Found problems: 85335
Let $\Omega(n)$ be the number of prime factors of $n$. Define $f(1)=1$ and $f(n)=(-1)^{\Omega(n)}.$ Furthermore, let
$$F(n)=\sum_{d|n} f(d).$$
Prove that $F(n)=0,1$ for all positive integers $n$. For which integers $n$ is $F(n)=1?$
Inside the equilateral triangle $ABC$ lies the point $T$. Prove that $TA$, $TB$ and $TC$ are the lengths of the sides of a triangle.
There are $n$ circles drawn on a piece of paper in such a way that any two circles intersect in two points, and no three circles pass through the same point. Turbo the snail slides along the circles in the following fashion. Initially he moves on one of the circles in clockwise direction. Turbo always keeps sliding along the current circle until he reaches an intersection with another circle. Then he continues his journey on this new circle and also changes the direction of moving, i.e. from clockwise to anticlockwise or $\textit{vice versa}$.
Suppose that Turbo’s path entirely covers all circles. Prove that $n$ must be odd.
[i]Proposed by Tejaswi Navilarekallu, India[/i]
For any positive integer $n$, define $f(n)$ to be the smallest positive integer that does not divide $n$. For example, $f(1)=2$, $f(6)=4$. Prove that for any positive integer $n$, either $f(f(n))$ or $f(f(f(n)))$ must be equal to $2$.
Let $ABCD$ be a tetrahedron and suppose that $M$ is a point inside it such that $\angle MAD=\angle MBC$ and $\angle MDB=\angle MCA$. Prove that $$MA\cdot MB+MC\cdot MD<\max(AD\cdot BC,AC\cdot BD).$$
For positive real numbers $ x,y,z$ the inequality
\[\frac1{x^2\plus{}1}\plus{}\frac1{y^2\plus{}1}\plus{}\frac1{z^2\plus{}1}\equal{}\frac12\]
holds. Prove the inequality
\[\frac1{x^3\plus{}2}\plus{}\frac1{y^3\plus{}2}\plus{}\frac1{z^3\plus{}2}<\frac13.\]
The given parabola $y=ax^2+bx+c$ doesn't intersect the $X$-axis and passes from the points $A(-2,1)$ and $B(2,9)$. Find all the possible values of the $x$ coordinates of the vertex of this parabola.
Let $p$ be a prime number and let further $p + 1$ rational numbers $a_0,...,a_p$ with the following property given: If one removes any of the $p + 1$ numbers, then the remaining may be split in at least two groups , which all have the same mean value (for different distant numbers, however, these mean values may be different). Prove that all $p + 1$ numbers are equal.
Let $E \subset [0,1]$ be a Lebesgue measurable set having Lebesgue measure $| E |<\frac{1}{2}$. Let
$$h (s) = \int _ {\overline {E}} \frac{dt}{{(s-t)}^2}$$
where $\overline {E} = [0,1] \backslash E$. Prove that there is one $t \in \overline {E}$ for which
$$\int_E \frac {ds} {h (s) {(s-t)} ^ 2} \leq c {| E |} ^ 2$$
with some absolute constant c .
$O$ is the circumcenter of $ABC$, and $H$ is the orthocenter of $ABC$. If $D$ is a midpoint of $AC$ and $E$ is the intersection of $BO$ and $ABC$'s circumcircle not $B$, show that three points $H, D, E$ are collinear.
Consider a board having 2 rows and $n$ columns. Thus there are $2n$ cells in the board. Each cell is to be filled in by $0$ or $1$ .
[list=a]
[*] In how many ways can this be done such that each row sum and each column sum is even?
[*] In how many ways can this be done such that each row sum and each column sum is odd?
[/list]
Let $T=\{a_1,a_2,\dots,a_{1000}\}$, where $a_1<a_2<\dots<a_{1000}$, be a uniformly randomly selected subset of $\{1,2,\dots,2018\}$ with cardinality $1000$. The expected value of $a_7$ can be written in reduced form as $\tfrac{m}{n}$. Find $m+n$.
Determine all the functions $f : \mathbb{R}\rightarrow\mathbb{R}$ that satisfies the following.
$f(xf(x)+f(x)f(y)+y-1)=f(xf(x)+xy)+y-1$
The circumference of a circle is divided into eight arcs by a convex quadrilateral $ABCD$ with four arcs lying inside the quadrilateral and the remaining four lying outside it. The lengths of the arcs lying inside the quadrilateral are denoted by $p,q,r,s$ in counter-clockwise direction. Suppose $p+r = q+s$. Prove that $ABCD$ is cyclic.
