Found problems: 85335
Side lengths $AB,BC,CD,AD$ of convex quadrilateral $ABCD$ are equal $16,13,14,17$ respectively. Circles $w_1,w_2,w_3,w_4$ are drawn with centers $A,B,C,D$ and radii $2,6,3,9$ respectively. Common external tangents to circles $w_1,w_2$; $w_2,w_3$; $w_3,w_4$; $w_4,w_1$ intersect at $A_1,B_1,C_1,D_1$ respectively.
Prove that lines $AA_1,BB_1,CC_1,DD_1$ are concurrent.
[i]Aliaksei Vaidzelevich[/i]
Call a polynomial $p(x)$ with positive integer roots [i]corrupt[/i] if there exists an integer that cannot be expressed as a sum of (not necessarily positive) multiples of its roots. The polynomial $A(x)$ is monic, corrupt, and has distinct roots. As well, $A(0)$ has $7$ positive divisors. Find the least possible value of $|A(1)|$.
Let positive integers $a,\ b,\ c$ be pairwise coprime. Denote by $g(a, b, c)$ the maximum integer not representable in the form $xa+yb+zc$ with positive integral $x,\ y,\ z$. Prove that
\[ g(a, b, c)\ge \sqrt{2abc}\]
[i](M. Ivanov)[/i]
[hide="Remarks (containing spoilers!)"]
1. It can be proven that $g(a,b,c)\ge \sqrt{3abc}$.
2. The constant $3$ is the best possible, as proved by the equation $g(3,3k+1,3k+2)=9k+5$.
[/hide]
Proce that number
$$\underbrace{11...11}_{2n \,\, digits}-\underbrace{22 ... 22}_{n \,\, digits}$$
is, for every natural $n$, a perfect square.
Given a set of $n>2$ planar vectors. A vector from this set is called [i]long[/i], if its length is not less than the length of the sum of other vectors in this set. Prove that if each vector is long, then the sum of all vectors equals to zero.
[i]N. Agakhanov[/i]
Consider the acute-angled triangle $ABC$, with orthocentre $H$ and circumcentre $O$. $D$ is the intersection point of lines $AH$ and $BC$ and $E$ lies on $\overline{AH}$ such that $AE=DH$.
Suppose $EO$ and $BC$ meet at $F$. Prove that $BD=CF$.
[i] (Călin Pop & Vlad Robu) [/i]
Prove that for $N>1$ that $(N^{2})^{2014} - (N^{11})^{106}$ is divisible by $N^6 + N^3 +1$
Is this just a proof by induction or is there a more elegant method? I don't think calculating $N = 2$ was expected.
Diagonals of a quadrilateral $ABCD$ are equal and meet at point $O$. The perpendicular bisectors to segments $AB$ and $CD$ meet at point $P$, and the perpendicular bisectors to $BC$ and $AD$ meet at point $Q$. Find angle $\angle POQ$.
by A.Zaslavsky
Suppose that $ f(x\plus{}3)\equal{}3x^2\plus{}7x\plus{}4$ and $ f(x)\equal{}ax^2\plus{}bx\plus{}c$. What is $ a\plus{}b\plus{}c$?
$ \textbf{(A)}\minus{}\!1 \qquad
\textbf{(B)}\ 0 \qquad
\textbf{(C)}\ 1 \qquad
\textbf{(D)}\ 2 \qquad
\textbf{(E)}\ 3$
Let $x_1$ and $x_2$ be roots of the equation $x^2 - 6x + 1 = 0$. Prove that for any integer $n \ge 1$ the number $x_1^n + x_2^n$ is integer and is not divisible by $5$.
A class has $25$ students. The teacher wants to stock $N$ candies, hold the Olympics and give away all $N$ candies for success in it (those who solve equally tasks should get equally, those who solve less get less, including, possibly, zero candies). At what smallest $N$ this will be possible, regardless of the number of tasks on Olympiad and the student successes?
Let $ A_1,B_1,C_1 $ be the feet of the heights of an acute triangle $ ABC. $ On the segments $ B_1C_1,C_1A_1,A_1B_1, $ take the points $ X,Y, $ respectively, $ Z, $ such that
$$ \left\{\begin{matrix}\frac{C_1X}{XB_1} =\frac{b\cos\angle BCA}{c\cos\angle ABC} \\ \frac{A_1Y}{YC_1} =\frac{c\cos\angle BAC}{a\cos\angle BCA} \\ \frac{B_1Z}{ZA_1} =\frac{a\cos\angle ABC}{b\cos\angle BAC} \end{matrix}\right. . $$
Show that $ AX,BY,CZ, $ are concurrent.
Let $P$ be a point inside a regular tetrahedron ABCD with edge length $1$. Show that $$d(P,AB)+d(P,AC)+d(P,AD)+d(P,BC)+d(P,BD)+d(P,CD) \ge \frac{3}{2} \sqrt2$$ , with equality only when $P$ is the centroid of $ABCD$.
Here $d(P,XY)$ denotes the distance from point $P$ to line $XY$.
