Found problems: 85335
Find all positive integers $a,b$ such that $\frac{a^2+b}{b^2-a}$ and $\frac{b^2+a}{a^2-b}$ are also integers.
Tim has a multiset of positive integers. Let $c_i$ be the number of occurrences of numbers that are [i]at least[/i] $i$ in the multiset. Let $m$ be the maximum element of the multiset. Tim calls a multiset [i]spicy[/i] if $c_1, \dots, c_m$ is a sequence of strictly decreasing powers of $3$. Tim calls the [i]hotness[/i] of a spicy multiset the sum of its elements. Find the sum of the hotness of all spicy multisets that satisfy $c_1 = 3^{2020}$. Give your answer $\pmod{1000}$. (Note: a multiset is an unordered set of numbers that can have repeats)
[i]Proposed by Timothy Qian[/i]
Two ants, one starting at $ (-1, 1) $, the other at $ (1, 1) $, walk to the right along the parabola $ y = x^2 $ such that their midpoint moves along the line $ y = 1 $ with constant speed $1$. When the left ant first hits the line $ y = \frac {1}{2} $, what is its speed?
A square and a rectangle of the same perimeter have a common corner. Prove that the intersection point of the diagonals of the rectangle lies on the diagonal of the square.
(Yu. Blinkov)
Quadratic trinomials $f$ and $g$ with integer coefficients obtain only positive values and the inequality $\dfrac{f(x)}{g(x)}\geq
\sqrt{2}$ is true $\forall x\in\mathbb{R}$. Prove that $\dfrac{f(x)}{g(x)}>\sqrt{2}$ is true $\forall x\in\mathbb{R}$
[I]Proposed by A. Khrabrov[/i]
Find all one-to-one mappings $f:\mathbb{N}\to\mathbb{N}$ such that for all positive integers $n$ the following relation holds:
\[ f(f(n)) \leq \frac {n+f(n)} 2 . \]
If $\frac{4^x}{2^{x+y}}=8$ and $\frac{9^{x+y}}{3^{5y}}=243$, $x$ and $y$ are real numbers, then $xy$ equals:
$\text{(A)} \ \frac{12}{5} \qquad \text{(B)} \ 4 \qquad \text{(C)} \ 6 \qquad \text{(D)} \ 12 \qquad \text{(E)} \ -4$
Let $w, x, y, z$ be integers from $0$ to $3$ inclusive. Find the number of ordered quadruples of $(w, x, y, z)$ such that $5x^2 + 5y^2 + 5z^2 - 6wx-6wy -6wz$ is divisible by $4$.
[i]Proposed by Timothy Qian[/i]
Compute the sum of all positive integers whose digits form either a strictly increasing or strictly decreasing sequence.
Arthur thought of a positive integer and noticed that the sum of its three smallest divisors is $17$ and that the sum of its three largest divisors is $3905$. Indicate all the numbers that Arthur may have thought of.
$ABCDE$ is a regular pentagon. What is the degree measure of the acute angle at the intersection of line segments $AC$ and $BD$?
Let $\tau(n)$ denote the number of positive divisors of the positive integer $n$. Prove that there exist infinitely many positive integers $a$ such that the equation $ \tau(an)=n $ does not have a positive integer solution $n$.
Find the least positive integer $N$ such that there exist positive real numbers $a_1,a_2,\dots,a_N$ such that
\[ \sum_{k=1}^{N}ka_k=1 \quad \text{and} \quad \sum_{k=1}^{N}\frac{a_k^2}{k}\leq \frac{1}{2022^2}. \]
p1. Circle $M$ is the incircle of ABC, while circle $N$ is the incircle of $ACD$. Circles $M$ and $N$ are tangent at point $E$. If side length $AD = x$ cm, $AB = y$ cm, $BC = z$ cm, find the length of side $DC$ (in terms of $x, y$, and $z$).
[img]https://cdn.artofproblemsolving.com/attachments/d/5/66ddc8a27e20e5a3b27ab24ff1eba3abee49a6.png[/img]
p2. The address of the house on Jalan Bahagia will be numbered with the following rules:
$\bullet$ One side of the road is numbered with consecutive even numbers starting from number $2$.
$\bullet$ The opposite side is numbered with an odd number starting from number $3$.
$\bullet$ In a row of even numbered houses, there is some land vacant house that has not been built.
$\bullet$ The first house numbered $2$ has a neighbor next door.
When the RT management ordered the numbers of the house, it is known that the cost of making each digit is $12.000$ Rp. For that, the total cost to be incurred is $1.020.000$ Rp. It is also known that the cost of all even-sided house numbers is $132.000$ Rp. cheaper than the odd side. When the land is empty later a house has been built, the number of houses on the even and odd sides is the same.
Determine the number of houses that are now on Jalan Bahagia .
p3. Given the following problem: Each element in the set $A = \{10, 11, 12,...,2008\}$ multiplied by each element in the set $B = \{21, 22, 23,...,99\}$. The results are then added together to give value of $X$. Determine the value of $X$. Someone answers the question by multiplying $2016991$ with $4740$. How can you explain that how does that person make sense?
p4. Let $P$ be the set of all positive integers between $0$ and $2008$ which can be expressed as the sum of two or more consecutive positive integers . (For example: $11 = 5 + 6$, $90 = 29 + 30 + 31$, $100 = 18 + 19 +20 + 21 + 22$. So $11, 90, 100$ are some members of $P$.) Find the sum of of all members of $P$.
p5. A four-digit number will be formed from the numbers at $0, 1, 2, 3, 4, 5$ provided that the numbers in the number are not repeated, and the number formed is a multiple of $3$. What is the probability that the number formed has a value less than $3000$?
