Found problems: 85335
Find all quadruple $ (m,n,p,q) \in \mathbb{Z}^4$ such that \[ p^m q^n \equal{} (p\plus{}q)^2 \plus{} 1.\]
Does there exist a prime number whose decimal representation is of the form $3811\cdots11$ (that is, consisting of the digits $3$ and $8$ in that order, followed by one or more digits $1$)?
Given $6$ points in a plane, assume that each two of them are connected by a segment. Let $D$ be the length of the longest, and $d$ the length of the shortest of these segments. Prove that $\frac Dd\ge\sqrt3$.
Nine points are drawn on the surface of a regular tetrahedron with an edge of $1$ cm. Prove that among these points there are two located at a distance (in space) no greater than $0.5$ cm.
Consider fractions $\frac{a}{b}$ where $a$ and $b$ are positive integers.
(a) Prove that for every positive integer $n$, there exists such a fraction $\frac{a}{b}$ such that $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}+1$.
(b) Show that there are infinitely many positive integers $n$ such that no such fraction $\frac{a}{b}$ satisfies $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}$.
Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a continuous function having a primitive $ F $ having the property that $ f-F $ is positive globally. Calculate $ \lim_{x\to\infty } f(x) . $
[i]Florian Dumitrel[/i]
For $2n$ positive integers a matching (i.e. dividing them into $n$ pairs) is called {\it non-square} if the product of two numbers in each pair is not a perfect square. Prove that if there is a non-square matching, then there are at least $n!$ non-square matchings.
(By $n!$ denote the product $1\cdot 2\cdot 3\cdot \ldots \cdot n$.)
A positive integer $b$ and a sequence $a_0,a_1,a_2,\dots$ of integers $0\le a_i<b$ is given. It is known that $a_0\neq 0$ and the sequence $\{a_i\}$ is eventually periodic but has infinitely many nonzero terms. Let $S$ be the set of positive integers $n$ so that $n\mid (a_0a_1\dots a_n)_b$. Given that $S$ is infinite, show that there are infinitely many primes that divide at least one element of $S$.
[i]Proposed by Carl Schildkraut and Holden Mui[/i]
Thirteen birds arrive and sit down in a plane. It's known that from each 5-tuple of birds, at least four birds sit on a circle. Determine the greatest $M \in \{1, 2, ..., 13\}$ such that from these 13 birds, at least $M$ birds sit on a circle, but not necessarily $M + 1$ birds sit on a circle. (prove that your $M$ is optimal)
Prove that for any odd positive integer $n$, $n^{12}-n^8-n^4+1$ is divisible by $2^9$.
Consider a curve $C$ on the $x$-$y$ plane expressed by $x=\tan \theta ,\ y=\frac{1}{\cos \theta}\left (0\leq \theta <\frac{\pi}{2}\right)$.
For a constant $t>0$, let the line $l$ pass through the point $P(t,\ 0)$ and is perpendicular to the $x$-axis,intersects with the curve $C$ at $Q$. Denote by $S_1$ the area of the figure bounded by the curve $C$, the $x$-axis, the $y$-axis and the line $l$, and denote by $S_2$ the area of $\triangle{OPQ}$. Find $\lim_{t\to\infty} \frac{S_1-S_2}{\ln t}.$
Let be two matrices $ A,B\in M_2\left(\mathbb{R}\right) $ that satisfy the equality $ \left( A-B\right)^2 =O_2. $
[b]a)[/b] Show that $ \det\left( A^2-B^2\right) =\left( \det A -\det B\right)^2. $
[b]b)[/b] Demonstrate that $ \det\left( AB-BA\right) =0\iff \det A=\det B. $
For non-negative integer $n$, the function $f$ is given by
\[f(x)=\begin{cases}
\frac{x}{2} & \text{if $n$ is even} \\
x-1 & \text{if $n$ is odd.}
\end{cases}
\]
Furthermore, let $h(n)$ be the smallest $k$ for which $f^k(n)=0$. Compute
\[\sum_{n=1}^{1024} h(n).\]
In a certain city, exactly 3 streets converge at any intersection. The streets are painted in three colors so that they converge at each intersection streets of three different colors. Three roads leave the city. Prove that they have different colors.
Given distinct positive integer $ a_1,a_2,…,a_{2020} $. For $ n \ge 2021 $, $a_n$ is the smallest number different from $a_1,a_2,…,a_{n-1}$ which doesn't divide $a_{n-2020}...a_{n-2}a_{n-1}$. Proof that every number large enough appears in the sequence.
We have $3$ circles such that any $2$ of them are externally tangent. Let $a$ be length of the outer tangent common to a pair of them. The lengths $b$ and $c$ are defined similarly. If $T$ is the sum of the areas of such circles, show that $\pi (a + b + c)^2 \le 12T $.