If the ratio of the legs of a right triangle is $ 1: 2$, then the ratio of the corresponding segments of the hypotenuse made by a perpendicular upon it from the vertex is:
$ \textbf{(A)}\ 1: 4 \qquad
\textbf{(B)}\ 1: \sqrt{2} \qquad
\textbf{(C)}\ 1: 2 \qquad
\textbf{(D)}\ 1: \sqrt{5} \qquad
\textbf{(E)}\ 1: 5$
A chess-board $6\times 6$ is tiled with the $2\times 1$ dominos. Prove that you can cut the board onto two parts by a straight line that does not cut dominos.
In a triangle $\triangle ABC$ with$\angle B>\angle C$, the altitude, the angle bisector, and the median from $A$ intersect $BC$ at $H, L$ and $D$, respectively. Show that $\angle HAL=\angle DAL$ if and only if $\angle BAC=90^{\circ}$.
A family of sets $F$ is called perfect if the following condition holds: For every triple of sets $X_1, X_2, X_3\in F$, at least one of the sets $$ (X_1\setminus X_2)\cap X_3,$$ $$(X_2\setminus X_1)\cap X_3$$ is empty. Show that if $F$ is a perfect family consisting of some subsets of a given finite set $U$, then $\left\lvert F\right\rvert\le\left\lvert U\right\rvert+1$.
[i]Proposed by Michał Pilipczuk[/i]
Let $S$ be the set of all rational numbers $x \in [0, 1]$ with repeating base $6$ expansion $$x = 0.\overline{a_1a_2 ... a_k} = 0.a_1a_2...a_ka_1a_2...a_k...$$ for some finite sequence $\{a_i\}^{k}_{i=1}$ of distinct nonnegative integers less than $6$. What is the sum of all numbers that can be written in this form? (Put your answer in base $10$.)
Take $r$ such that $1\le r\le n$, and consider all subsets of $r$ elements of the set $\{1,2,\ldots,n\}$. Each subset has a smallest element. Let $F(n,r)$ be the arithmetic mean of these smallest elements. Prove that: \[ F(n,r)={n+1\over r+1}. \]
There are $100$ numbers from $(0,1)$ on the board. On every move we replace two numbers $a,b$ with roots of $x^2-ax+b=0$(if it has two roots). Prove that process is not endless.
Two ants sit at the vertex of the parabola $y = x^2$. One starts walking northeast (i.e., upward along the line $y = x$ and the other starts walking northwest (i.e., upward along the line $y = -x$). Each time they reach the parabola again, they swap directions and continue walking. Both ants walk at the same speed. When the ants meet for the eleventh time (including the time at the origin), their paths will enclose 10 squares. What is the total area of these squares?
Let $ABC$ be an acute triangle and $M$ be the midpoint of $AB$. A circle through the points $B$ and $C$ intersects the segments $CM$ and $BM$ at points $P$ and $Q$ respectively. Point $K$ is symmetric to $P$ with respect to point $M$. The circumcircles of $\triangle AKM$ and $\triangle CQM$ intersect for the second time at $X$. The circumcircles of $\triangle AMC$ and $\triangle KMQ$ intersect for the second time at $Y$. The segments $BP$ and $CQ$ intersect at point $T$. Prove that the line $MT$ is tangent to the circumcircle of $\triangle MXY$.
We say a finite set $S$ of points in the plane is [i]very[/i] if for every point $X$ in $S$, there exists an inversion with center $X$ mapping every point in $S$ other than $X$ to another point in $S$ (possibly the same point).
(a) Fix an integer $n$. Prove that if $n \ge 2$, then any line segment $\overline{AB}$ contains a unique very set $S$ of size $n$ such that $A, B \in S$.
(b) Find the largest possible size of a very set not contained in any line.
(Here, an [i]inversion[/i] with center $O$ and radius $r$ sends every point $P$ other than $O$ to the point $P'$ along ray $OP$ such that $OP\cdot OP' = r^2$.)
[i]Proposed by Sammy Luo[/i]
For given positive integers $n$ and $p$, find neaessary and sufficient conditions for the system of equations
$$x + py = n , \\ x + y = p^2$$
to have a solution $(x, y, z)$ of positive integers. Prove also that there is at most one such solution.
(a) On a long pavement, a sequence of $999$ integers is written in chalk. The numbers need not be in increasing order and need not be distinct. Merlijn encircles $500$ of the numbers with red chalk. From left to right, the numbers circled in red are precisely the numbers $1, 2, 3, ...,499, 500$. Next, Jeroen encircles $500$ of the numbers with blue chalk. From left to right, the numbers circled in blue are precisely the numbers $500, 499, 498, ...,2,1$.
Prove that the middle number in the sequence of $999$ numbers is circled both in red and in blue.
(b) Merlijn and Jeroen cross the street and find another sequence of $999$ integers on the pavement. Again Merlijn circles $500$ of the numbers with red chalk. Again the numbers circled in red are precisely the numbers $1, 2, 3, ...,499, 500$ from left to right. Now Jeroen circles $500$ of the numbers, not necessarily the same as Merlijn, with green chalk. The numbers circled in green are also precisely the numbers $1, 2, 3, ...,499, 500$ from left to right.
Prove: there is a number that is circled both in red and in green that is not the middle number of the sequence of $999$ numbers.