Found problems: 85335
Find all $(a, b, c, d)\in \mathbb{R}$ satisfying
\[\begin{aligned}
\begin{cases}
a+b+c+d=0,\\
a^2+b^2+c^2+d^2=12,\\
abcd=-3.\\
\end{cases}
\end{aligned}\]
In the acute triangle $ABC$ the point $M$ is on the side $BC$. The inscribed circle of triangle $ABM$ touches the sides $BM$, $MA$ and $AB$ in points $D$, $E$ and $F$, and the inscribed circle of triangle $ACM$ touches the sides $CM$, $MA$ and $AC$ in points $X$, $Y$ and $Z$. The lines $FD$ and $ZX$ intersect in point $H$. Prove that lines $AH$, $XY$ and $DE$ are concurrent.
For every $a \in \mathbb N$ denote by $M(a)$ the number of elements of the set
\[ \{ b \in \mathbb N | a + b \text{ is a divisor of } ab \}.\]
Find $\max_{a\leq 1983} M(a).$
On the sides $AB$ and $AD$ of the square $ABCD$, the points $N$ and $P$ are selected respectively such that $NC=NP$. The point $Q$ is chosen on the segment $AN$ so that $\angle QPN = \angle NCB$. Prove that $2\angle BCQ = \angle AQP$.
Find all polynomials $P\in \mathbb C[X]$ such that \[P(X^{2})=P(X)^{2}+2P(X)\]
Let $n, k, l$ be positive integers satisfying $n \ge 3$, $l \le n - 2, l - k \le \frac{n-3}{2}$. Suppose that $a_1, a_2, \dots, a_k$ are integers chosen from $\{1, 2, \dots, n\}$ such that the set of remainders of the subset sums over all subsets of $a_i$ when divided by $n$ is exactly $\{1, 2, \dots, l\}$. Show that \[ a_1 + a_2 + \dots + a_k = l. \]
Let $k$ be a positive integer, let $z_1,z_2, \ldots, z_k \in \mathbb{C}$ be distinct and let $u_1,u_2,\ldots,u_k \in \mathbb{C}$ be such that the set $\big\{a_n=u_1z_1^n+u_2z_2^n+\ldots+u_kz_k^n : n \in \mathbb{Z}_{>0} \big\}$ is finite. Prove that there exists a positive integer $p$ such that $a_n=a_{n+p},$ for any positive integer $n.$
Let points $A, B, C, A', B', C'$ be chosen in the plane such that no three of them are collinear, and let lines $AA'$, $BB'$ and $CC'$ be tangent to a given equilateral hyperbola at points $A$, $B$ and $C$, respectively. Assume that the circumcircle of $A'B'C'$ is the same as the nine-point circle of triangle $ABC$. Let $s(A')$ be the Simson line of point $A'$ with respect to the orthic triangle of $ABC$. Let $A^*$ be the intersection of line $B'C'$ and the perpendicular on $s(A')$ from the point $A$. Points $B^*$ and $C^*$ are defined in a similar manner. Prove that points $A^*$, $B^*$ and $C^*$ are collinear.
[i]Submitted by Áron Bán-Szabó, Budapest[/i]
a) Find all integer pairs $(x,y)$ satisfying $x^4+(y+2)^3=(x+2)^4.$
b) Prove that: if $p$ is a prime of the form $p=4k+3$ $(k$ is a non-negative number$),$ then there doesn's exist $p-1$ consecutive non-negative integers such that we can divide the set of these numbers into $2$ distinct subsets so that the product of all the numbers in one subset is equal to that in the remained subset.
For an integer $n \geq 3$, let $\mathcal M$ be the set $\{(x, y) | x, y \in \mathbb Z, 1 \leq x \leq n, 1 \leq y \leq n\}$ of points in the plane.
What is the maximum possible number of points in a subset $S \subseteq \mathcal M$ which does not contain three distinct points being the vertices of a right triangle?
There are some checkers in $n\cdot n$ size chess board.Known that for all numbers $1\le i,j\le n$ if checkwork in the intersection of $i$ th row and $j$ th column is empty,so the number of checkers that are in this row and column is at least $n$.Prove that there are at least $\frac{n^2}{2}$ checkers in chess board.
The sequence $(a_n)$ is defined by $a_1=1,a_2=\frac{1}{2}$,$$n(n+1)
a_{n+1}a_{n}+na_{n}a_{n-1}=(n+1)^2a_{n+1}a_{n-1}(n\ge 2).$$
Prove that $$\frac{2}{n+1}<\sqrt[n]{a_n}<\frac{1}{\sqrt{n}}(n\ge 3).$$
If $\frac{2+4+6}{1+3+5}-\frac{1+3+5}{2+4+6} = \frac{m}{n}$ for relatively prime integers $m$ and $n$, compute $100m+n$.
