Found problems: 85335
Determine the least odd number $a > 5$ satisfying the following conditions: There are positive integers $m_1,m_2, n_1, n_2$ such that $a=m_1^2+n_1^2$, $a^2=m_2^2+n_2^2$, and $m_1-n_1=m_2-n_2.$
In an urn there are n balls numbered $1, 2, \cdots, n$. They are drawn at random one by one without replacement and the numbers are recorded. What is the probability that the resulting random permutation has only one local maximum?
A term in a sequence is a local maximum if it is greater than all its neighbors.
The escalator of the department store, which at any given time can be seen at $75$ steps section, moves up one step in $2$ seconds. At time $0$, Juku is standing on an escalator step equidistant from each end, facing the direction of travel. He goes by a certain rule: one step forward, two steps back, then again one step forward, two back, etc., taking one every second in increments of one step. Which end will Juku finally get out and at what point will it happen?
Let $x$ and $y$ be distinct positive integers below $15$. For any two distinct numbers $a, b$ from the set $\{2, x,y\}$, $ab + 1$ is always a positive square. Find all possible values of the square $xy + 1$.
The leader of an IMO team chooses positive integers $n$ and $k$ with $n > k$, and announces them to the deputy leader and a contestant. The leader then secretly tells the deputy leader an $n$-digit binary string, and the deputy leader writes down all $n$-digit binary strings which differ from the leader’s in exactly $k$ positions. (For example, if $n = 3$ and $k = 1$, and if the leader chooses $101$, the deputy leader would write down $001, 111$ and $100$.) The contestant is allowed to look at the strings written by the deputy leader and guess the leader’s string. What is the minimum number of guesses (in terms of $n$ and $k$) needed to guarantee the correct answer?
$n$ vectors are on the plane. We can move each vector forward and backeard on the line that the vector is on it. If there are 2 vectors that their endpoints concide we can omit them and replace them with their sum (If their sum is nonzero). Suppose with these operations with 2 different method we reach to a vector. Prove that these vectors are on a common line
Let $n$ be a positive integer. $2n+1$ tokens are in a row, each being black or white. A token is said to be [i]balanced[/i] if the number of white tokens on its left plus the number of black tokens on its right is $n$. Determine whether the number of [i]balanced[/i] tokens is even or odd.
Prove that any rational $r \in (0, 1)$ can be written uniquely in the form $$r=\frac{a_1}{1!}+\frac{a_2}{2!}+\frac{a_3}{3!}+\cdots+\frac{a_k}{k!}$$ where $a_i\text{’s}$ are nonnegative integers with $a_i\leq i-1$ for all $i$.
Let $(u_n)$ be a sequence of real numbers which satisfies
$$u_{n+2}=|u_{n+1}|-u_n\qquad\text{for all }n\in\mathbb N.$$Prove that there exists a positive integer $p$ such that $u_n=u_{n+p}$ holds for all $n\in\mathbb N$.
All $ 2n\ge 2 $ squares of a $ 2\times n $ rectangle are colored with three colors. We say that a color has a [i]cut[/i] if there is some column (from all $ n $) that has both squares colored with it. Determine:
[b]a)[/b] the number of colorings that have no cuts.
[b]b)[/b] the number of colorings that have a single cut.
Let $a_1,a_2,\cdots ,a_n$ be real numbers, not all zero. Prove that the equation:
\[\sqrt{1+a_1x}+\sqrt{1+a_2x}+\cdots +\sqrt{1+a_nx}=n\]
has at most one real nonzero root.
A positive integer will be called [i]typical[/i] if the sum of its decimal digits is a multiple of $2011$.
a) Show that there are infinitely many [i]typical[/i] numbers, each having at least $2011$ multiples which are also typical numbers.
b) Does there exist a positive integer such that each of its multiples is typical?
Let $n>2$ be a positive integer and let $p$ be a prime. Suppose that the nonzero integers are colored in $n$ colors. Let $a_1,a_2,\ldots,a_{n}$ be integers such that for all $1\le i\le n$, $p^i\nmid a_i$ and $p^{i-1}\mid a_i$. In terms of $n$, $p$, and $\{a_i\}_{i=1}^{n}$, determine if there must exist integers $x_1,x_2,\ldots,x_{n}$ of the same color such that $a_1x_1+a_2x_2+\cdots+a_{n}x_{n}=0$.
