Found problems: 1782
Find all functions $f: \mathbb{Q}\to \mathbb{Q}$ such that for all $x,y \in \mathbb{Q}$: \[f(x+y)+f(x-y)=2(f(x)+f(y)).\]
The sequence $a_1, a_2,..., a_{2000}$ of real numbers satisfies the condition
\[a_1^3+a_2^3+...+a_n^3=(a_1+a_2+...+a_n)^2\]
for all $n$, $1\leq n \leq 2000$. Prove that every element of the sequence is an integer.
Let $S_n$ be the number of sequences $(a_1, a_2, \ldots, a_n),$ where $a_i \in \{0,1\},$ in which no six consecutive blocks are equal. Prove that $S_n \rightarrow \infty$ when $n \rightarrow \infty.$
Find all functions $ f: \mathbb{R} \to \mathbb{R}$ satisfying
\[ f\left(\frac {x \plus{} y}{x \minus{} y}\right) \equal{} \frac {f\left(x\right) \plus{} f\left(y\right)}{f\left(x\right) \minus{} f\left(y\right)}
\]
for all $ x \neq y$.
Determine the positive integers $n$ such that the inequality \[n! > \sqrt{n^n}\] holds.
Let $ (a_{n})_{n\ge 1}$ be a sequence of positive integers satisfying $ (a_{m},a_{n}) = a_{(m,n)}$ (for all $ m,n\in N^ +$). Prove that for any $ n\in N^ + ,\prod_{d|n}{a_{d}^{\mu (\frac {n}{d})}}$ is an integer. where $ d|n$ denotes $ d$ take all positive divisors of $ n.$ Function $ \mu (n)$ is defined as follows: if $ n$ can be divided by square of certain prime number, then $ \mu (1) = 1;\mu (n) = 0$; if $ n$ can be expressed as product of $ k$ different prime numbers, then $ \mu (n) = ( - 1)^k.$
Find all positive real numbers $r<1$ such that there exists a set $\mathcal{S}$ with the given properties:
i) For any real number $t$, exactly one of $t, t+r$ and $t+1$ belongs to $\mathcal{S}$;
ii) For any real number $t$, exactly one of $t, t-r$ and $t-1$ belongs to $\mathcal{S}$.
a) Evaluate
\[\lim_{n\to \infty} \underbrace{\sqrt{a+\sqrt{a+\ldots+\sqrt{a+\sqrt{b}}}}}_{n\ \text{square roots}}\]
with $a,b>0$.
b)Let $(a_n)_{n\ge 1}$ and $(x_n)_{n\ge 1}$ such that $a_n>0$ and
\[x_n=\sqrt{a_n+\sqrt{a_{n-1}+\ldots+\sqrt{a_2+\sqrt{a_1}}}},\ \forall n\in \mathbb{N}^*\]
Prove that:
1) $(x_n)_{n\ge 1}$ is bounded if and only if $(a_n)_{n\ge 1}$ is bounded.
2) $(x_n)_{n\ge 1}$ is convergent if and only if $(a_n)_{n\ge 1}$ is convergent.
[i]Valentin Matrosenco[/i]
Let $\{b_n\}_{n\geq 1}^{\infty}$ be a sequence of positive integers. The sequence $\{a_n\}_{n\geq 1}^{\infty}$ is defined as follows: $a_1$ is a fixed positive integer and
\[a_{n+1}=a_n^{b_n}+1 ,\qquad \forall n\geq 1.\]
Find all positive integers $m\geq 3$ with the following property: If the sequence $\{a_n\mod m\}_{n\geq 1 }^{\infty}$ is eventually periodic, then there exist positive integers $q,u,v$ with $2\leq q\leq m-1$, such that the sequence $\{b_{v+ut}\mod q\}_{t\geq 1}^{\infty}$ is purely periodic.
A triangular array of numbers has a first row consisting of the odd integers $ 1,3,5,\ldots,99$ in increasing order. Each row below the first has one fewer entry than the row above it, and the bottom row has a single entry. Each entry in any row after the top row equals the sum of the two entries diagonally above it in the row immediately above it. How many entries in the array are multiples of $ 67$?
