Found problems: 85335
For every positive integer $n\ge 3$, let $\phi_n$ be the set of all positive integers less than and coprime to $n$. Consider the polynomial:
$$P_n(x)=\sum_{k\in\phi_n} {x^{k-1}}.$$
a. Prove that $P_n(x)=(x^{r_n}+1)Q_n(x)$ for some positive integer $r_n$ and polynomial $Q_n(x)\in\mathbb{Z}[x]$ (not necessary non-constant polynomial).
b. Find all $n$ such that $P_n(x)$ is irreducible over $\mathbb{Z}[x]$.
let m and n be natural numbers such that: $3m|(m+3)^n+1$
Prove that $\frac{(m+3)^n+1}{3m}$ is odd
For a positive integer $n$, let $A_n$ and $B_n$ be the families of $n$-element subsets of $S_n=\{1,2,\ldots ,2n\}$ with respectively even and odd sums of elements. Compute $|A_n|-|B_n|$.
The numbers $1, 2,\ldots ,50$ are written on a blackboard. Ana performs the following operations: she chooses any three numbers $a, b$ and $c$ from the board and replaces them with their sum $a + b + c$ and writes the number $(a + b) (b + c) (c + a)$ in the notebook. Ana performs these operations until there are only two numbers left on the board ($24$ operations in total). Then, she calculates the sum of the numbers written down in her notebook. Let $M$ and $m$ be the maximum and minimum possible of the sums obtained by Ana.
Find the value of $\frac{M}{m}$.
Let $f(x) = x^3 + 3x^2 + 1$. There is a unique line of the form $y = mx + b$ such that $m > 0$ and this line intersects $f(x)$ at three points, $A, B, C$ such that $AB = BC = 2$. Find $\lfloor 100m \rfloor$.
Find $[ \sqrt{19992000}]$ where $[a]$ is the greatest integer less than or equal to $x$.
Parallelogram $ABCD$ is the base of a pyramid $SABCD$. Planes determined by triangles $ASC$ and $BSD$ are mutually perpendicular. Find the area of the side $ASD$, if areas of sides $ASB,BSC$ and $CSD$ are equal to $x,y$ and $z$, respectively.
Let $n>1$ be a positive integer. Claire writes $n$ distinct positive real numbers $x_1, x_2, \dots, x_n$ in a row on a blackboard. In a $\textit{move},$ William can erase a number $x$ and replace it with either $\tfrac{1}{x}$ or $x+1$ at the same location. His goal is to perform a sequence of moves such that after he is done, the number are strictly increasing from left to right.
[list]
[*]Prove that there exists a positive constant $A,$ independent of $n,$ such that William can always reach his goal in at most $An \log n$ moves.
[*]Prove that there exists a positive constant $B,$ independent of $n,$ such that Claire can choose the initial numbers such that William cannot attain his goal in less than $Bn \log n$ moves.
[/list]
Let $\vartriangle ABC$ be a triangle with $AB > AC$, its incircle is tangent to $BC$ at $D$. Let $DE$ be a diameter of the incircle, and let $F$ be the intersection between line $AE$ and side $BC$. Find the ratio between the areas of $\vartriangle DEF$ and $\vartriangle ABC$ in terms of the three side lengths of$\vartriangle ABC$.
A person moves in the $x-y$ plane moving along points with integer co-ordinates $x$ and $y$ only. When she is at a point $(x,y)$, she takes a step based on the following rules:
(a) if $x+y$ is even she moves to either $(x+1,y)$ or $(x+1,y+1)$;
(b) if $x+y$ is odd she moves to either $(x,y+1)$ or $(x+1,y+1)$.
How many distinct paths can she take to go from $(0,0)$ to $(8,8)$ given that she took exactly three steps to the right $((x,y)$ to $(x+1,y))$?
A circle of radius 1 is tangent to a circle of radius 2. The sides of $ \triangle ABC$ are tangent to the circles as shown, and the sides $ \overline{AB}$ and $ \overline{AC}$ are congruent. What is the area of $ \triangle ABC$?
[asy]defaultpen(black+linewidth(0.7));
size(7cm);
real t=2^0.5;
D((0,0)--(4*t,0)--(2*t,8)--cycle, black);
D(CR((2*t,2),2), black);
D(CR((2*t,5),1), black);
dot(origin^^(4t,0)^^(2t,8));
label("B", (0,0), SW);
label("C", (4*t,0), SE);
label("A", (2*t,8), N);
D((2*t,2)--(2*t,4), black); D((2*t,5)--(2*t,6), black);
MP('2', (2*t,3), W); MP('1',(2*t, 5.5), W);[/asy]
$ \textbf{(A) } \frac {35}2 \qquad \textbf{(B) } 15\sqrt {2} \qquad \textbf{(C) } \frac {64}3 \qquad \textbf{(D) } 16\sqrt {2} \qquad \textbf{(E) } 24$
Prove that for every positive integer $ n$,
$ n^n \le (n!)^2 \le \left( \frac{(n\plus{}1)(n\plus{}2)}{6} \right) ^n.$
Fill in each of the ten boxes with a 3-digit number so that the following conditions are satisfied.
[list=1]
[*]Every number has three distinct digits that sum to $15$. $0$ may not be a leading digit. One digit of each number has been given to you.
[*]No two numbers in any pair of boxes use the same three digits. For example, it is not allowed for two different boxes to have the numbers $456$ and $645$.
[*]Two boxes joined by an arrow must have two numbers that share an equal hundreds digit, tens digit, or ones digit. Also, the smaller number must point to the larger.[/list]
You do not need to prove that your configuration is the only one possible; you merely need to find a configuration that satisfies the constraints above. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.)
