Found problems: 85335
Let $f(x) =x^4+14x^3+52x^2+56x+16.$ Let $z_1, z_2, z_3, z_4$ be the four roots of $f$. Find the smallest possible value of $|z_az_b+z_cz_d|$ where $\{a,b,c,d\}=\{1,2,3,4\}$.
What is the smallest prime number $p$ such that $p^3+4p^2+4p$ has exactly $30$ positive divisors ?
Determine all the pairs of positive integers $(a,b),$ such that $$14\varphi^2(a)-\varphi(ab)+22\varphi^2(b)=a^2+b^2,$$ where $\varphi(n)$ is Euler's totient function.
Let $ABC$ be a triangle with $\angle BAC > 90^o$ and with $AB < AC$. Let $r$ be the internal bisector of $\angle ACB$ and let $s$ be the perpendicular, through $A$, on $r$. Denote by $F$ the intersection of $r$ and $ s$, and denote by $E$ the intersection of $s$ with the segment $BC$. Let also $D$ be the symmetric of $A$ with respect to the line $BF$. Assuming that the circumcircle of triangle $EAC$ is tangent to line $AB$ and $ D$ lies on $r$, determine the value of $\angle CDB$.
Determine all natural numbers $n$ and $k > 1$ such that $k$ divides each of the numbers$\binom{n}{1}$,$\binom{n}{2}$,..........,$\binom{n}{n-1}$
Let triangle $ABC$ be an acute scalene triangle with orthocenter $H$. The foot of perpendicular from $A$ to $BC$ is $O$, and denote $K,L$ by the midpoints of $AB, AC$, respectively. For a point $D(\neq O,B,C)$ on segment $BC$, let $E,F$ be the orthocenters of triangles $ABD, ACD$, respectively, and denote $M,N$ by the midpoints of $DE,DF$. The perpendicular line from $M$ to $KH$ cuts the perpendicular line from $N$ to $LH$ at $P$. If $Q$ is the midpoint of $EF$, and $S$ is the orthocenter of triangle $HPQ$, then prove that as $D$ varies on $BC$, the ratio $\frac{OS}{OH}$, $\frac{OQ}{OP}$ remains constant.
Suppose that $a$ is an even positive integer and $A=a^{n}+a^{n-1}+\ldots +a+1,n\in \mathbb{N^{*}}$ is a perfect square.Prove that $8\mid a$.
Given an isosceles trapezoid $ABCD$. The bisector of angle $B$ intersects the base $AD$ at point $L$. Prove that the center of the circle circumscribed around triangle $BLD$ lies on the circle circumscribed around the trapezoid.
(Yu. Blinkov)
Let $N$ be the number of non-empty subsets $T$ of $S = \{1,2, 3,4,...,2020\}$ satisfying $max (T) >1000$. Compute the largest integer $k$ such that $3^k$ divides $N$.
Let $ABC$ be a triangle. Prove that there is a line $\ell$ (in the plane of triangle $ABC$) such that the intersection of the interior of triangle $ABC$ and the interior of its reflection $A'B'C'$ in $\ell$ has area more than $\frac23$ the area of triangle $ABC$.
The sequence $\{y_{n}\}_{n \ge 1}$ is defined by \[y_{1}=y_{2}=1,\;\; y_{n+2}= (4k-5)y_{n+1}-y_{n}+4-2k.\] Determine all integers $k$ such that each term of this sequence is a perfect square.
Consider a 10-dimensional \(10 \times 10 \times \cdots \times 10 \) cube consisting of \(10^{10}\) unit cubes, such that one cube \(A\) is centered at the origin, and one cube \(B\) is centered at \((9, 9, 9, 9, 9, 9, 9, 9, 9, 9)\). Paint \(A\) red and remove \(B\), leaving an empty space. Let a move consist of taking a cube adjacent to the empty space and placing it into the empty space, leaving the space originally contained by the cube empty. What is the minimum number of moves required to result in a configuration where the cube centered at \((9, 9, 9, 9, 9, 9, 9, 9, 9, 9)\) is red?