Let $ABC$ to be a triangle and $\Gamma$ its circumcircle. Also, let $D, F, G$ and $E$, in this order, on the arc $BC$ which does not contain $A$ satisfying $\angle BAD = \angle CAE$ and $\angle BAF = \angle CAG$. Let $D`, F`, G`$ and $E`$ to be the intersections of $AD, AF, AG$ and $AE$ with $BC$, respectively. Moreover, $X$ is the intersection of $DF`$ with $EG`$, $Y$ is the intersection of $D`F$ with $E`G$, $Z$ is the intersection of $D`G$ with $E`F$ and $W$ is the intersection of $EF`$ with $DG`$.
Prove that $X, Y$ and $A$ are collinear, such as $W, Z$ and $A$. Moreover, prove that $\angle BAX = \angle CAZ$.
Let $(a_n)^{+\infty}_{n=1}$ be a sequence defined recursively as follows: $a_1=1$ and $$a_{n+1}=1 + \sum\limits_{k=1}^{n}ka_k$$
For every $n > 1$, prove that $\sqrt[n]{a_n} < \frac {n+1}{2}$.
Find the smallest $a$ for which the equation $x^2-ax +21 = 0$ has a root that is a natural number.
The exam has $25$ topics, each of which has $8$ questions. On a test, there are $4$ questions of different topics.
Is it possible to make $50$ tests so that each question was asked exactly once, and for any two topics there is a test where are questions of both topics?
Let $ABC$ and $DEF$ be two triangles such that $A+ D = 120^{\circ}$ and $B+E = 120^{\circ}$. Suppose they have the same circumradius. Prove that they have the same 'Fermat length'.
In $ \triangle ABC$, $ \angle ABC \equal{} 45^\circ$. Point $ D$ is on $ \overline{BC}$ so that $ 2 \cdot BD \equal{} CD$ and $ \angle DAB \equal{} 15^\circ$. Find $ \angle ACB$.
[asy]
pair A, B, C, D;
A = origin;
real Bcoord = 3*sqrt(2) + sqrt(6);
B = Bcoord/2*dir(180);
C = sqrt(6)*dir(120);
draw(A--B--C--cycle);
D = (C-B)/2.4 + B;
draw(A--D);
label("$A$", A, dir(0));
label("$B$", B, dir(180));
label("$C$", C, dir(110));
label("$D$", D, dir(130));
[/asy]
$ \textbf{(A)} \ 54^\circ \qquad \textbf{(B)} \ 60^\circ \qquad \textbf{(C)} \ 72^\circ \qquad \textbf{(D)} \ 75^\circ \qquad \textbf{(E)} \ 90^\circ$
We define the binary operation $\times$ on elements of $\mathbb{Z}^2$ as \[(a,b)\times(c,d)=(ac+bd,ad+bc)\] for all integers $a,b,c,$ and $d$. Compute the number of ordered six-tuples $(a_1,a_2,a_3,a_4,a_5,a_6)$ of integers such that \[[[[[(1,a_1)\times (2,a_2)]\times (3,a_3)]\times (4,a_4)]\times (5,a_5)]\times (6,a_6)=(350,280).\]
[i]Proposed by Michael Ren and James Lin[/i]
There are $1979$ equilateral triangles: $T_1,T_2, . . . ,T_{1979}$. A side of triangle $T_k$ is equal to $\frac{1}{k}$, $k = 1,2, . . . ,1979$. At what values of a number $a$ can one place all these triangles into the equilateral triangle with side length $a$ so that they don’t intersect (points of contact are allowed)?
For $x, y$ positive integers, $x^2-4y+1$ is a multiple of $(x-2y)(1-2y)$. Prove that $|x-2y|$ is a square number.
Find all positive integers $k$ such that there exist two positive integers $m$ and $n$ satisfying
\[m(m + k) = n(n + 1).\]
Which of the following has the greatest number of line of symmetry?
$ \textbf{(A)}\ \text{ Equilateral Triangle}$
$\textbf{(B)}\ \text{Non-square rhombus} $
$\textbf{(C)}\ \text{Non-square rectangle}$
$\textbf{(D)}\ \text{Isosceles Triangle}$
$\textbf{(E)}\ \text{Square} $