Found problems: 85335
Every brick has $5$ holes in a line. The holes can be filled with bolts (fi tting in one hole) and braces (fi tting into two neighboring holes). No hole may remain free.
One puts $n$ of these bricks in a line to form a pattern from left to right. In this line no two braces and no three bolts may be adjacent.
How many diff erent such patterns can be produced with $n$ bricks?
Find all functions $f:\mathbb{R} \to \mathbb{R}$ so that
\[f(f(x)+x+y) = f(x+y) + y f(y)\]
for all real numbers $x, y$.
Let $p(k)$ be the smallest prime not dividing $k$. Put $q(k) = 1$ if $p(k) = 2$, or the product of all primes $< p(k)$ if $p(k) > 2$. Define the sequence $x_0, x_1, x_2, ...$ by $x_0 = 1$, $x_{n+1} = \frac{x_np(x_n)}{q(x_n)}$. Find all $n$ such that $x_n = 111111$
A triangle $ ABC$ can be divided into two isosceles triangles by a line through $ A$. Given that one of the angles of the triangles is $ 30^{\circ}$, find all possible values of the other two angles.
Find all functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy the following relationship for all $x,y\in\mathbb{R}:$\[f(f(y-x)-xf(y))+f(x)=y\cdot(1-f(x)).\]
Let $m, n > 1$ be coprime odd integers. Show that
$$\big \lfloor \frac{m^{\phi (n)+1} + n^{\phi (m)+1}}{mn} \rfloor$$
is an even integer, where $\phi$ is Euler’s totient function.
Find all pairs of real numbers $(x, y)$, that satisfy the system of equations: $$\left\{\begin{matrix} 6(1-x)^2=\dfrac{1}{y} \\ \\6(1-y)^2=\dfrac{1}{x}.\end{matrix}\right.$$
Let $P(x)$ be a polynomial with real coefficients. Prove that there exist positive integers $n$ and $k$ such that $k$ has $n$ digits and more than $P(n)$ positive divisors.
Let $f(x)$ be a polynomial with positive integer coefficients. For every $n\in\mathbb{N}$, let $a_{1}^{(n)}, a_{2}^{(n)}, \dots , a_{n}^{(n)}$ be fixed positive integers that give pairwise different residues modulo $n$ and let
\[g(n) = \sum\limits_{i=1}^{n} f(a_{i}^{(n)}) = f(a_{1}^{(n)}) + f(a_{2}^{(n)}) + \dots + f(a_{n}^{(n)})\]
Prove that there exists a constant $M$ such that for all integers $m>M$ we have $\gcd(m, g(m))>2023^{2023}$.
Maya lists all the positive divisors of $ 2010^2$. She then randomly selects two distinct divisors from this list. Let $ p$ be the probability that exactly one of the selected divisors is a perfect square. The probability $ p$ can be expressed in the form $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m \plus{} n$.
Let $S_1, S_2,S_3$ be spheres of radii $a, b, c$ respectively whose centers lie on a line $l$. Sphere $S_2$ is externally tangent to $S_1$ and $S_3$, whereas $S_1$ and $S_3$ have no common points. A straight line t touches each of the spheres,
Find the sine of the angle between $l$ and $t$
If three of the roots of $x^4+ax^2+bx+c=0$ are $1$, $2$, and $3$, then the value of $a+c$ is:
$\text{(A)}\ 35 \qquad
\text{(B)}\ 24\qquad
\text{(C)}\ -12\qquad
\text{(D)}\ -61 \qquad
\text{(E)}\ -63$
Prove that, for $k \in {\mathbb Z^+}$
$$k(k+1)(k+2)(k+3)$$
is not a perfect square.
Prove that for $\forall n>1$, $n\in \mathbb{N}$ , there exist infinitely many pairs of positive irrational numbers $a$ and $b$, such that $a^n=b$.
Given a regular tetrahedron $ABCD$ with edge length $1$ and a point $P$ inside it.
What is the maximum value of $\left|PA\right|+\left|PB\right|+\left|PC\right|+\left|PD\right|$.
A function $f(x)$ is such that for any integer $x$, $f(x)+xf(2-x)=6$. Compute $-2019f(2020)$.
Let $k>1$ be a fixed positive integer. Prove that if $n$ is a sufficiently large positive integer, there exists a sequence of integers with the following properties:
[list=disc]
[*]Each element of the sequence is between $1$ and $n$, inclusive.
[*]For any two different contiguous subsequence of the sequence with length between $2$ and $k$ inclusive, the multisets of values in those two subsequences is not the same.
[*]The sequence has length at least $0.499n^2$
[/list]
[b] Problem 6.[/b] Let $m\geq 5$ and $n$ are given natural numbers, and $M$ is regular $2n+1$-gon. Find the number of the convex $m$-gons with vertices among the vertices of $M$, who have at least one acute angle.
[i]Alexandar Ivanov[/i]
Victor writes down all $7-$digit numbers using the digits $1, 2, 3, 4, 5, 6,$ and $7$ exactly once. Prove that there are no two numbers among them where one is a multiple of the other.
Suppose $x$ and $y$ are real numbers such that \[x^2+xy+y^2=2\qquad\text{and}\qquad x^2-y^2=\sqrt5.\] The sum of all possible distinct values of $|x|$ can be written in the form $\textstyle\sum_{i=1}^n\sqrt{a_i}$, where each of the $a_i$ is a rational number. If $\textstyle\sum_{i=1}^na_i=\frac mn$ where $m$ and $n$ are positive realtively prime integers, what is $100m+n$?
[i] Proposed by David Altizio [/i]
Find the smallest positive integer $n$ with the following property:
For any integer $m$ with $0 < m < 2004$, there exists an integer $k$ such that
\[\frac{m}{2004}<\frac{k}{n}<\frac{m+1}{2005}.\]
A triangular piece of paper of area $1$ is folded along a line parallel to one of the sides and pressed flat. What is the minimum possible area of the resulting figure?
Students from three middle schools worked on a summer project.
*Seven students from Allen school worked for $3$ days.
*Four students from Balboa school worked for $5$ days.
*Five students from Carver school worked for $9$ days.
The total amount paid for the students' work was $ \$774$. Assuming each student received the same amount for a day's work, how much did the students from Balboa school earn altogether?
$\text{(A)}\ 9.00\text{ dollars} \qquad \text{(B)}\ 48.38\text{ dollars} \qquad \text{(C)}\ 180.00\text{ dollars} \qquad \text{(D)}\ 193.50\text{ dollars} \qquad \text{(E)}\ 258.00\text{ dollars}$
Determine the smallest natural number which can be represented both as the sum of $2002$ positive integers with the same sum of decimal digits, and as the sum of $2003$ integers with the same sum of decimal digits.
In the quadrilateral $ABCD$ is $AD \parallel BC$ and the angles $\angle A$ and $\angle D$ are acute. The diagonals intersect in $P$. The circumscribed circles of $\vartriangle ABP$ and $\vartriangle CDP$ intersect the line $AD$ again at $S$ and $T$ respectively. Call $M$ the midpoint of $[ST]$. Prove that $\vartriangle BCM$ is isosceles.
[img]https://1.bp.blogspot.com/-C5MqC0RTqwY/Xy1fAavi_aI/AAAAAAAAMSM/2MXMlwb13McCYTrOHm1ZzWc0nkaR1J6zQCLcBGAsYHQ/s0/flanders%2B2016%2Bp1.png[/img]