Found problems: 85335
Find all positive integers $n$ such that the set $\{n,n+1,n+2,n+3,n+4,n+5\}$ can be partitioned into two subsets so that the product of the numbers in each subset is equal.
Let a point $P$ lie inside a triangle $ABC$. The rays starting at $P$ and crossing the sides $BC$, $AC$, $AB$ under the right angle meet the circumcircle of $ABC$ at $A_{1}$, $B_{1}$, $C_{1}$ respectively. It is known that lines $AA_{1}$, $BB_{1}$, $CC_{1}$ concur at point $Q$. Prove that all such lines $PQ$ concur.
Suppose $\triangle ABC$ has $D$ as the midpoint of $BC$ and orthocenter $H$. Let $P$ be an arbitrary point on the nine point circle of $ABC$. The line through $P$ perpendicular to $AP$ intersects $BC$ at $Q$. The line through $A$ perpendicular to $AQ$ intersects $PQ$ at $X$. If $M$ is the midpoint of $AQ$, show that $HX \perp DM$.
How many pairs $(a, b)$ for integers $1 \le a, b \le 2015$ which satisfy that $a$ is divisible by $b + 1$ and $2016 - a$ is divisible by $b$.
Let $CD$ be a chord of a circle $\Gamma_1$ and $AB$ a diameter of $\Gamma_1$ perpendicular to $CD$ at $N$ with $AN > NB$. A circle $\Gamma_2$ centered at $C$ with radius $CN$ intersects $\Gamma_1$ at points $P$ and $Q$. The line $PQ$ intersects $CD$ at $M$ and $AC$ at $K$; and the extension of $NK$ meets $\Gamma_2$ at $L$. Prove that $PQ$ is perpendicular to $AL$
Let $n$ be a positive integer. Let $T_n$ be a set of positive integers such that:
\[{T_n={ \{11(k+h)+10(n^k+n^h)| (1 \leq k,h \leq 10)}}\}\]
Find all $n$ for which there don't exist two distinct positive integers $a, b \in T_n$ such that $a\equiv b \pmod{110}$
Let $a$ and $b$ be distinct positive real numbers, such that $a -\sqrt{ab}$ and $b -\sqrt{ab}$ are both rational numbers. Prove that $a$ and $b$ are rational numbers.
A semicircle has diameter $\overline{AD}$ with $AD = 30$. Points $B$ and $C$ lie on $\overline{AD}$, and points $E$ and $F$ lie on the arc of the semicircle. The two right triangles $\vartriangle BCF$ and $\vartriangle CDE$ are congruent. The area of $\vartriangle BCF$ is $m\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m + n$.
[img]https://cdn.artofproblemsolving.com/attachments/b/c/c10258e2e15cab74abafbac5ff50b1d0fd42e6.png[/img]
Given points $A$, $B$, $C$, and $D$ on circle $\omega$ such that lines $AB$ and $CD$ intersect on point $T$ where $A$ is between $B$ and $T$, moreover $D$ is between $C$ and $T$. It is known that the line passing through $D$ which is parallel to $AB$ intersects $\omega$ again on point $E$ and line $ET$ intersects $\omega$ again on point $F$. Let $CF$ and $AB$ intersect on point $G$, $X$ be the midpoint of segment $AB$, and $Y$ be the reflection of point $T$ to $G$.
Prove that $X$, $Y$, $C$, and $D$ are concyclic.
A number is written in each corner of the cube. On each step, each number is replaced with the average of three numbers in the three adjacent corners (all the numbers are replaced simultaneously). After ten such steps, every number returns to its initial value. Must all numbers have been originally equal?
