Found problems: 85335
Prove that any natural number smaller or equal than the factorial of a natural number $ n $ is the sum of at most $ n $ distinct divisors of the factorial of $ n. $
In the country of Sikinia there are finitely many cities. From each city, exactly three roads go out and each road goes to another Sikinian city. A tourist starts a trip from city $A$ and drives according to the following rule: he turns left at the first city, then right at the next city, and so on, alternately. Show that he will eventually return to $A.$
Find all functions $f$ from the interval $(1,\infty)$ to $(1,\infty)$ with the following property: if $x,y\in(1,\infty)$ and $x^2\le y\le x^3,$ then $(f(x))^2\le f(y) \le (f(x))^3.$
The area of the figure enclosed by the $x$-axis, $y$-axis, and line $7x + 8y = 15$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
In convex hexagon $ABCDEF$, we know that $\angle BCA = \angle DEC = \angle AFB = \angle CBD = \angle EDF.$ Prove that $AB = CD = EF.$
Let $u_1=1,u_2=2,u_3=24,$ and
$u_{n+1}=\frac{6u_n^2 u_{n-2}-8u_nu_{n-1}^2}{u_{n-1}u_{n-2}}, n \geq 3.$
Prove that the elements of the sequence are natural numbers and that $n\mid u_n$ for all $n$.
$2010$ ships deliver bananas, lemons and pineapples from South America to Russia. The total number of bananas on each ship equals the number of lemons on all other ships combined, while the total number of lemons on each ship equals the total number of pineapples on all other ships combined. Prove that the total number of fruits is a multiple of $31$.
Pyramid $EARLY$ has rectangular base $EARL$ and apex $Y$, and all of its edges are of integer length. The four edges from the apex have lengths $1, 4, 7, 8$ (in no particular order), and EY is perpendicular to $YR$. Find the area of rectangle $EARL$.
Consider a unit circle with center $O$. Let $P$ be a point outside the circle such that the two line segments passing through $P$ and tangent to the circle form an angle of $60^\circ$. Compute the length of $OP$.
Let $ABCD$ be a quadrilateral with $\angle DAB = \angle ABC = 120^o$. If $AB = 3$, $BC = 2$, and $AD = 4$, what is the length of $CD$?
Consider the set of grid points $(m,n)$ in the plane, $m,n$ integers. Let $\sigma$ be a finite subset and define
\[S(\sigma)=\sum_{(m,n)\in\sigma}(100-|m|-|n|) \]
Find the maximum of $S$, taken over the set of all such subsets $\sigma$.
Let $m, n \ge 2$ be positive integers, and let $a_{1}, a_{2}, \cdots,a_{n}$ be integers, none of which is a multiple of $m^{n-1}$. Show that there exist integers $e_{1}, e_{2}, \cdots, e_{n}$, not all zero, with $\vert e_i \vert<m$ for all $i$, such that $e_{1}a_{1}+e_{2}a_{2}+ \cdots +e_{n}a_{n}$ is a multiple of $m^n$.
An infinite arithmetic progression whose terms are positive integers contains the square of an integer and the cube of an integer. Show that it contains the sixth power of an integer.
Determine all pairs $(a, b)$ of integers which satisfy the equality $\frac{a + 2}{b + 1} +\frac{a + 1}{b + 2} = 1 +\frac{6}{a + b + 1}$
Let $x$ and $y$ be integers between $0$ and $5$, inclusive. For the system of modular congruences $$ \begin{cases}
x + 3y \equiv 1 \,\,(mod \, 2) \\
4x + 5y \equiv 2 \,\,(mod \, 3) \end{cases}$$, find the sum of all distinct possible values of $x + y$
Let $a$ be an odd positive integer. Prove that if $d$ divides $a^2+2$, then $d\equiv1\pmod8$ or $d\equiv3\pmod8$.
Let $a_1$, $a_2$, $a_3$, $a_4$, $a_5$ be real numbers whose sum is $20$. Determine with proof the smallest possible value of \[ \displaystyle\sum_{1\le i \le j \le 5} \lfloor a_i + a_j \rfloor. \]
Find all positive integers $n$ such that $2^n-n^2+1$ is a perfect square.
Let $n$ be a positive integer. A [i]partition [/i] of $n$ is a multiset (set with repeated elements) whose sum of elements is $n$. For example, the partitions of $3$ are $\{1, 1, 1\}, \{1, 2\}$ and $\{3\}$. Each partition of $n$ is written as a non-descending sequence. For example, for $n = 3$, the list is $(1, 1, 1)$, $(1, 2)$ and $(3)$. For each sequence $x = (x_1, x_2, ..., x_k)$, define $f(x)=\prod_{i=1}^{k-1} {x_{i+1} \choose x_ i}$ . Furthermore, the $f$ of partition $\{n\}$ is $f((n)) = 1$. Prove that the sum of all $f$'s in the list is $2^{n-1}.$
Let $n{}$ be a natural number. The pairwise distinct nonzero integers $a_1,a_2,\ldots,a_n$ have the property that the number \[(k+a_1)(k+a_2)\cdots(k+a_n)\]is divisible by $a_1a_2\cdots a_n$ for any integer $k{}.$ Find the largest possible value of $a_n.$
[i]Proposed by F. Petrov and K. Sukhov[/i]
Pieces are placed in some squares of an $8 \times 8$ board sothat:
a) There is at least one token in any rectangle with sides $2 \times 1$ or $1\times 2$.
b) There are at least two neighboring pieces in any rectangle with sides $7\times 1$ or $1\times 7$.
Find the smallest number of tokens that can be taken to fulfill with both conditions.
Let $C_k=\frac{1}{k+1}\binom{2k}{k}$ denote the $k^{\text{th}}$ Catalan number and $p$ be an odd prime. Prove that exactly half of the numbers in the set
\[\left\{\sum_{k=1}^{p-1}C_kn^k\,\middle\vert\, n\in\{1,2,\ldots,p-1\}\right\}\]
are divisible by $p$.
[i]Tristan Shin[/i]
We define the number $s$ as
\[s=\sum^{\infty}_{i=1} \dfrac{1}{10^i-1}=\dfrac{1}{9}+\dfrac{1}{99}+\dfrac{1}{999}+\dfrac{1}{9999}+\ldots=0.12232424...\]
We can determine the $n$th digit right of the decimal point of $s$ without summing the entire infinite series because after summing the first $n$ terms of the series, the rest of the series sums to less than $\dfrac{2}{10^{n+1}}$. Determine the smallest prime number $p$ for which the $p$th digit right of the decimal point of $s$ is greater than 2. Justify your answer.
Is there a solution of the Cauchy problem $y\partial u/\partial x+\sin x\partial u/\partial y=y$, $u|_{x=0}=y^4$ on the whole $(x,y)$ plane? Is it unique?
In $\vartriangle ABC$, two straight lines drawn from an arbitrary point $D$ on $AB$ are parallel to $AC$ , $BC$ and intersect $BC$ , $AC$ at $F$ , $G$, respectively. Prove that the sum of the circumferences of the circles circumscribed around $\vartriangle ADG$ and $\vartriangle BDF$ is equal to the circumference of the circle circumscribed around $\vartriangle ABC$.