The integers $ 1,2,\dots,20$ are written on the blackboard. Consider the following operation as one step: [i]choose two integers $ a$ and $ b$ such that $ a\minus{}b \ge 2$ and replace them with $ a\minus{}1$ and $ b\plus{}1$[/i]. Please, determine the maximum number of steps that can be done.
[i]Yudi Satria, Jakarta[/i]
We say that a positive integer $n$ is [i]fantastic[/i] if there exist positive rational numbers $a$ and $b$ such that
$$ n = a + \frac 1a + b + \frac 1b.$$
[b](a)[/b] Prove that there exist infinitely many prime numbers $p$ such that no multiple of $p$ is fantastic.
[b](b)[/b] Prove that there exist infinitely many prime numbers $p$ such that some multiple of $p$ is fantastic.
[i]Proposed by Walther Janous, Austria[/i]
Two sets $M=\{(x,y)|x^2+2y^2=3\},N=\{(x,y)|y=mx+b\}$. For all $m\in\mathbb{R}$, $M\cap N\neq\varnothing$, then the range value of $b$ is
$\text{(A)}\left[-\frac{\sqrt6}{2},\frac{\sqrt6}{2}\right]\qquad\text{(B)}\left(-\frac{\sqrt6}{2},\frac{\sqrt6}{2}\right)\qquad\text{(C)}\left(-\frac{2\sqrt3}{3},\frac{2\sqrt3}{3}\right]\qquad\text{(D)}\left[-\frac{2\sqrt3}{3},\frac{2\sqrt3}{3}\right]$
On the Cartesian plane consider the set $V$ of all vectors with integer coordinates. Determine all functions $f : V \rightarrow \mathbb{R}$ satisfying the conditions:
(i) $f(v) = 1$ for each of the four vectors $v \in V$ of unit length.
(ii) $f(v+w) = f(v)+f(w)$ for every two perpendicular vectors $v, w \in V$
(Zero vector is considered to be perpendicular to every vector).
In a dance, a group $S$ of $1994$ students stand in a big circle. Each student claps the hands of each of his two neighbours a number of times. For each student $x,$ let $f(x)$ be the total number of times $x$ claps the hands of his neighbours. As an example, suppose there are $3$ students $A, B$ and $C$. A claps hand with $B$ two times, $B$ claps hand with $C$ three times and $C$ claps hand with $A$ five times. Then $f(A) = 7, f(B) = 5$ and $f(C) = 8.$
(i) Prove that $\{f(x) | x \in S\}\ne\{n | n$ is an integer, $2 \le n \le 1995\}$.
(ii) Find an example in which $\{f(x) | x \in S\} = \{n | n$ is an integer, $n \ne 3, 2 \le n \le 1996\}$
At a gathering of $30$ people, there are $20$ people who all know each other and $10$ people who know no one. People who know each other hug, and people who do not know each other shake hands. How many handshakes occur?
$\textbf{(A)}\ 240\qquad\textbf{(B)}\ 245\qquad\textbf{(C)}\ 290\qquad\textbf{(D)}\ 480\qquad\textbf{(E)}\ 490$
Find all continuous function $f:\mathbb{R}^{\geq 0}\rightarrow \mathbb{R}^{\geq 0}$ such that :
\[f(xf(y))+f(f(y)) = f(x)f(y)+2 \: \: \forall x,y\in \mathbb{R}^{\geq 0}\]
[i]Proposed by Mohammad Ahmadi[/i]
A square floor is tiled with congruent square tiles. The tiles on the two diagonals of the floor are black. The rest of the tiles are white. If there are 101 black tiles, then the total number of tiles is
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[/asy]
$ \textbf{(A)}\ 121\qquad\textbf{(B)}\ 625\qquad\textbf{(C)}\ 676\qquad\textbf{(D)}\ 2500\qquad\textbf{(E)}\ 2601 $
Let $a_1,a_2,\dots,a_{2023}$ be positive integers such that
[list=disc]
[*] $a_1,a_2,\dots,a_{2023}$ is a permutation of $1,2,\dots,2023$, and
[*] $|a_1-a_2|,|a_2-a_3|,\dots,|a_{2022}-a_{2023}|$ is a permutation of $1,2,\dots,2022$.
[/list]
Prove that $\max(a_1,a_{2023})\ge 507$.
$(BEL 1)$ A parabola $P_1$ with equation $x^2 - 2py = 0$ and parabola $P_2$ with equation $x^2 + 2py = 0, p > 0$, are given. A line $t$ is tangent to $P_2.$ Find the locus of pole $M$ of the line $t$ with respect to $P_1.$
$24! = 620,448,401,733,239,439,360,000$ ends in four zeros, and $25!=15,511,210,043,330,985,984,000,000$ ends in six zeros. Thus, there is no integer $n$ such that $n!$ ends in exactly five zeros. Let $S$ be the set of all $k$ such that for no integer n does $n!$ end in exactly $k$ zeros. If the numbers in $S$ are listed in increasing order, 5 will be the first number. Find the 100th number in that list.