[asy]size(200);
defaultpen(fontsize(10));
label("1", origin);
label("3", (2,0));
label("5", (4,0));
label("$\cdots$", (6,0));
label("97", (8,0));
label("99", (10,0));
label("4", (1,-1));
label("8", (3,-1));
label("12", (5,-1));
label("196", (9,-1));
label(rotate(90)*"$\cdots$", (6,-2));[/asy]
Let $ f(x) \equal{} \frac{x^2\plus{}1}{2x}$ for $ x \neq 0.$ Define $ f^{(0)}(x) \equal{} x$ and $ f^{(n)}(x) \equal{} f(f^{(n\minus{}1)}(x))$ for all positive integers $ n$ and $ x \neq 0.$ Prove that for all non-negative integers $ n$ and $ x \neq \{\minus{}1,0,1\}$
\[ \frac{f^{(n)}(x)}{f^{(n\plus{}1)}(x)} \equal{} 1 \plus{} \frac{1}{f \left( \left( \frac{x\plus{}1}{x\minus{}1} \right)^{2n} \right)}.\]
Prove that for integer $n$ we have:
\[n! \le \left( \frac{n+1}{2} \right)^n\]
[size=75][i](please note that the pupils in the competition never heard of AM-GM or alikes, it is intended to be solved without any knowledge on inequalities)[/i][/size]
Let $a,b,c$ be distinct positive real numbers, and let $k$ be a positive integer greater than $3$. Show that
\[\left\lvert\frac{a^{k+1}(b-c)+b^{k+1}(c-a)+c^{k+1}(a-b)}{a^k(b-c)+b^k(c-a)+c^k(a-b)}\right\rvert\ge \frac{k+1}{3(k-1)}(a+b+c)\]
and
\[\left\lvert\frac{a^{k+2}(b-c)+b^{k+2}(c-a)+c^{k+2}(a-b)}{a^k(b-c)+b^k(c-a)+c^k(a-b)}\right\rvert\ge \frac{(k+1)(k+2)}{3k(k-1)}(a^2+b^2+c^2).\]
[i]Calvin Deng.[/i]
Find all numbers $N=\overline{a_1a_2\ldots a_n}$ for which $9\times \overline{a_1a_2\ldots a_n}=\overline{a_n\ldots a_2a_1}$ such that at most one of the digits $a_1,a_2,\ldots ,a_n$ is zero.
You are given a set of $n$ blocks, each weighing at least $1$; their total weight is $2n$. Prove that for every real number $r$ with $0 \leq r \leq 2n-2$ you can choose a subset of the blocks whose total weight is at least $r$ but at most $r + 2$.
Let $ f(x)$ be a $ n \minus{}$degree polynomial all of whose coefficients are equal to $ \pm 1$, and having $ x \equal{} 1$ as its $ m$ multiple root. If $ m\ge 2^k (k\ge 2,k\in N)$, then $ n\ge 2^{k \plus{} 1} \minus{} 1.$
function $f:\mathbb{N}^{\star} \times \mathbb{N}^{\star} \rightarrow \mathbb{N}^{\star}$ ($\mathbb{N}^{\star}=\mathbb{N}\cup \{0\}$)with these conditon:
1- $f(0,x)=x+1$
2- $f(x+1,0)=f(x,1)$
3- $f(x+1,y+1)=f(x,f(x+1,y))$(romania 1997)
find $f(3,1997)$
Let $\, |U|, \, \sigma(U) \,$ and $\, \pi(U) \,$ denote the number of elements, the sum, and the product, respectively, of a finite set $\, U \,$ of positive integers. (If $\, U \,$ is the empty set, $\, |U| = 0, \, \sigma(U) = 0, \, \pi(U) = 1$.) Let $\, S \,$ be a finite set of positive integers. As usual, let $\, \binom{n}{k} \,$ denote $\, n! \over k! \, (n-k)!$. Prove that \[ \sum_{U \subseteq S} (-1)^{|U|} \binom{m - \sigma(U)}{|S|} = \pi(S) \] for all integers $\, m \geq \sigma(S)$.
Prove that any positive integer can be represented as a sum of Fibonacci numbers, no two of which are consecutive.
Let $\alpha(n)$ be the number of digits equal to one in the dyadic representation of a positive integer $n$. Prove that [list=a] [*] the inequality $\alpha(n^2 ) \le \frac{1}{2} \alpha(n) (1+\alpha(n))$ holds, [*] equality is attained for infinitely $n\in\mathbb{N}$, [*] there exists a sequence $\{n_i\}$ such that $\lim_{i \to \infty} \frac{ \alpha({n_{i}}^2 )}{ \alpha(n_{i}) } = 0$.[/list]
Consider a function $f$ on nonnegative integers such that $f(0)=1, f(1)=0$ and $f(n)+f(n-1)=nf(n-1)+(n-1)f(n-2)$ for $n \ge 2$. Show that
\[\frac{f(n)}{n!}=\sum_{k=0}^n \frac{(-1)^k}{k!}\]
Let $(G,\cdot)$ be a finite group with the identity element, $e$. The smallest positive integer $n$ with the property that $x^{n}= e$, for all $x \in G$, is called the [i]exponent[/i] of $G$.
