Found problems: 85335
For which values of positive integer $m$ is it possible to find polynomials $P, Q\in\mathbb{C} [x]$, with degrees at least two, such that \[x(x+1)\cdots(x+m-1)=P(Q(x)).\][i]Proposed by Navid Safaei, Tehran[/i]
Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.
Pairwise distinct prime numbers $p, q, r$ satisfy the equality $$rp^3 + p^2 + p = 2rq^2 +q^2 + q.$$
Determine all possible values of the product $pqr$.
Suppose $f$ is a nonconstant polynomial with integer coefficients with the following property:
[list]
[*]$f(0)$ and $f(1)$ are both odd.
[*]Define a sequence of integers with $a_k = f(1)f(2) \cdots f(k)+1$
[/list]
Prove that there are infinitely many prime numbers dividing at least one element of the sequence.
[i]Proposed by Sayandeep Shee[/i]
$a,b,c,t$ are antural numbers and $k=c^{t}$ and $n=a^{k}-b^{k}$.
a) Prove that if $k$ has at least $q$ different prime divisors, then $n$ has at least $qt$ different prime divisors.
b)Prove that $\varphi(n)$ id divisible by $2^{\frac{t}{2}}$
Let be the matrix $ A=\begin{pmatrix} 1& 2& -1\\ 2&2 &0\\1& 4& -3 \end{pmatrix} . $
[b]a)[/b] Show that the equation $ AX=\begin{pmatrix} 2\\ 1\\5 \end{pmatrix} $ has infinite solutions in $ \mathcal{M}_1^3\left( \mathbb{C} \right) . $
[b]b)[/b] Find the rank of the adugate of $ A. $
Let $p: \mathbb C \to \mathbb C$ be a polynomial with degree $n$ and complex coefficients which satisfies
\[x \in \mathbb R \iff p(x) \in \mathbb R.\]
Show that $n=1$
Let $x_0 = x_{101} = 0$. The numbers $x_1, x_2,...,x_{100}$ are chosen at random from the interval $[0, 1]$ uniformly and independently. Compute the probability that $2x_i \ge x_{i-1} + x_{i+1}$ for all $i = 1, 2,..., 100.$
We say that a set $S$ of integers is [i]rootiful[/i] if, for any positive integer $n$ and any $a_0, a_1, \cdots, a_n \in S$, all integer roots of the polynomial $a_0+a_1x+\cdots+a_nx^n$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^a - 2^b$ for positive integers $a$ and $b$.
A triangle $ABC$ is given. A circle $\Gamma$, passing through $A$, is tangent to side $BC$ at point $P$ and intersects sides $AB$ and $AC$ at $M$ and $N$ respectively. Prove that the smaller arcs $MP$ and $NP$ of $\Gamma$ are equal iff $\Gamma$ is tangent to the circumcircle of $\Delta ABC$ at $A$.
Let $ABCD$ be a convex quadrileteral such that $m(\widehat{ABD})=40^\circ$, $m(\widehat{DBC})=70^\circ$, $m(\widehat{BDA})=80^\circ$, and $m(\widehat{BDC})=50^\circ$. What is $m(\widehat{CAD})$?
$
\textbf{(A)}\ 25^\circ
\qquad\textbf{(B)}\ 30^\circ
\qquad\textbf{(C)}\ 35^\circ
\qquad\textbf{(D)}\ 38^\circ
\qquad\textbf{(E)}\ 40^\circ
$
Ada and Emily are playing a game that ends when either player wins, after some number of rounds. Each round, either nobody wins, Ada wins, or Emily wins. The probability that neither player wins each round is $\frac{1}{5}$ and the probability that Emily wins the game as a whole is $\frac{3}{4}.$ If the probability that in a given round Emily wins is $\frac{m}{n}$ such that $m$ and $n$ are relatively prime integers, then find $m+n.$
[i]Proposed by Ada Tsui[/i]
Let $ A, B$ and $ C$ be non-collinear points. Prove that there is a unique point $ X$ in the plane of $ ABC$ such that \[ XA^2 \plus{} XB^2 \plus{} AB^2 \equal{} XB^2 \plus{} XC^2 \plus{} BC^2 \equal{} XC^2 \plus{} XA^2 \plus{} CA^2.\]
The population of a small town is $480$. The graph indicates the number of females and males in the town, but the vertical scale-values are omitted. How many males live in the town?