[i]Proposed by Evan Chen[/i]
Let $ABCD$ be a convex quadrilateral. Construct the points $P,Q,R,$ and $S$ on continue of $AB,BC,CD,$ and $DA$, respectively, such that
\[BP=CQ=DR=AS.\]
Show that if $PQRS$ is a square, then $ABCD$ is also a square.
Let $f$ be a function $f: \mathbb{N} \cup \{0\} \mapsto \mathbb{N},$ and satisfies the following conditions:
(1) $f(0) = 0, f(1) = 1,$
(2) $f(n+2) = 23 \cdot f(n+1) + f(n), n = 0,1, \ldots.$
Prove that for any $m \in \mathbb{N}$, there exist a $d \in \mathbb{N}$ such that $m | f(f(n)) \Leftrightarrow d | n.$
Let $V$ be the set of non-collinear pairs of vectors in $\mathbb{R}^3$, and $H$ be the set of lines passing through the origin. Is is true that for every continuous map $f\colon V\rightarrow H$ there exists a continuous map $g\colon V\rightarrow \mathbb{R}^3\,\backslash\,\{ 0\}$ such that $g(v)\in f(v)$ for all $v\in V$?
(translated by Miklós Maróti)
Oliver is at a carnival. He is offered to play a game where he rolls a fair dice and receives $\$1$ if his roll is a $1$ or $2$, receives $\$2$ if his roll is a $3$ or $4$, and receives $\$3$ if his roll is a $5$ or $6$. Oliver plays the game repeatedly until he has received a total of at least $\$2$. What is the probability that he ends with $\$3$?
Let $ S$ be a set that contains $ n$ elements. Let $ A_{1},A_{2},\cdots,A_{k}$ be $ k$ distinct subsets of $ S$, where $ k\geq 2, |A_{i}| \equal{} a_{i}\geq 1 ( 1\leq i\leq k)$. Prove that the number of subsets of $ S$ that don't contain any $ A_{i} (1\leq i\leq k)$ is greater than or equal to $ 2^n\prod_{i \equal{} 1}^k(1 \minus{} \frac {1}{2^{a_{i}}}).$
Let $ABCD$ be a cyclic quadrilateral. Prove that there exists a point $X$ on segment $\overline{BD}$ such that $\angle BAC=\angle XAD$ and $\angle BCA=\angle XCD$ if and only if there exists a point $Y$ on segment $\overline{AC}$ such that $\angle CBD=\angle YBA$ and $\angle CDB=\angle YDA$.
A tree with $n\geq 2$ vertices is given. (A tree is a connected graph without cycles.) The vertices of the tree have real numbers $x_1,x_2,\dots,x_n$ associated with them. Each edge is associated with the product of the two numbers corresponding to the vertices it connects. Let $S$ be a sum of number across all edges. Prove that \[\sqrt{n-1}\left(x_1^2+x_2^2+\dots+x_n^2\right)\geq 2S.\]
(Author: V. Dolnikov)
Study if it there exist an strictly increasing sequence of integers $0=a_0<a_1<a_2<...$ satisfying the following conditions
$i)$ Any natural number can be written as the sum of two terms of the sequence (not necessarily distinct).
$ii)$For any positive integer $n$ we have $a_n > \frac{n^2}{16}$
An equilateral pentagon $AMNPQ$ is inscribed in triangle $ABC$ such that $M\in\overline{AB}$, $Q\in\overline{AC}$, and $N,P\in\overline{BC}$.
Suppose that $ABC$ is an equilateral triangle of side length $2$, and that $AMNPQ$ has a line of symmetry perpendicular to $BC$. Then the area of $AMNPQ$ is $n-p\sqrt{q}$, where $n, p, q$ are positive integers and $q$ is not divisible by the square of a prime. Compute $100n+10p+q$.
[i]Proposed by Michael Ren[/i]
The population of the United States in 1980 was $226,504,825$. The area of the country is $3,615,122$ square miles. The are $(5280)^{2}$ square feet in one square mile. Which number below best approximates the average number of square feet per person?
$ \textbf{(A)}\ 5,000\qquad\textbf{(B)}\ 10,000\qquad\textbf{(C)}\ 50,000\qquad\textbf{(D)}\ 100,000\qquad\textbf{(E)}\ 500,000 $
A white plane is partitioned onto cells (in a usual way). A finite number of cells are coloured black. Each black cell has an even (0, 2 or 4) adjacent (by the side) white cells. Prove that one may colour each white cell in green or red such that every black cell will have equal number of red and green adjacent cells.
Find the smallest number with exactly 28 divisors.