Found problems: 85335
We say that a square-free positive integer $n$ is [i]almost prime[/i] if
\[n \mid x^{d_1}+x^{d_2}+\dots+x^{d_k}-kx\]
for all integers $x$, where $1=d_1<d_2<\dots<d_k=n$ are all the positive divisors of $n$. Suppose that $r$ is a Fermat prime (i.e. it is a prime of the form $2^{2^m}+1$ for an integer $m \ge 0$), $p$ is a prime divisor of an almost prime integer $n$, and $p \equiv 1 \pmod{r}$. Show that, with the above notation, $d_i \equiv 1 \pmod{r}$ for all $1 \le i \le k$.
(An integer $n$ is called [i]square-free[/i] if it is not divisible by $d^2$ for any integer $d>1$.)
Prove the equality
$${n \choose 0}^2+ {n \choose 1}^2+ {n \choose 2}^2+...+{n \choose n}^2={2n \choose n}$$
Peter and Paul gamble as follows. For each natural number, successively, they determine its largest odd divisor and compute its remainder when divided by $4$. If this remainder is $1$, then Peter gives Paul a coin; otherwise, Paul
gives Peter a coin. After some time they stop playing and balance the accounts. Prove that Paul wins.
Let $a,b,c,d$ be positive real numbers such that $\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}+\frac{1}{d+1}=2$. Prove that $$\sum_{cyc} \sqrt{\frac{a^2+1}{2}} \geq (3.\sum_{cyc} \sqrt{a}) -8$$
In triangle $ABC$, $AB=AC$. If there is a point $P$ strictly between $A$ and $B$ such that $AP=PC=CB$, then $\angle A =$
[asy]
draw((0,0)--(8,0)--(4,12)--cycle);
draw((8,0)--(1.6,4.8));
label("A", (4,12), N);
label("B", (0,0), W);
label("C", (8,0), E);
label("P", (1.6,4.8), NW);
dot((0,0));
dot((4,12));
dot((8,0));
dot((1.6,4.8));
[/asy]
$ \textbf{(A)}\ 30^{\circ} \qquad\textbf{(B)}\ 36^{\circ} \qquad\textbf{(C)}\ 48^{\circ} \qquad\textbf{(D)}\ 60^{\circ} \qquad\textbf{(E)}\ 72^{\circ} $
There are $n$ people, any two of them have called each other at most once. In any group of $n-2$ of them, anyone of the group has called with other people in this group for $3^k$ times, where $k$ is a non-negative integer (the value of $k$ is fixed). Find all possible integers $n$.
Let $ABCD$ be a trapezoid with $AD$ parallel to $BC$, $AD = 2$, and $BC = 1$. Let $M$ be the midpoint of $AD$, and let $P$ be the intersection of $BD$ with $CM$. Extend $AP$ to meet segment $CD$ at point $Q$. If the ratio $CQ/QD = a/b$, where $a$ and $b$ are positive integers and $\text{gcd}(a, b) = 1$, find $a + b$.
Among the $n$ inhabitants of an island, every two are either friends or enemies. Some day, the chief of the island orders that each inhabitant (including himself) makes and wears a necklace consisting of marbles, in such a way that the necklaces of two friends have at least one marble of the same type and that the necklaces of two enemies differ at all marbles. (A necklace may also be marbleless). Show that the chief’s order can be achieved by using $\left\lfloor\frac{n^2}4\right\rfloor$ different types of stones, but not necessarily by using fewer types.
Triangle $ABC$ is inscribed in circle $\Omega$. The interior angle bisector of angle $A$ intersects side $BC$ and $\Omega$ at $D$ and $L$ (other than $A$), respectively. Let $M$ be the midpoint of side $BC$. The circumcircle of triangle $ADM$ intersects sides $AB$ and $AC$ again at $Q$ and $P$ (other than $A$), respectively. Let $N$ be the midpoint of segment $PQ$, and let $H$ be the foot of the perpendicular from $L$ to line $ND$. Prove that line $ML$ is tangent to the circumcircle of triangle $HMN$.
When a positive integer $N$ is divided by $60$, the remainder is $49$. When $N$ is divided by $15$, the remainder is
$ \mathrm{(A) \ } 0 \qquad \mathrm{(B) \ } 3 \qquad \mathrm {(C) \ } 4 \qquad \mathrm{(D) \ } 5 \qquad \mathrm{(E) \ } 8$
In a perpendicular triangle the perimeter is 60 and the altitude on the hypotenuse is 12. Then, the length of the hypotenuse is
$ \text{(A)}\ 24 \qquad \text{(B)}\ 25 \qquad \text{(C)}\ 26 \qquad \text{(D)}\ 27 \qquad \text{(E)}\ 28$
Let $ n $ be a positive integer. Consider words of length $n$ composed of letters from the set $ \{ M, E, O \} $. Let $ a $ be the number of such words containing an even number (possibly 0) of blocks $ ME $ and an even number (possibly 0) blocks of $ MO $ . Similarly let $ b $ the number of such words containing an odd number of blocks $ ME $ and an odd number of blocks $ MO $. Prove that $ a>b $.
