2}-\cdots-x-1=0$$ where $n$ is a positive integer, show that $a=1$ or $-1<a<0$.
Prove that for an arbitrary pair of vectors $f$ and $g$ in the space the inequality
\[af^2 + bfg +cg^2 \geq 0\]
holds if and only if the following conditions are fulfilled:
\[a \geq 0, \quad c \geq 0, \quad 4ac \geq b^2.\]
The six-digit number $\underline{2}\,\underline{0}\,\underline{2}\,\underline{1}\,\underline{0}\,\underline{A}$ is prime for only one digit $A.$ What is $A?$
$(\textbf{A})\: 1\qquad(\textbf{B}) \: 3\qquad(\textbf{C}) \: 5 \qquad(\textbf{D}) \: 7\qquad(\textbf{E}) \: 9$
When four gallons are added to a tank that is one-third full, the tank is then one-half full. The capacity of the tank in gallons is
$\text{(A)}\ 8 \qquad \text{(B)}\ 12 \qquad \text{(C)}\ 20 \qquad \text{(D)}\ 24 \qquad \text{(E)}\ 48$
Let $S$ be a set consisting of $m$ pairs $(a,b)$ of positive integers with the property that $1 \leq a < b \leq n$. Show that there are at least
\[ 4m \cdot \dfrac{(m - \dfrac{n^2}{4})}{3n} \]
triples $(a,b,c)$ such that $(a,b)$, $(a,c)$, and $(b,c)$ belong to $S$.
Prove that for any positive integer $ n$, there exists only $ n$ degree polynomial $ f(x),$ satisfying $ f(0) \equal{} 1$ and $ (x \plus{} 1)[f(x)]^2 \minus{} 1$ is an odd function.
Find the number of $7$-tuples of positive integers $(a,b,c,d,e,f,g)$ that satisfy the following systems of equations:
\begin{align*}
abc&=70,\\
cde&=71,\\
efg&=72.
\end{align*}
Determine all triples $(m , n , p)$ satisfying :
\[n^{2p}=m^2+n^2+p+1\]
where $m$ and $n$ are integers and $p$ is a prime number.
Let $n$ be a positive integer with $n = p^{2010}q^{2010}$ for two odd primes $p$ and $q$. Show that there exist exactly $\sqrt[2010]{n}$ positive integers $x \le n$ such that $p^{2010}|x^p - 1$ and $q^{2010}|x^q - 1$.
One side of a rectangle has length 18. The area plus the perimeter of the rectangle is 2016. Find the perimeter of the rectangle.
There are $N{}$ points marked on the plane. Any three of them form a triangle, the values of the angles of which in are expressed in natural numbers (in degrees). What is the maximum $N{}$ for which this is possible?
[i]Proposed by E. Bakaev[/i]
How many $2$-digit factors does $555555$ have?
$\boxed{N2}$ Let $p$ be a prime numbers and $x_1,x_2,...,x_n$ be integers.Show that if
\[x_1^n+x_2^n+...+x_p^n\equiv 0 \pmod{p}\]
for all positive integers n then $x_1\equiv x_2 \equiv...\equiv x_p \pmod{p}.$
The coordinate of $ P$ at time $ t$, moving on a plane, is expressed by $ x = f(t) = \cos 2t + t\sin 2t,\ y = g(t) = \sin 2t - t\cos 2t$.
(1) Find the acceleration vector $ \overrightarrow{\alpha}$ of $ P$ at time $ t$ .
(2) Let $ L$ denote the line passing through the point $ P$ for the time $ t%Error. "neqo" is a bad command.
$, which is parallel to the acceleration vector $ \overrightarrow{\alpha}$ at the time. Prove that $ L$ always touches to the unit circle with center the origin, then find the point of tangency $ Q$.
(3) Prove that $ f(t)$ decreases in the interval $ 0\leq t \leqq \frac {\pi}{2}$.
(4) When $ t$ varies in the range $ \frac {\pi}{4}\leq t\leq \frac {\pi}{2}$, find the area $ S$ of the figure formed by moving the line segment $ PQ$.
Consider the space $ \{0,1 \} ^{N} $ with the product topology (where $\{0,1 \}$ is a discrete space). Let $ T: \{0,1 \} ^ {\mathbb {N}} \rightarrow \{0,1 \} ^ {\mathbb {N}} $ be the left-shift, ie $ (Tx) (n) = x (n+1) $ for every $ n \in \mathbb {N} $.
Can a finite number of Borel sets be given: $ B_ {1}, \ldots, B_ {m} \subset \{0,1 \} ^ {N} $ such that
$$
\left \{T ^ {i} \left (B_ {j} \right) \mid i \in \mathbb {N}, 1 \leq j \leq m \right \}
$$the $ \sigma $-algebra generated by the set system coincides with the Borel set system?
we have an isogonal triangle $ABC$ such that $BC=AB$. take a random $P$ on the altitude from $B$ to $AC$.
The circle $(ABP)$ intersects $AC$ second time in $M$. Take $N$ such that it's on the segment $AC$ and $AM=NC$ and $M \neq N$.The second intersection of $NP$ and circle $(APB)$ is $X$ , ($X \neq P$) and the second intersection of $AB$ and circle $(APN)$ is $Y$ ,($Y \neq A$).The tangent from $A$ to the circle $(APN)$ intersects the altitude from $B$ at $Z$.
Prove that $CZ$ is tangent to circle $(PXY)$.
Find the smallest real number $x$ for which exist two non-congruent triangles, whose sides have integer lengths and the numerical value of the area of each triangle is $x$.