Let $K{}$ be an equilateral triangle of unit area, and choose $n{}$ independent random points uniformly from $K{}$. Let $K_n$ be the intersection of all translations of $K{}$ that contain all the selected points. Determine the expected value of the area of $K_n.$
Let $ABCD$ be a rhombus, let $APQC$ be a parallelogram such that the point $B$ lies inside it and the side $AP$ is equal to the side of the rhombus. Prove that $B$ is the orthocenter of the triangle $DPQ$.
Egor Bakaev
It is known that $m$ and $n$ are positive integers, $m > n^{n-1}$, and all the numbers $m+1$, $m+2$, \dots, $m+n$ are composite. Prove that there exist such different primes $p_1$, $p_2$, \dots, $p_n$ that $p_k$ divides $m+k$ for $k = 1$, 2, \dots, $n$.
[i]Proposed by C. A. Grimm [/i]
A plane is cut into unit squares, which are then colored in $n$ colors. A polygon $P$ is created from $n$ unit squares that are connected by their sides. It is known that any cell polygon created by $P$ with translation, covers $n$ unit squares in different colors. Prove that the plane can be covered with copies of $P$ so that each cell is covered exactly once.
Let $f:\mathbb{R} \to \mathbb{R}$ be a continuous function satisfying
\[f(x+2y)=2f(x)f(y)\]
for every $x,y \in \mathbb{R}$. Prove that $f$ is constant.
Given a triangle $ABC$, let $A',B',C'$ be the perpendicular feet dropped from the centroid $G$ of the triangle $ABC$ onto the sides $BC,CA,AB$ respectively. Reflect $A',B',C'$ through $G$ to $A'',B'',C''$ respectively. Prove that the lines $AA'',BB'',CC''$ are concurrent.
For a fixed positive integer $n \geq 3$ we are given a $n$ $\times$ $n$ board with all unit squares initially white. We define a [i]floating plus [/i]as a $5$-tuple $(M,L,R,A,B)$ of unit squares such that $L$ is in the same row and left of $M$, $R$ is in the same row and right of $M$, $A$ is in the same column and above $M$ and $B$ is in the same column and below $M$. It is possible for $M$ to form a floating plus with unit squares that are not next to it. Find the largest positive integer $k$ (depending on $n$) such that we can color some $k$ squares black in such a way that there is no black colored floating plus.
[i]Authored by Nikola Velov[/i]
If $(x, y)$ is a solution to the system
\[ xy=6 \qquad \text{and} \qquad x^2y+xy^2+x+y=63, \]
find $x^2+y^2.$
$ \textbf{(A)}\ 13 \qquad\textbf{(B)}\ \frac{1173}{32} \qquad\textbf{(C)}\ 55 \qquad\textbf{(D)}\ 69 \qquad\textbf{(E)}\ 81 $
How many ordered pairs of integers $(x, y)$ satisfy
\[8(x^3+x^2y+xy^2+y^3) = 15(x^2+y^2+xy+1)?\]
A group of people is divided over two busses in such a way that there are as many seats in total as people. The chance that two friends are seated on the same bus is $\frac{1}{2}$.
a) Show that the number of people in the group is a square.
b) Show that the number of seats on each bus is a triangular number.
In a country with $5$ distinct cities, there may or may not be a road between each pair of cities. It’s possible to get from any city to any other city through a series of roads, but there is no set of three cities $\{A,B,C\}$ such that there are roads between $A$ and $B$, $B$ and $C$, and $C$ and $A$. How many road systems between the five cities are possible?
The Occidentalia bank issues coins with denominations of $1$ peso, $8$ pesos, $27$ pesos... and any amount that is a perfect cube ($n^3$) of pesos. Determine what is the least amount $k$ of coins needed to give $2017$ pesos. For that amount, find all the possible ways to give $2017$ pesos using exactly $k$ currency.
Let $ABC$ be a scalene triangle and $I$ be its incenter. Suppose the incircle $\omega$ touches $BC$ at a point $D$, and $N$ lies on $\omega$ such that $ND$ is a diameter of $\omega$. Let $X$ and $Y$ be points on lines $AC$ and $AB$ respectively such that $\angle BIX = \angle CIY = 90^\circ$. Let $V$ be the feet of perpendicular from $I$ onto line $XY$. Prove that the points $I$, $V$, $A$, $N$ are concyclic.
[i](Proposed by Ivan Chan Guan Yu)[/i]