Found problems: 85335
Let $n, m, k$ and $l$ be positive integers with $n \neq 1$ such that $n^k + mn^l + 1$ divides $n^{k+l} - 1$. Prove that
[list]
[*]$m = 1$ and $l = 2k$; or
[*]$l|k$ and $m = \frac{n^{k-l}-1}{n^l-1}$.
[/list]
Determine all pairs $ (n,p)$ of positive integers, where $ p$ is prime, such that $ 3^p\minus{}np\equal{}n\plus{}p$.
Given is a fixed regular pentagon $ABCDE$ with side $1$. Let $M$ be an arbitrary point inside or on it. Let the distance from $M$ to the closest vertex be $r_1$, to the next closest be $r_2$ and so on, so that the distances from $M$ to the five vertices satisfy $r_1\le r_2\le r_3\le r_4\le r_5$. Find (a) the locus of $M$ which gives $r_3$ the minimum possible value, and (b) the locus of $M$ which gives $r_3$ the maximum possible value.
A truncated trigonal pyramid is circumscribed around a sphere touching its bases at points $T_1, T_2$. Let $h$ be the altitude of the pyramid, $R_1, R_2$ be the circumradii of its bases, and $O_1, O_2$ be the circumcenters of the bases. Prove that $$R_1R_2h^2 = (R_1^2-O_1T_1^2)(R_2^2-O_2T_2^2).$$
A convex polyhedron has $n$ faces that are all congruent triangles with angles $36^{\circ}, 72^{\circ}$, and $72^{\circ}$. Determine, with proof, the maximum possible value of $n$.
Solve the equation
[b]a)[/b] $ \log_2^2 +(x-1)\log_2 x =6-2x $ in $ \mathbb{R} . $
[b]b)[/b] $ 2^{x+1}+3^{x+1} +2^{1/x^2}+3^{1/x^2}=18 $ in $ (0,\infty ) . $
[i]Cristinel Mortici[/i]
Let $ f \ne 0$ be a polynomial with real coefficients. Define the sequence $ f_{0}, f_{1}, f_{2}, \ldots$ of polynomials by $ f_{0}= f$ and $ f_{n+1}= f_{n}+f_{n}'$ for every $ n \ge 0$. Prove that there exists a number $ N$ such that for every $ n \ge N$, all roots of $ f_{n}$ are real.
Prove that for all $X > 1$, there exists a triangle whose sides have lengths $P_1(X) = X^4+X^3+2X^2+X+1, P_2(X) = 2X^3+X^2+2X+1$, and $P_3(X) = X^4-1$. Prove that all these triangles have the same greatest angle and calculate it.
Show that there exist two consecutive squares such that there are at least $1000$ primes between them.
Real numbers $a,b$ and $c$ have arithmetic mean $0$. The arithmetic mean of $a^2, b^2$ and $c^2$ is $10$. What is the arithmetic mean of $ab, ac$ and $bc$?
$
\textbf{(A) }-5 \qquad
\textbf{(B) }-\frac{10}{3} \qquad
\textbf{(C) }-\frac{10}{9} \qquad
\textbf{(D) }0 \qquad
\textbf{(E) }\frac{10}{9} \qquad
$
Find maximal numbers of planes, such there are $6$ points and
1) $4$ or more points lies on every plane.
2) No one line passes through $4$ points.
Triangle $\vartriangle ABC$ has side lengths $AB = 26$, $BC = 34$, and $CA = 24\sqrt2$. A fourth point $D$ makes a right angle $\angle BDC$. What is the smallest possible length of $\overline{AD}$?
We say that a binary string $s$ [i]contains[/i] another binary string $t$ if there exist indices $i_1,i_2,\ldots,i_{|t|}$ with $i_1 < i_2 < \ldots < i_{|t|}$ such that $$s_{i_1}s_{i_2}\ldots s_{i_{|t|}} = t.$$ (In other words, $t$ is found as a not necessarily contiguous substring of $s$.) For example, $110010$ contains $111$. What is the length of the shortest string $s$ which contains the binary representations of all the positive integers less than or equal to $2048$?
Guy has 17 cards. Each of them has an integer written on it (the numbers are not necessarily positive, and not necessarily different from each other). Guy noticed that for each card, the square of the number written on it equals the sum of the numbers on the 16 other cards.
What are the numbers on Guy's cards? Find all of the options.
Determine all positive integers $ n$ such that it is possible to tile a $ 15 \times n$ board with pieces shaped like this:
[asy]size(100); draw((0,0)--(3,0)); draw((0,1)--(3,1)); draw((0,2)--(1,2)); draw((2,2)--(3,2)); draw((0,0)--(0,2)); draw((1,0)--(1,2)); draw((2,0)--(2,2)); draw((3,0)--(3,2)); draw((5,0)--(6,0)); draw((4,1)--(7,1)); draw((4,2)--(7,2)); draw((5,3)--(6,3)); draw((4,1)--(4,2)); draw((5,0)--(5,3)); draw((6,0)--(6,3)); draw((7,1)--(7,2));[/asy]
If $b_{1}, b_{2}, \ldots, b_{n}$ are non-negative reals not all zero, then prove that the polynomial \[x^{n}-b_{1}x^{n-1}-b_{2}x^{n-2}-\ldots-b_{n}=0\] has only one positive root $p$, which is simple. Moreover prove that any root of the polynomial does not exceed $p$ in absolute value.