Note: In In the case of externally tangent circles, the common external tangent is the segment tangent to them that touches them at different points.
Consider any five points in the interior of square $S$ of side length $1$. Prove that at least one of the distances between these points is less than $\sqrt{2} \slash 2.$ Can this constant be replaced by a smaller number?
Find all increasing sequences $a_1,a_2,a_3,...$ of natural numbers such that for each $i,j\in \mathbb N$, number of the divisors of $i+j$ and $a_i+a_j$ is equal. (an increasing sequence is a sequence that if $i\le j$, then $a_i\le a_j$.)
Consider the function $f:[0,1)\rightarrow[0,1)$ defined by $f(x)=2x-\lfloor 2x\rfloor$, where $\lfloor 2x\rfloor$ is the greatest integer less than or equal to $2x$. Find the sum of all values of $x$ such that $f^{17}(x)=x.$
[i]Proposed by Matthew Weiss
A simplified model of a bicycle of mass $M$ has two tires that each comes into contact with the ground at a point. The wheelbase of this bicycle (the distance between the points of contact with the ground) is $w$, and the center of mass of the bicycle is located midway between the tires and a height h above the ground. The bicycle is moving to the right, but slowing down at a constant rate. The acceleration has a magnitude $a$. Air resistance may be ignored.
[asy]
size(175);
pen dps = linewidth(0.7) + fontsize(4); defaultpen(dps);
draw(circle((0,0),1),black+linewidth(2.5));
draw(circle((3,0),1),black+linewidth(2.5));
draw((1.5,0)--(0,0)--(1,1.5)--(2.5,1.5)--(1.5,0)--(1,1.5),black+linewidth(1));
draw((3,0)--(2.4,1.8),black+linewidth(1));
filldraw(circle((1.5,2/3),0.05),gray);
draw((1.3,1.6)--(0.7,1.6)--(0.7,1.75)--cycle,black+linewidth(1));
label("center of mass of bicycle",(2.5,1.9));
draw((1.55,0.85)--(1.8,1.8),BeginArrow);
draw((4.5,-1)--(4.5,2/3),BeginArrow,EndArrow);
label("$h$",(4.5,-1/6),E);
draw((1.5,2/3)--(4.5,2/3),dotted);
draw((0,-1)--(4.5,-1),dotted);
draw((0,-5/4)--(3,-5/4),BeginArrow,EndArrow);
label("$w$",(3/2,-5/4),S);
draw((0,-1)--(0,-6/4),dotted);
draw((3,-1)--(3,-6/4),dotted);
[/asy]
Case 1 ([b][u]Questions 28 - 29[/u][/b]): Assume that the coefficient of sliding friction between each tire and the ground is $\mu$, and that both tires are skidding: sliding without rotating. Express your answers in terms of $w$, $h$, $M$, and $g$.
What is the maximum value of $a$ so that both tires remain in contact with the ground?
$ \textbf{(A)}\ \frac{wg}{h} $
$ \textbf{(B)}\ \frac{wg}{2h}$
$ \textbf{(C)}\ \frac{hg}{2w} $
$ \textbf{(D)}\ \frac{h}{2wg} $
$ \textbf{(E)}\ \text{none of the above} $
A well-known puzzle asks for a partition of a cross into four parts which are to be reassembled into a square. One solution is exhibited on the picture.
[img]https://cdn.artofproblemsolving.com/attachments/9/1/3b8990baf5e37270c640e280c479b788d989ba.png[/img]
Show that there are infinitely many solutions. (Some solutions split the cross into four equal parts!)
Does there exist a hexagon (not necessarily convex) with side lengths $1, 2, 3, 4, 5, 6$ (not necessarily in this order) that can be tiled with a) $31$ b) $32$ equilateral triangles with side length $1?$
Given an integer $m\ge 2$, find the smallest integer $k > m$ such that for any partition of the set $\{m,m + 1,..,k\}$ into two classes $A$ and $B$ at least one of the classes contains three numbers $a,b,c$ (not necessarily distinct) such that $a^b = c$.
$1994$ points from the plane are given so that any $100$ of them can be selected $98$ that can be rounded (some points may be at the border of the circle) with a diameter of $1$. Determine the smallest number of circles with radius $1$, sufficient to cover all $1994$
We have a deck of $2n$ cards. Each shuffling changes the order from $a_1,a_2,\ldots,a_n,b_1,b_2,\ldots,b_n$ to $a_1,b_1,a_2,b_2,\ldots,a_n,b_n$. Determine all even numbers $2n$ such that after shuffling the deck $8$ times the original order is restored.