(a) For all primes $p \geq 3$, prove that the multiplicative group $\mathcal G_{p}$ of the matrices of the form $\begin{pmatrix}\hat 1 & \hat a & \hat b \\ \hat 0 & \hat 1 & \hat c \\ \hat 0 & \hat 0 & \hat 1 \end{pmatrix}$, with $\hat a, \hat b, \hat c \in \mathbb Z \slash p \mathbb Z$, is not commutative and has [i]exponent[/i] $p$.
(b) Prove that if $\left( G, \circ \right)$ and $\left( H, \bullet \right)$ are finite groups of [i]exponents[/i] $m$ and $n$, respectively, then the group $\left( G \times H, \odot \right)$ with the operation given by $(g,h) \odot \left( g^\prime, h^\prime \right) = \left( g \circ g^\prime, h \bullet h^\prime \right)$, for all $\left( g,h \right), \, \left( g^\prime, h^\prime \right) \in G \times H$, has the [i]exponent[/i] equal to $\textrm{lcm}(m,n)$.
(c) Prove that any $n \geq 3$ is the [i]exponent[/i] of a finite, non-commutative group.
[i]Ion Savu[/i]
Let $t$ and $n$ be fixed integers each at least $2$. Find the largest positive integer $m$ for which there exists a polynomial $P$, of degree $n$ and with rational coefficients, such that the following property holds: exactly one of \[ \frac{P(k)}{t^k} \text{ and } \frac{P(k)}{t^{k+1}} \] is an integer for each $k = 0,1, ..., m$.
[i]Proposed by Michael Kural[/i]
Prove that
\[\frac{1995}{2}-\frac{1994}{3}+\frac{1993}{4}-\ldots -\frac{2}{1995}+\frac{1}{1996}=\frac{1}{999}+\frac{3}{1000}+\ldots +\frac{1995}{1996}\]
We define a [i]chessboard polygon[/i] to be a polygon whose sides are situated along lines of the form $ x \equal{} a$ or $ y \equal{} b$, where $ a$ and $ b$ are integers. These lines divide the interior into unit squares, which are shaded alternately grey and white so that adjacent squares have different colors. To tile a chessboard polygon by dominoes is to exactly cover the polygon by non-overlapping $ 1 \times 2$ rectangles. Finally, a [i]tasteful tiling[/i] is one which avoids the two configurations of dominoes shown on the left below. Two tilings of a $ 3 \times 4$ rectangle are shown; the first one is tasteful, while the second is not, due to the vertical dominoes in the upper right corner.
[asy]size(300); pathpen = linewidth(2.5);
void chessboard(int a, int b, pair P){
for(int i = 0; i < a; ++i) for(int j = 0; j < b; ++j)
if((i+j) % 2 == 1) fill(shift(P.x+i,P.y+j)*unitsquare,rgb(0.6,0.6,0.6));
D(P--P+(a,0)--P+(a,b)--P+(0,b)--cycle);
}
chessboard(2,2,(2.5,0));fill(unitsquare,rgb(0.6,0.6,0.6));fill(shift(1,1)*unitsquare,rgb(0.6,0.6,0.6)); chessboard(4,3,(6,0)); chessboard(4,3,(11,0)); MP("\mathrm{Distasteful\ tilings}",(2.25,3),fontsize(12));
/* draw lines */
D((0,0)--(2,0)--(2,2)--(0,2)--cycle); D((1,0)--(1,2)); D((2.5,1)--(4.5,1)); D((7,0)--(7,2)--(6,2)--(10,2)--(9,2)--(9,0)--(9,1)--(7,1)); D((8,2)--(8,3)); D((12,0)--(12,2)--(11,2)--(13,2)); D((13,1)--(15,1)--(14,1)--(14,3)); D((13,0)--(13,3));[/asy] a) Prove that if a chessboard polygon can be tiled by dominoes, then it can be done so tastefully.
b) Prove that such a tasteful tiling is unique.