Let $ABC$ be an acute-angled triangle. Let $D$ and $E$ be the feet of the altitudes on the sides $BC$ or $AC$. Points $F$ and $G$ are located on the lines $AD$ and $BE$ in such a way that$ \frac{AF}{FD}=\frac{BG}{GE}$. The line passing through $C$ and $F$ intersects $BE$ at point $H$, and the line passing through $C$ and $G$ intersects $AD$ at point $I$. Prove that points $F, G, H$ and $I$ lie on a circle.
(Walther Janous)
Non-negative numbers $a$, $b$, $c$ satisfy
$a^2+b^2+c^2\geq 3$. Prove the inequality
$$
(a+b+c)^3\geq 9(ab+bc+ca).
$$
Let n be an odd positive integer.Let M be a set of $n$ positive integers ,which are 2x2 different. A set $T$ $\in$ $M$ is called "good" if the product of its elements is divisible by the sum of the elements in $M$, but is not divisible by the square of the same sum. Given that $M$ is "good",how many "good" subsets of $M$ can there be?
Philena and Nathan are playing a game. First, Nathan secretly chooses an ordered pair $(x, y)$ of positive integers such that $x \leq 20$ and $y \leq 23$. (Philena knows that Nathan’s pair must satisfy $x \leq 20$ and $y \leq 23$.) The game then proceeds in rounds; in every round, Philena chooses an ordered pair $(a, b)$ of positive integers and tells it to Nathan; Nathan says YES if $x \leq a$ and $y \leq b$, and NO otherwise. Find, with proof, the smallest positive integer $N$ for which Philena has a strategy that guarantees she can be certain of Nathan’s pair after at most $N$ rounds.
An equation $P(x)=Q(y)$ is called [b]Interesting[/b] if $P$ and $Q$ are polynomials with degree at least one and integer coefficients and the equations has an infinite number of answers in $\mathbb{N}$.
An interesting equation $P(x)=Q(y)$ [b]yields in[/b] interesting equation $F(x)=G(y)$ if there exists polynomial $R(x) \in \mathbb{Q} [x]$ such that $F(x) \equiv R(P(x))$ and $G(x) \equiv R(Q(x))$.
(a) Suppose that $S$ is an infinite subset of $\mathbb{N} \times \mathbb{N}$.$S$ [i]is an answer[/i] of interesting equation $P(x)=Q(y)$ if each element of $S$ is an answer of this equation. Prove that for each $S$ there's an interesting equation $P_0(x)=Q_0(y)$ such that if there exists any interesting equation that $S$ is an answer of it, $P_0(x)=Q_0(y)$ yields in that equation.
(b) Define the degree of an interesting equation $P(x)=Q(y)$ by $max\{deg(P),deg(Q)\}$. An interesting equation is called [b]primary[/b] if there's no other interesting equation with lower degree that yields in it.
Prove that if $P(x)=Q(y)$ is a primary interesting equation and $P$ and $Q$ are monic then $(deg(P),deg(Q))=1$.
Time allowed for this question was 2 hours.
You are given positive integers $m, n>1$. Vasyl and Petryk play the following game: they take turns marking on the coordinate plane yet unmarked points of the form $(x, y)$, where $x, y$ are positive integers with $1 \leq x \leq m, 1 \leq y \leq n$. The player loses if after his move there are two marked points, the distance between which is not a positive integer. Who will win this game if Vasyl moves first and each player wants to win?
[i]Proposed by Mykyta Kharin[/i]
Two thousand points are given on a circle. Label one of the points 1. From this point, count 2 points in the clockwise direction and label this point 2. From the point labeled 2, count 3 points in the clockwise direction and label this point 3. (See figure.) Continue this process until the labels $1, 2, 3, \dots, 1993$ are all used. Some of the points on the circle will have more than one label and some points will not have a label. What is the smallest integer that labels the same point as 1993?
[asy]
int x=101, y=3*floor(x/4);
draw(Arc(origin, 1, 360*(y-3)/x, 360*(y+4)/x));
int i;
for(i=y-2; i<y+4; i=i+1) {
dot(dir(360*i/x));
}
label("3", dir(360*(y-2)/x), dir(360*(y-2)/x));
label("2", dir(360*(y+1)/x), dir(360*(y+1)/x));
label("1", dir(360*(y+3)/x), dir(360*(y+3)/x));[/asy]
Alice and Bob play the following game. To start, Alice arranges the numbers $1,2,\ldots,n$ in some order in a row and then Bob chooses one of the numbers and places a pebble on it. A player's [i]turn[/i] consists of picking up and placing the pebble on an adjacent number under the restriction that the pebble can be placed on the number $k$ at most $k$ times. The two players alternate taking turns beginning with Alice. The first player who cannot make a move loses. For each positive integer $n$, determine who has a winning strategy.
Let $\{x_i\}^{\infty}_{i=1}$ and $\{y_i\}^{\infty}_{i=1}$ be sequences of real numbers such that $x_1=y_1=\sqrt{3}$, $$x_{n+1}=x_n+\sqrt{1+x_n^2}\quad\text{and}\quad y_{n+1}=\frac{y_n}{1+\sqrt{1+y_n^2}}$$
for all $n\geq 1$. Prove that $2<x_ny_n<3$ for all $n>1$.