[i]Ravi Jagadeesan.[/i]
Let $a$ and $b$ be positive integers such that $2a - 9b + 18ab = 2018$. Find $b - a$.
Given real numbers $x_1, x_2, \dots , x_n \ (n \geq 2)$, with $x_i \geq \frac 1n \ (i = 1, 2, \dots, n)$ and with $x_1^2+x_2^2+\cdots+x_n^2 = 1$ , find whether the product $P = x_1x_2x_3 \cdots x_n$ has a greatest and/or least value and if so, give these values.
Let $a_1,a_2,\ldots$ be a sequence of integers with infinitely many positive and negative terms. Suppose that for every positive integer $n$ the numbers $a_1,a_2,\ldots,a_n$ leave $n$ different remainders upon division by $n$.
Prove that every integer occurs exactly once in the sequence $a_1,a_2,\ldots$.
a) Prove that for each positive integer $n$ there is a unique positive integer $a_n$ such that $$(1 + \sqrt5)^n =\sqrt{a_n} + \sqrt{a_n+4^n} . $$
b) Prove that $a_{2010}$ is divisible by $5\cdot 4^{2009}$ and find the quotient
In each cell of a $2025 \times 2025$ board, a nonnegative real number is written in such a way that the sum of the numbers in each row is equal to $1$, and the sum of the numbers in each column is equal to $1$. Define $r_i$ to be the largest value in row $i$, and let $R = r_1 + r_2 + ... + r_{2025}$. Similarly, define $c_i$ to be the largest value in column $i$, and let $C = c_1 + c_2 + ... + c_{2025}$.
What is the largest possible value of $\frac{R}{C}$?
[i]Proposed by Paulius Aleknavičius, Lithuania, and Anghel David Andrei, Romania[/i]
Julia bakes a cake in the shape of a unit square. Each minute, Julia makes two cuts through the cake as follows:
[list]
[*] she picks a [b]square[/b] piece $\mathcal{S}$ of the cake with no cuts through its interior; then
[*] she slices the entire cake along the two lines parallel to the sides of the cake passing through the center of $\mathcal{S}$.
[/list]
She does not move any pieces of cake during this process. After eight minutes, she has a grid of $9^2 = 81$ pieces of cake. (The pieces can be various sizes.) Compute the number of distinct grids that she could have ended up with. Two grids are the same if they have the same set of cuts; in particular, two grids that differ by a rotation or reflection are distinct.
[i]Proposed by Sean Li[/i]
USAYNO: \url{http://goo.gl/wVR25} % USAYNO link: http://goo.gl/wVR25
[i]Proposed by Lewis Chen, Evan Chen, Eugene Chen[/i]
There exists a certain right triangle with the smallest area in the $2$D coordinate plane such that all of its vertices have integer coordinates but none of its sides are parallel to the $x$- or $y$-axis. Additionally, all of its sides have distinct, integer lengths. What is the area of this triangle?
Suppose that $n$ is a positive integer and let \[d_{1}<d_{2}<d_{3}<d_{4}\] be the four smallest positive integer divisors of $n$. Find all integers $n$ such that \[n={d_{1}}^{2}+{d_{2}}^{2}+{d_{3}}^{2}+{d_{4}}^{2}.\]
Let $f$ be a polynomial with real coefficients of degree $n$. Suppose that $\displaystyle \frac{f(x)-f(y)}{x-y}$ is an integer for all $0 \leq x<y \leq n$. Prove that $a-b | f(a)-f(b)$ for all distinct integers $a,b$.
For positive real numbers $ a$, $ b$, $ c$ and $ d$ such that $ a^2 \plus{} b^2 \plus{} c^2 \plus{} d^2 \equal{} 1$ prove that
$ a^2b^2cd \plus{} \plus{}ab^2c^2d \plus{} abc^2d^2 \plus{} a^2bcd^2 \plus{} a^2bc^2d \plus{} ab^2cd^2 \le 3/32,$
and determine the cases of equality.
Determine all triples of integers $(x, y, z)$ that satisfy the equation
\[x^z + y^z = z.\]