Found problems: 230
Let $ABC$ be a triangle with all angles $\leq 120^{\circ}$. Let $F$ be the Fermat point of triangle $ABC$, that is, the interior point of $ABC$ such that $\angle AFB = \angle BFC = \angle CFA = 120^\circ$. For each one of the three triangles $BFC$, $CFA$ and $AFB$, draw its Euler line - that is, the line connecting its circumcenter and its centroid.
Prove that these three Euler lines pass through one common point.
[i]Remark.[/i] The Fermat point $F$ is also known as the [b]first Fermat point[/b] or the [b]first Toricelli point[/b] of triangle $ABC$.
[i]Floor van Lamoen[/i]
Two coprime positive integers $ a, b $ are given. Integer sequence $ \{ a_n \}, \{b_n \} $ satisties
\[ (a+b \sqrt2 )^{2n} = a_n + b_n \sqrt2 \]
Find all prime numbers $ p $ such that there exist positive integer $ n \le p $ satisfying $ p | b_n $.
Let $P$ be a given point inside the triangle $ABC$. Suppose $L,M,N$ are the midpoints of $BC, CA, AB$ respectively and \[PL: PM: PN= BC: CA: AB.\] The extensions of $AP, BP, CP$ meet the circumcircle of $ABC$ at $D,E,F$ respectively. Prove that the circumcentres of $APF, APE, BPF, BPD, CPD, CPE$ are concyclic.
Let $ k$ be a positive real number. Triangle ABC is acute and not isosceles, O is its circumcenter and AD,BE,CF are the internal bisectors. On the rays AD,BE,CF, respectively, let points L,M,N such that $ \frac {AL}{AD} \equal{} \frac {BM}{BE} \equal{} \frac {CN}{CF} \equal{} k$. Denote $ (O_1),(O_2),(O_3)$ be respectively the circle through L and touches OA at A, the circle through M and touches OB at B, the circle through N and touches OC at C.
1) Prove that when $ k \equal{} \frac{1}{2}$, three circles $ (O_1),(O_2),(O_3)$ have exactly two common points, the centroid G of triangle ABC lies on that common chord of these circles.
2) Find all values of k such that three circles $ (O_1),(O_2),(O_3)$ have exactly two common points
The nine-point Euler circle of triangle $ABC$ is tangent to the excircles in the angle $A,B,C$ at $Fa, Fb, Fc$ respectively. Prove that $AF_a$ bisects the angle $\angle CAB$ if and only if $AFa$ bisects the angle $\angle F_bAF_c$.
Đỗ Thanh Sơn
Let $ABC$ be an acute-angled triangle with $AB<AC$ and let $O$ be the centre of its circumcircle $\omega$. Let $D$ be a point on the line segment $BC$ such that $\angle BAD = \angle CAO$. Let $E$ be the second point of intersection of $\omega$ and the line $AD$. If $M$, $N$ and $P$ are the midpoints of the line segments $BE$, $OD$ and $AC$, respectively, show that the points $M$, $N$ and $P$ are collinear.
A student took a rectangular piece of paper with length equal to one meter and width equal to five centimeters. The student brought the ends together, turning one end 180 degrees and gluing the surfaces to create a figure called a Möbius strip. On one side of this strip, the student placed a flea and an ant. It is known that if the flea and the ant move in different directions on the Möbius strip, they will meet each other in 2 minutes. However, if they move in the same direction, they will meet in 7 minutes. Given that the flea is faster than the ant and both move at constant speeds, determine the speed of the flea.
[i]Proposed by Lia Chitishvili, Georgia[/i]
Let $n$ be largest number such that \[ \frac{2014^{100!}-2011^{100!}}{3^n} \] is still an integer. Compute the remainder when $3^n$ is divided by $1000$.
Maybe well known:
$p$ a prime number, $n$ an integer. Prove that $n$ divides $\phi(p^n-1)$ where $\phi(x)$ is the Euler function.
Show that there exists a positive integer $ k$ such that $ k \cdot 2^{n} \plus{} 1$ is composite for all $ n \in \mathbb{N}_{0}$.
Find the maximum value of \[ \sin{2\alpha} + \sin{2\beta} + \sin{2\gamma} \] where $\alpha,\beta$ and $\gamma$ are positive and $\alpha + \beta + \gamma = 180^{\circ}$.
Consider a 3 × 4 rectangular table where each cell can be colored using one of three available colors. Determine the number of different ways the table can be colored such that no two cells sharing a common side have the same color. It is not necessary to use all three colors in each coloring.
