Found problems: 1340
Prove that for each positive integer $n$, the sum of the numbers of digits of $4^n$ and of $25^n$ (in the decimal system) is odd.
Let $M$ be the maximum value of $(6x-3y-8z)$, subject to $2x^2+3y^2+4z^2 = 1$. Find $[M]$.
For a real number $a$, $[a]$ denotes the largest integer not exceeding $a$.
Find all positive real numbers $x$ satisfying the equation
$$x\cdot [x]+2022=[x^2]$$
Let $a, b, c, d$ be integers. Prove that for any positive integer $n$, there are at least $\left \lfloor{\frac{n}{4}}\right \rfloor $ positive integers $m \leq n$ such that $m^5 + dm^4 + cm^3 + bm^2 + 2023m + a$ is not a perfect square.
[i]Proposed by Ilir Snopce[/i]
Show that for all positive integers $n$, \[\lfloor \sqrt{n}+\sqrt{n+1}\rfloor =\lfloor \sqrt{4n+1}\rfloor =\lfloor \sqrt{4n+2}\rfloor =\lfloor \sqrt{4n+3}\rfloor.\]
Let ${a_n}$ be a sequence defined by $a_1=2$, $a_{n+1}=\left[ \frac {3a_n}{2}\right]$ $\forall n \in \mathbb N$
$0.a_1a_2...$ rational or irrational?
Determine all natural numbers $n$ such that for each natural number $a$ relatively prime with $n$ and $a \le 1 + \left\lfloor \sqrt{n} \right\rfloor$ there exists some integer $x$ with $a \equiv x^2 \mod n$.
Remark: "Natural numbers" is the set of positive integers.
What is the sum of real numbers satisfying the equation $\left \lfloor \frac{6x+5}{8} \right \rfloor = \frac{15x-7}{5}$?
$
\textbf{(A)}\ 2
\qquad\textbf{(B)}\ \frac{81}{90}
\qquad\textbf{(C)}\ \frac{7}{15}
\qquad\textbf{(D)}\ \frac{4}{5}
\qquad\textbf{(E)}\ \frac{19}{15}
$
Determine the number of distinct positive real solutions to $$\lfloor x \rfloor ^{\{x\}} = \frac{1}{2022}x^2$$
.
Note: $\lfloor x \rfloor$ is known as the floor function, which returns the greatest integer less than or equal to $x$. Furthermore, $\{x\}$ is defined as $x - \lfloor x \rfloor$.
Two positive real numbers $\alpha, \beta$ satisfies that for any positive integers $k_1,k_2$, it holds that $\lfloor k_1 \alpha \rfloor \neq \lfloor k_2 \beta \rfloor$, where $\lfloor x \rfloor$ denotes the largest integer less than or equal to $x$. Prove that there exist positive integers $m_1,m_2$ such that $\frac{m_1}{\alpha}+\frac{m_2}{\beta}=1$.
Let $p$ be a prime greater than $3$. Prove that
\[ p^2\Big| \sum_{i=1}^{\left\lfloor\frac{2p}{3}\right\rfloor}\dbinom{p}{i}. \]
You are playing a game in which you have $3$ envelopes, each containing a uniformly random amount of money between $0$ and $1000$ dollars. (That is, for any real $0 \leq a < b \leq 1000$, the probability that the amount of money in a given envelope is between $a$ and $b$ is $\frac{b-a}{1000}$.) At any step, you take an envelope and look at its contents. You may choose either to keep the envelope, at which point you finish, or discard it and repeat the process with one less envelope. If you play to optimize your expected winnings, your expected winnings will be $E$. What is $\lfloor E\rfloor,$ the greatest integer less than or equal to $E$?
[i]Author: Alex Zhu[/i]
Let $n$ be a fixed natural number.
[b]a)[/b] Find all solutions to the following equation :
\[ \sum_{k=1}^n [\frac x{2^k}]=x-1 \]
[b]b)[/b] Find the number of solutions to the following equation ($m$ is a fixed natural) :
\[ \sum_{k=1}^n [\frac x{2^k}]=x-m \]
Let $S$ be a set of 100 integers. Suppose that for all positive integers $x$ and $y$ (possibly equal) such that $x + y$ is in $S$, either $x$ or $y$ (or both) is in $S$. Prove that the sum of the numbers in $S$ is at most 10,000.
