Found problems: 85335
Find all sets of three primes $p, q$, and $r$ such that $p + q = r$ and $(r -p)(q - p) - 27p$ is a perfect square.
Upper content of a subset $E$ of the plane $\mathbb{R}^{2}$ is defined as
$$\mathcal{C}(E)=\inf\{\sum_{i=1}^{n} \text{diam}(E_{i})\}$$
where $\inf$ is taken over all finite families of sets $E_{1},\dots,E_{n}$ $n\in \mathbb{N}$, in $\mathbb{R}^{2}$
such that $E\subset \bigcup_{i=1}^{n}E_{i}$.
Lower content of $E$ is defined as
$$\mathcal{K}(E)=\sup\{\text{length}(L) |\, L \text{ is a closed line segment onto which $E$ can be contracted}\}$$.
Prove that
i) $\mathcal{C}(L)=\text{length}(L)$ if $L$ is a closed line segment;
ii) $\mathcal{C}(E) \geq \mathcal{K}(E)$;
iii) the equality in ii) is not always true even if $E$ is compact.
In a board $8$x$8$, the unit squares have numbers $1-64$, as shown. The unit square with a multiple of $3$ on it are red. Find the minimum number of chess' bishops that we need to put on the board such that any red unit square either has a bishop on it or is attacked by at least one bishop.
Note: A bishops moves diagonally.
[img]https://i.imgur.com/03baBwp.jpeg[/img]
The quadratic polynomials $f$ and $g$ with real coefficients are such that if $g(x)$ is an integer for some $x>0$, then so is $f(x)$. Prove that there exist integers $m,n$ such that $f(x)=mg(x)+n$ for all $x$.
Determine all functions $f: \mathbb{R} \to \mathbb{R}$ such that
$$ f(x^3) + f(y)^3 + f(z)^3 = 3xyz $$
for all real numbers $x$, $y$ and $z$ with $x+y+z=0$.
Different points $C$ and $D$ are chosen on a circle with center $O$ and diameter $AB$ so that they are on the same side of the diameter $AB$. On the diameter $AB$ is chosen a point $P$ different from the point $O$ such that the points $P, O, D, C$ are on the same circle. Prove that $\angle APC = \angle BPD$.
We are given a positive integer $s \ge 2$. For each positive integer $k$, we define its [i]twist[/i] $k’$ as follows: write $k$ as $as+b$, where $a, b$ are non-negative integers and $b < s$, then $k’ = bs+a$. For the positive integer $n$, consider the infinite sequence $d_1, d_2, \dots$ where $d_1=n$ and $d_{i+1}$ is the twist of $d_i$ for each positive integer $i$.
Prove that this sequence contains $1$ if and only if the remainder when $n$ is divided by $s^2-1$ is either $1$ or $s$.
Let $A$ be a set of $n\ge2$ positive integers, and let $\textstyle f(x)=\sum_{a\in A}x^a$. Prove that there exists a complex number $z$ with $\lvert z\rvert=1$ and $\lvert f(z)\rvert=\sqrt{n-2}$.
Let $a$ be a positive real number.
Evaluate $I=\int_0^{+\infty} \frac{\sin x\cos x}{x(x^2+a^2)}dx.$
For any real number $t$, denote by $[t]$ the greatest integer which is less than or equal to $t$. For example: $[8] = 8$, $[\pi] = 3$, and $[-5/2] = -3$. Show that the equation
\[[x] + [2x] + [4x] + [8x] + [16x] + [32x] = 12345\]
has no real solution.
The functions $ f(x) ,\ g(x)$ satify that $ f(x) \equal{} \frac {x^3}{2} \plus{} 1 \minus{} x\int_0^x g(t)\ dt,\ g(x) \equal{} x \minus{} \int_0^1 f(t)\ dt$.
Let $ l_1,\ l_2$ be the tangent lines of the curve $ y \equal{} f(x)$, which pass through the point $ (a,\ g(a))$ on the curve $ y \equal{} g(x)$.
Find the minimum area of the figure bounded by the tangent tlines $ l_1,\ l_2$ and the curve $ y \equal{} f(x)$ .
Mari and Yuri play the next play. At first, there are two piles on the table, with $m$ and $n$ candies, respectively. At each turn, players eats one pile of candy from the table and distribute another pile of candy into two non-empty parts ,. Everything is done in turn and wins the player who can no longer share the pile (when there is only one candy left). Which player will win if both use the optimal strategy and Mari makes the first move?
Two circles have radii $15$ and $95$. If the two external tangents to the circles intersect at $60$ degrees, how far apart are the centers of the circles?
Daniel, Clarence, and Matthew split a \$20.20 dinner bill so that Daniel pays half of what Clarence pays. If Daniel pays \$6.06, what is the ratio of Clarence's pay to Matthew's pay?
[i]Proposed by Henry Ren[/i]
Find the minimum area bounded by the graphs of $ y\equal{}x^2$ and $ y\equal{}kx(x^2\minus{}k)\ (k>0)$.
Show that if an infinite arithmetic progression of positive integers contains a square and a cube, it must contain a sixth power.
The side lengths of a scalene triangle are roots of the polynomial $$x^3-20x^2+131x-281.3.$$ Find the square of the area of the triangle.
Let $f:[0;+\infty)\to \mathbb R$ be a continuous function such that $\lim\limits_{x\to +\infty} f(x)=L$ exists (it may be finite or infinite). Prove that $$ \lim\limits_{n\to\infty}\int\limits_0^{1}f(nx)\,\mathrm{d}x=L. $$
Consider a cake in the shape of a circle. It's been divided to some inequal parts by its radii. Arash and Bahram want to eat this cake. At the very first, Arash takes one of the parts. In the next steps, they consecutively pick up a piece adjacent to another piece formerly removed. Suppose that the cake has been divided to 5 parts. Prove that Arash can choose his pieces in such a way at least half of the cake is his.
Let $P(x)$ and $Q(x)$ be integer polynomials of degree $p$ and $q$ respectively. Assume that $P(x)$ divides $Q(x)$ and all their coefficients are either $1$ or $2002$. Show that $p+1$ is a divisor of $q+1$.
[i]Mihai Cipu[/i]
The points $P(a,b)$ and $Q(0,c)$ are on the curve $\dfrac{y}{c} = \cosh{(\dfrac{x}{c})}.$ The line through $Q$ parallel to the normal at $P$ cuts the $x-$axis at $R.$ Prove that $QR = b.$
Compute $3^{3^{\ldots^3}} \mod{333},$ where there are $3^{3^3}$ $3$'s in the exponent.
If $x=\sqrt2+\sqrt3+\sqrt6$ is a root of $x^4+ax^3+bx^2+cx+d=0$ where $a,b,c,d$ are integers, what is the value of $|a+b+c+d|$?
Prove that in a scalene acute-angled triangle, the orthocenter, the incenter, and the circumcenter are not collinear.
Given any real number $N_0,$ if $N_{j+1}= \cos N_j ,$ prove that $\lim_{j\to \infty} N_j$ exists and is independent of $N_0.$