[asy]
size(200);
defaultpen(linewidth(0.8));
path arrow;
pair squares[]={(2,4),(6,4),(10,4),(0,0),(4,0),(8,0),(12,0),(2,-4),(6,-4),(10,-4)};
pair horizarrows[]={(4,4),(2,0),(6,0),(10,0),(4,-4),(8,-4)};
bool isLeft[]={false,false,true,false,false,false};
pair diagarrows[]={(1,2),(7,2),(9,2),(1,-2),(5,-2),(11,-2)};
bool isDown[]={true,false,true,false,false,true};
for(int i=0;i<=9;i=i+1)
{
draw(box(squares[i]-(1,1),squares[i]+(1,1)));
label("$"+(string)i+"$",squares[i]);
}
for(int j=0;j<=5;j=j+1)
{
if(isLeft[j])
arrow=(horizarrows[j].x-1,horizarrows[j].y)--(horizarrows[j].x+1,horizarrows[j].y);
else
arrow=(horizarrows[j].x+1,horizarrows[j].y)--(horizarrows[j].x-1,horizarrows[j].y);
draw(arrow,BeginArrow(size=7));
}
for(int k=0;k<=5;k=k+1)
{
if(isDown[k])
arrow=(diagarrows[k].x-1/3,diagarrows[k].y-1)--(diagarrows[k].x+1/3,diagarrows[k].y+1);
else
arrow=(diagarrows[k].x-1/3,diagarrows[k].y+1)--(diagarrows[k].x+1/3,diagarrows[k].y-1);
draw(arrow,BeginArrow(size=7));
}
[/asy]
Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ that satisfy
\[f(f(x)+y)=2x+f(f(y)-x)\quad\text{for all real}\ x,y. \]
Given a trapezium with two parallel sides of lengths $m$ and $n$, where $m$, $n$ are integers, prove that it is
possible to divide the trapezium into several congruent triangles.
A cart of mass $m$ moving at $12 \text{ m/s}$ to the right collides elastically with a cart of mass $4.0 \text{ kg}$ that is originally at rest. After the collision, the cart of mass $m$ moves to the left with a velocity of $6.0 \text{ m/s}$. Assuming an elastic collision in one dimension only, what is the velocity of the center of mass ($v_{\text{cm}}$) of the two carts before the collision?
$\textbf{(A) } v_{\text{cm}} = 2.0 \text{ m/s}\\
\textbf{(B) } v_{\text{cm}}=3.0 \text{ m/s}\\
\textbf{(C) } v_{\text{cm}}=6.0 \text{ m/s}\\
\textbf{(D) } v_{\text{cm}}=9.0 \text{ m/s}\\
\textbf{(E) } v_{\text{cm}}=18.0 \text{ m/s}$
An equilateral triangle is divided into $n^2$ congruent equilateral triangles. A spider stands at one of the vertices, a fly at another. Alternately each of them moves to a neighbouring vertex. Prove that the spider can always catch the fly.
Let $ n \geq 3$ be an odd integer. We denote by $ [\minus{}n,n]$ the set of all integers greater or equal than $ \minus{}n$ and less or equal than $ n$.
Player $ A$ chooses an arbitrary positive integer $ k$, then player $ B$ picks a subset of $ k$ (distinct) elements from $ [\minus{}n,n]$. Let this subset be $ S$.
If all numbers in $ [\minus{}n,n]$ can be written as the sum of exactly $ n$ distinct elements of $ S$, then player $ A$ wins the game. If not, $ B$ wins.
Find the least value of $ k$ such that player $ A$ can always win the game.
A natural number is called [i]chaotigal [/i] if it and its successor both have the sum of their digits divisible by $2021$. How many digits are in the smallest chaotigal number?
Show that the following equation has finitely many solutions $(t,A,x,y,z)$ in positive integers
$$\sqrt{t(1-A^{-2})(1-x^{-2})(1-y^{-2})(1-z^{-2})}=(1+x^{-1})(1+y^{-1})(1+z^{-1})$$
We are given $n (n \ge 5)$ circles in a plane. Suppose that every three of them have a common point. Prove that all $n$ circles have a common point.
Let $p,n$ be positive integers, such that $p$ is prime and $p <n$.
If $p$ divides $n + 1$ and $ \left(\left[\frac{n}{p}\right], (p-1)!\right) = 1$, then prove that $p\cdot \left[\frac{n}{p}\right]^2$ divides ${n \choose p} -\left[\frac{n}{p}\right]$ .
(Here $[x]$ represents the integer part of the real number $x$.)
Find all positive integers $a, b, c, d$ with $a \le b$ and $c \le d$ such that $\begin{cases} a + b = cd \\
c + d = ab \end{cases}$ .
Let $a, b, c$ be positive reals. Prove that $\sqrt{2a^2+bc}+\sqrt{2b^2+ac}+\sqrt{2c^2+ab}\ge 3 \sqrt{ab+bc+ca}$
Define mapping $F : \mathbb{R}^4\rightarrow \mathbb{R}^4$ as $F(x,\ y,\ z,\ w)=(xy,\ y,\ z,\ w)$ and let mapping $f : S^3\rightarrow \mathbb{R}^4$ be restriction of $F$ to 3 dimensional ball $S^3=\{(x,\ y,\ z,\ w)\in{\mathbb{R}^4} | x^2+y^2+z^2+w^2=1\}$.
Find the rank of $df_p$, or the differentiation of $f$ at every point $p$ in $S^3$.