There's infinity of the following blocks on the table:$1*1 , 1*2 , 1*3 ,.., 1*n$. We have a $n*n$ table and Ali chooses some of these blocks so that the sum of their area is at least $n^2$. Then , Amir tries to cover the $n*n$ table so that none of blocks go out of the table and they don't overlap and he wanna maximize the covered area in the $n*n$ table with those blocks chosen by Ali. Let $k$ be the maximum coverable area independent of Ali's choice. Prove that:
$$n^2 - \lceil \frac{n^2}{4} \rceil \leq k \leq n^2 - \lfloor \frac{n^2}{8} \rfloor$$
*Note : the blocks can be placed only vertically or horizontally.
Determine all functions $f : \mathbb{R} \to \mathbb{R}$ such that the following equation holds for every real $x,y$:
\[ f(f(x) + y) = \lfloor x + f(f(y)) \rfloor. \]
[b]Note:[/b] $\lfloor x \rfloor$ denotes the greatest integer not greater than $x$.
Divide the plane into an infinite square grid by drawing all the lines $x=m$ and $y=n$ for $m,n \in \mathbb Z$. Next, if a square's upper-right corner has both coordinates even, color it black; otherwise, color it white (in this way, exactly $1/4$ of the squares are black and no two black squares are adjacent). Let $r$ and $s$ be odd integers, and let $(x,y)$ be a point in the interior of any white square such that $rx-sy$ is irrational. Shoot a laser out of this point with slope $r/s$; lasers pass through white squares and reflect off black squares. Prove that the path of this laser will form a closed loop.
Compute $ \sum_{n \equal{} 1}^\infty\sum_{k \equal{} 1}^{n \minus{} 1}\frac {k}{2^{n \plus{} k}}$.
Consider the polynomial
$P_n(x) = \binom {n}{2}+\binom {n}{5}x+\binom {n}{8}x^2 + \cdots + \binom {n}{3k+2}x^{3k}$
where $n \ge 2$ is a natural number and $k = \left\lfloor \frac{n-2}{3} \right \rfloor$
[b](a)[/b] Prove that $P_{n+3}(x)=3P_{n+2}(x)-3P_{n+1}(x)+(x+1)P_n(x)$
[b](b)[/b] Find all integer numbers $a$ such that $P_n(a^3)$ is divisible by $3^{ \lfloor \frac{n-1}{2} \rfloor}$ for all $n \ge 2$
For $ n\geq2$ let $ a_1, a_2, \ldots a_n$ be positive real numbers such that
\[ (a_1 \plus{} a_2 \plus{} \cdots \plus{} a_n)\left(\frac {1}{a_1} \plus{} \frac {1}{a_2} \plus{} \cdots \plus{} \frac {1}{a_n}\right) \leq \left(n \plus{} \frac {1}{2}\right)^2.
\]
Prove that $ \max(a_1, a_2, \ldots, a_n)\leq 4\min(a_1, a_2, \ldots, a_n)$.
Determine all infinite sets $A$ of positive integers with the following propety:
If $a,b \in A$ and $a \ge b$ then $\left\lfloor \frac{a}{b} \right\rfloor \in A$
Suppose $A\subset \{(a_1,a_2,\dots,a_n)\mid a_i\in \mathbb{R},i=1,2\dots,n\}$. For any $\alpha=(a_1,a_2,\dots,a_n)\in A$ and $\beta=(b_1,b_2,\dots,b_n)\in A$, we define
\[ \gamma(\alpha,\beta)=(|a_1-b_1|,|a_2-b_2|,\dots,|a_n-b_n|), \] \[ D(A)=\{\gamma(\alpha,\beta)\mid\alpha,\beta\in A\}. \] Please show that $|D(A)|\geq |A|$.
A sample of 121 integers is given, each between 1 and 1000 inclusive, with repetitions allowed. The sample has a unique mode (most frequent value). Let $D$ be the difference between the mode and the arithmetic mean of the sample. What is the largest possible value of $\lfloor D\rfloor$? (For real $x$, $\lfloor x\rfloor$ is the greatest integer less than or equal to $x$.)
Find all pairs $(x,y)$ of positive real numbers such that $xy$ is an integer and $x+y = \lfloor x^2 - y^2 \rfloor$.
Let $n_1,n_2, \cdots, n_{26}$ be pairwise distinct positive integers satisfying
(1) for each $n_i$, its digits belong to the set $\{1,2\}$;
(2) for each $i,j$, $n_i$ can't be obtained from $n_j$ by adding some digits on the right.
Find the smallest possible value of $\sum_{i=1}^{26} S(n_i)$, where $S(m)$ denotes the sum of all digits of a positive integer $m$.