[i]Proposed by Prudencio Guerrero Fernández, Cuba[/i]
Let $ABC$ be an acute angled triangle satisfying the conditions $AB>BC$ and $AC>BC$. Denote by $O$ and $H$ the circumcentre and orthocentre, respectively, of the triangle $ABC.$ Suppose that the circumcircle of the triangle $AHC$ intersects the line $AB$ at $M$ different from $A$, and the circumcircle of the triangle $AHB$ intersects the line $AC$ at $N$ different from $A.$ Prove that the circumcentre of the triangle $MNH$ lies on the line $OH$.
In triangle $ABC$, $I$ is the incenter. We have chosen points $P,Q,R$ on segments $IA,IB,IC$ respectively such that $IP\cdot IA=IQ \cdot IB=IR\cdot IC$.
Prove that the points $I$ and $O$ belong to Euler line of triangle $PQR$ where $O$ is circumcenter of $ABC$.
Two triangles $ABC, A’B’C’$ have the same orthocenter $H$ and the same circumcircle with center $O$. Letting $PQR$ be the triangle formed by $AA’, BB’, CC’$, prove that the circumcenter of $PQR$ lies on $OH$.
$m(m>1)$ is an intenger, define $(a_n)$:
$a_0=m,a_{n}=\varphi(a_{n-1})$ for all positive intenger $n$.
If for all nonnegative intenger $k$, $a_{k+1}\mid a_k$, find all $m$ that is not larger than $2016$.
Note: $\varphi(n)$ means Euler Function.
For how many integers $1\leq n\leq 9999$ is there a solution to the congruence \[\phi(n)\equiv 2\,\,\,\pmod{12},\] where $\phi(n)$ is the Euler phi-function?
Let in-circle of $ABC$ touch $AB$, $BC$, $AC$ in $C_1$, $A_1$, $B_1$ respectively.
Let $H$- intersection point of altitudes in $A_1B_1C_1$, $I$ and $O$-be in-center and circumcenter of $ABC$ respectively.
Prove, that $I, O, H$ lies on one line.
Let $n$ be an odd positive integer such that both $\phi(n)$ and $\phi (n+1)$ are powers of two. Prove $n+1$ is power of two or $n=5$.
Consider a polyhedron with at least five faces such that exactly three edges emerge from each of its vertices. Two players play the following game: Each, in turn, signs his or her name on a previously unsigned face. The winner is the player who first succeeds in signing three faces that share a common vertex. Show that the player who signs first will always win by playing as well as possible.
Let $\varphi$ denote the Euler phi-function. Prove that for every positive integer $n$
$$2^{n(n+1)} | 32 \cdot \varphi \left( 2^{2^n} - 1 \right).$$
Let $n$ be a positive integer. The Georgian folk dance team consists of $2n$ dancers, with $n$ males and $n$ females. Each dancer, both male and female, is assigned a number from 1 to $n$. During one of their dances, all the dancers line up in a single line. Their wish is that, for every integer $k$ from 1 to $n$, there are exactly $k$ dancers positioned between the $k$th numbered male and the $k$th numbered female. Prove the following statements:
a) If $n \equiv 1 \text{ or } 2 \mod{4}$, then the dancers cannot fulfill their wish.
b) If $n \equiv 0 \text{ or } 3 \mod{4}$, then the dancers can fulfill their wish.
[i]Proposed by Giorgi Arabidze, Georgia[/i]
Suppose that for some $m,n\in\mathbb{N}$ we have $\varphi (5^m-1)=5^n-1$, where $\varphi$ denotes the Euler function. Show that $(m,n)>1$.
For a finite set $X$ define \[
S(X) = \sum_{x \in X} x \text{ and }
P(x) = \prod_{x \in X} x.
\] Let $A$ and $B$ be two finite sets of positive integers such that $\left\lvert A \right\rvert = \left\lvert B \right\rvert$, $P(A) = P(B)$ and $S(A) \neq S(B)$. Suppose for any $n \in A \cup B$ and prime $p$ dividing $n$, we have $p^{36} \mid n$ and $p^{37} \nmid n$. Prove that \[ \left\lvert S(A) - S(B) \right\rvert > 1.9 \cdot 10^{6}. \][i]Proposed by Evan Chen[/i]
Let $ABC$ be a triangle, $G$ it`s centroid, $H$ it`s orthocenter, and $M$ the midpoint of the arc $\widehat{AC}$ (not containing $B$). It is known that $MG=R$, where $R$ is the radius of the circumcircle. Prove that $BG\geq BH$.
[i]Proposed by F. Bakharev[/i]