Let $a=2019^{1009}, b=2019!$ and $c=1010^{2019}$, then which of the following is true?
(A)$c<b<a$
(B)$a<b<c$
(C)$b<a<c$
(D)$b<c<a$
Let $ A$ represent the set of all functions $ f : \mathbb{N} \rightarrow \mathbb{N}$ such that for all $ k \in \overline{1, 2007}$, $ f^{[k]} \neq \mathrm{Id}_{\mathbb{N}}$ and $ f^{[2008]} \equiv \mathrm{Id}_{\mathbb{N}}$.
a) Prove that $ A$ is non-empty.
b) Find, with proof, whether $ A$ is infinite.
c) Prove that all the elements of $ A$ are bijective functions.
(Denote by $ \mathbb{N}$ the set of the nonnegative integers, and by $ f^{[k]}$, the composition of $ f$ with itself $ k$ times.)
We consider the equation $x^2 + (a + b - 1)x + ab - a - b = 0$, where $a$ and $b$ are positive integers with $a \leq b$.
a) Show that the equation has $2$ distinct real solutions.
b) Prove that if one of the solutions is an integer, then both solutions are non-positive integers and $b < 2a.$
The measures (in degrees) of the interior angles of a convex hexagon form an arithmetic sequence of positive integers. Let $m^{\circ}$ be the measure of the largest interior angle of the hexagon. The largest possible value of $m^{\circ}$ is
$ \textbf{(A)}\ 165^{\circ}\qquad\textbf{(B)}\ 167^{\circ}\qquad\textbf{(C)}\ 170^{\circ}\qquad\textbf{(D)}\ 175^{\circ}\qquad\textbf{(E)}\ 179^{\circ} $
Let $ABC$ be a triangle with $AB = 13, BC = 14, $and$ CA = 15$. Suppose $PQRS$ is a square such that $P$ and $R$ lie on line $BC, Q$ lies on line $CA$, and $S$ lies on line $AB$. Compute the side length of this square.
For any positive integer $n$, let $\tau (n)$ denote the number of its positive divisors (including 1 and itself). Determine all positive integers $m$ for which there exists a positive integer $n$ such that $\frac{\tau (n^{2})}{\tau (n)}=m$.
Let $ n$ be a positive integer and let $ a_1,a_2,a_3,\ldots,a_k$ $ ( k\ge 2)$ be distinct integers in the set $ { 1,2,\ldots,n}$ such that $ n$ divides $ a_i(a_{i + 1} - 1)$ for $ i = 1,2,\ldots,k - 1$. Prove that $ n$ does not divide $ a_k(a_1 - 1).$
[i]Proposed by Ross Atkins, Australia [/i]
A convex quadrilateral $ABCD$ has perpendicular diagonals. The perpendicular bisectors of the sides $AB$ and $CD$ meet at a unique point $P$ inside $ABCD$. Prove that the quadrilateral $ABCD$ is cyclic if and only if triangles $ABP$ and $CDP$ have equal areas.
In the acute-angled triangle $ABC$, let $AP$ and $BQ$ be the altitudes, $CM$ be the median . Point $R$ is the midpoint of $CM$. Line $PQ$ intersects line $AB$ at $T$. Prove that $OR \perp TC$, where $O$ is the center of the circumscribed circle of triangle $ABC$.
Find all positive integers $k,m$ and $n$ such that $k!+3^m=3^n$
There are $ n$ markers, each with one side white and the other side black. In the beginning, these $ n$ markers are aligned in a row so that their white sides are all up. In each step, if possible, we choose a marker whose white side is up (but not one of the outermost markers), remove it, and reverse the closest marker to the left of it and also reverse the closest marker to the right of it. Prove that, by a finite sequence of such steps, one can achieve a state with only two markers remaining if and only if $ n \minus{} 1$ is not divisible by $ 3$.
[i]Proposed by Dusan Dukic, Serbia[/i]
The sequence $(a_n)_{n \ge 1}^\infty$ is given by: $a_1=2$ and $a_{n+1}=a_n^2+a_n$ for all $n \ge 1$.
For an integer $m \ge 2$, $L(m)$ denotes the greatest prime divisor of $m$. Prove that there exists some $k$, for which $L(a_k) > 1000^{1000}$.
[i]Proposed by Nikola Velov[/i]
Let $l$ be a line and $A$ be a point such that $A$ is not on $l.$ Let $P$ be a point on $l$ such that segment $AP$ and line $l$ for a $60^{\circ}$ angle and $AP=1.$ Extend segment $AP$ past $P$ to a point $B$ on the other side of $l.$ Then, let the perpendicular from $B$ to $l$ have foot $M,$ and extend $BM$ past $M$ to $C.$ Finally, extend $CP$ past $P$ to $D.$ Given that $\frac{BP}{AP}=\frac{CM}{BM}=\frac{DP}{CP}=2,$ determine the are of triangle $BPD.$