[asy]
draw((0,13)--(0,0)--(20,0));
draw((3,0)--(3,10)--(8,10)--(8,0));
draw((3,5)--(8,5));
draw((11,0)--(11,5)--(16,5)--(16,0));
label("$\textbf{POPULATION}$",(10,11),N);
label("$\textbf{F}$",(5.5,0),S);
label("$\textbf{M}$",(13.5,0),S);
[/asy]
$\text{(A)}\ 120 \qquad \text{(B)}\ 160 \qquad \text{(C)}\ 200 \qquad \text{(D)}\ 240 \qquad \text{(E)}\ 360$
Determine whether there exist distinct real numbers $a, b, c, t$ for which:
[i](i)[/i] the equation $ax^2 + btx + c = 0$ has two distinct real roots $x_1, x_2,$
[i](ii)[/i] the equation $bx^2 + ctx + a = 0$ has two distinct real roots $x_2, x_3,$
[i](iii)[/i] the equation $cx^2 + atx + b = 0$ has two distinct real roots $x_3, x_1.$
For any two convex polygons $P_1$ and $P_2$ with mutually distinct vertices, denote by $f(P_1, P_2)$ the total number of their vertices that lie on a side of the other polygon. For each positive integer $n \ge 4$, determine
\[ \max \{ f(P_1, P_2) ~ | ~ P_1 ~ \text{and} ~ P_2 ~ \text{are convex} ~ n \text{-gons} \}. \]
(We say that a polygon is convex if all its internal angles are strictly less than $180^\circ$.)
[i]Josef Tkadlec (Czech Republic)[/i]
Let $ABCDEF$ be a convex hexagon, such that $FA = AB$, $BC = CD$, $DE = EF$, and $\angle FAB = 2\angle EAC$. Suppose that the area of $ABC$ is $25$, the area of $CDE$ is $10$, the area of $EF A$ is $25$, and the area of $ACE$ is $x$. Find, with proof, all possible values of $x$.
Consider the operation "minus the reciprocal of," defined by $a\diamond b=a-\frac{1}{b}$. What is $((1\diamond2)\diamond3)-(1\diamond(2\diamond3))$?
$\textbf{(A) } -\dfrac{7}{30}
\qquad\textbf{(B) } -\dfrac{1}{6}
\qquad\textbf{(C) } 0
\qquad\textbf{(D) } \dfrac{1}{6}
\qquad\textbf{(E) } \dfrac{7}{30}
$
Let $\Gamma_1$, $\Gamma_2$, $\Gamma_3$, $\Gamma_4$ be distinct circles such that $\Gamma_1$, $\Gamma_3$ are externally tangent at $P$, and $\Gamma_2$, $\Gamma_4$ are externally tangent at the same point $P$. Suppose that $\Gamma_1$ and $\Gamma_2$; $\Gamma_2$ and $\Gamma_3$; $\Gamma_3$ and $\Gamma_4$; $\Gamma_4$ and $\Gamma_1$ meet at $A$, $B$, $C$, $D$, respectively, and that all these points are different from $P$. Prove that
\[
\frac{AB\cdot BC}{AD\cdot DC}=\frac{PB^2}{PD^2}.
\]
We say that the prime numbers $p_1,\dots,p_n$ construct the graph $G$ if we can assign to each vertex of $G$ a natural number whose prime divisors are among $p_1,\dots,p_n$ and there is an edge between two vertices in $G$ if and only if the numbers assigned to the two vertices have a common divisor greater than $1$. What is the minimal $n$ such that there exist prime numbers $p_1,\dots,p_n$ which construct any graph $G$ with $N$ vertices?
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that for all $x,y\in\mathbb{R}$,
$$f(xy+f(x))=f(xf(y))+x$$
Ben flips a coin 10 times and then records the absolute difference between the total number of heads and tails he's flipped. He then flips the coin one more time and records the absolute difference between the total number of heads and tails he's flipped. If the probability that the second number he records is greater than the first can be expressed as $\frac{a}{b}$ for positive integers $a,b$ with $\gcd (a,b) = 1$, then find $a + b$.
[i]Proposed by Vismay Sharan[/i]
Let $f(x)$ and $g(x)$ be mutually inverse decreasing functions on the interval $(0,\infty)$. Can it hold that $f(x)>g(x)$ for all $x\in(0,\infty)$?
If a square is partitioned into $ n$ convex polygons, determine the maximum possible number of edges in the obtained figure.
(You may wish to use the following theorem of Euler: If a polygon is partitioned into $ n$ polygons with $ v$ vertices and $ e$ edges in the resulting figure, then $ v\minus{}e\plus{}n\equal{}1$.)
Let $\overline{ABCD}$ be a $4$-digit number. What is the smallest possible positive value of $\overline{ABCD}- \overline{DCBA}$?