[i](3 points)[/i]
Let $ \sigma_1 ,\sigma_2 $ be two permutations of order $ n $ such that $ \sigma_1 (k)=\sigma_2 (n-k+1) $ for $ k=\overline{1,n} . $ Prove that the number of inversions of $ \sigma_1 $ plus the number of inversions of $ \sigma_2 $ is $ \frac{n(n+1)}{2} . $
Let $a_{1}, \ldots, a_{n}$ be an infinite sequence of strictly positive integers, so that $a_{k} < a_{k+1}$ for any $k.$ Prove that there exists an infinity of terms $ a_{m},$ which can be written like $a_m = x \cdot a_p + y \cdot a_q$ with $x,y$ strictly positive integers and $p \neq q.$
Prove that for any positive integer $n$ and any real numbers $a_1,a_2,\cdots,a_n,b_1,b_2,\cdots,b_n$ we have that the equation
\[a_1 \sin(x) + a_2 \sin(2x) +\cdots+a_n\sin(nx)=b_1 \cos(x)+b_2\cos(2x)+\cdots +b_n \cos(nx)\]
has at least one real root.
The number $A$ is a product of $n$ distinct natural numbers. Prove that $A$ has at least $\frac{n(n-1)}{2}+1$ distinct divisors (including 1 and $A$).
Find all triples $(a,b,c)$ of real numbers all different from zero that satisfies:
\begin{eqnarray} a^4+b^2c^2=16a\nonumber \\ b^4+c^2a^2=16b \nonumber\\ c^4+a^2b^2=16c \nonumber \end{eqnarray}
If $ \triangle ABC$ is inscribed in a semicircle whose diameter is $ AB$, then $ AC \plus{} BC$ must be
$ \textbf{(A)}\ \text{equal to }AB \qquad\textbf{(B)}\ \text{equal to }AB\sqrt {2} \qquad\textbf{(C)}\ \geq AB\sqrt {2}$
$ \textbf{(D)}\ \leq AB\sqrt {2} \qquad\textbf{(E)}\ AB^2$
Let square $ABCD$ and circle $\Omega$ be on the same plane, and $AA'$, $BB'$, $CC'$, $DD'$ be tangents to $\Omega$. Let $WXY Z$ be a convex quadrilateral with side lengths $WX = AA'$, $XY = BB'$, $Y Z = CC'$, and $ZW = DD'$. If $WXY Z$ has an inscribed circle, prove that the diagonals $WY$ and $XZ$ are perpendicular to each other.
On a blackboard the product $log_{( )}[ ]\times\dots\times log_{( )}[ ]$ is written (there are 50 logarithms in the product). Donald has $100$ cards: $[2], [3],\dots, [51]$ and $(52),\dots,(101)$. He is replacing each $()$ with some card of form $(x)$ and each $[]$ with some card of form $[y]$. Find the difference between largest and smallest values Donald can achieve.
Determine all positive integers $n$ such that numbers from $1$ to $n$ can be sorted in some order $x_1,x_2,...,x_n$ with the property that the number $x_1+x_2+...+x_k$ is divisible by $k$, for all $1\le k\le n$., that is $1$ is divides $x_1$, $2$ divides $x_1+x_2$, $3$ divides $x_1+x_2+x_3$, and so on until $n$ divides $x_1+x_2+...+x_n$.
For each non-negative $a$, consider the equation $$x^3 + ax - a^3 - 29 = 0.$$ Let $x_o$ be the positive root of this equation. Prove that for all $a > 0$ such a root exists. What is the smallest value of $x_o$?
Let $a$ and $b$ be two complex numbers. Prove the inequality
$$|1 + ab| + |a + b| \ge \sqrt{|a^2 - 1| \cdot |b^2 - 1|}$$
Let $p$ be an odd prime number. Show that the smallest positive quadratic nonresidue of $p$ is smaller than $\sqrt{p}+1$.
Find the largest divisor of $2800$ that is a perfect square.
Let $n\ge 2$ is a given integer , $x_1,x_2,\ldots,x_n $ be real numbers such that
$(1) x_1+x_2+\ldots+x_n=0 $,
$(2) |x_i|\le 1$ $(i=1,2,\cdots,n)$.
Find the maximum of Min$\{|x_1-x_2|,|x_2-x_3|,\cdots,|x_{n-1}-x_n|\}$.
Does the set of real numbers have a well-order $\prec$ such that the intersection of the subset $\{(x,y) : x\prec y\}$ of the plane with every line is Lebesgue measurable on the line?