Found problems: 85335
Prove that there exists a sequence of $100$ different integers such that the sum of the squares of any two consecutive terms is a perfect square.
(S Tokarev)
Let $p$ be a prime such that $\frac{p-1}{2}$ is also prime. A pair of integers $(x, y)$ with $1\le x, y \le p-1$ is called a [i]commuter[/i] if at least one of $x^y -y^x$ or $x^y +y^x$ is divisible by $p$. Show that the number of commuters is at most $4.2p\sqrt{p}$.
Find all real numbers $a,$ which satisfy the following condition:
For every sequence $a_1,a_2,a_3,\ldots$ of pairwise different positive integers, for which the inequality $a_n\leq an$ holds for every positive integer $n,$ there exist infinitely many numbers in the sequence with sum of their digits in base $4038,$ which is not divisible by $2019.$
Determine the natural number $a =\frac{p+q}{r}+\frac{q+r}{p}+\frac{r+p}{q}$ where $p, q$ and $r$ are prime positive numbers.
Let $f(x) = x^{1998} - x^{199}+x^{19}+ 1$. Prove that there is an infinite set of prime numbers, each dividing at least one of the integers $f(1), f(2), f(3), f(4), ...$
The tangents at four different points of an arc of a circle less than $180^o$ intersect forming a convex quadrilateral $ABCD$. Prove that two of the vertices belong to an ellipse whose foci to the other two vertices.
Let $f$ be the cubic polynomial
\[
f(x) = x^3 + bx^2 + cx + d,
\]
where $b$, $c$, and $d$ are real numbers. Let $x_1$, $x_2$, $\ldots\,$, $x_n$ be nonnegative numbers, and let $m$ be their average. Suppose that $m \ge - \dfrac{b}{2}\,$. Prove that
\[
\sum_{i = 1}^n f(x_i) \ge n f(m).
\]
(a) Prove that $p^2 + q^2 + 1 > p(q + 1)$ for any real numbers $p, q$, .
(b) Determine the largest real constant $b$ such that the inequality $p^2 + q^2 + 1 \ge bp(q + 1)$ holds for all real numbers $p, q$
(c) Determine the largest real constant c such that the inequality $p^2 + q^2 + 1 \ge cp(q + 1)$ holds for all integers $p, q$.
Let $a,b,c$ be positive reals satisfying :
\[ \frac{a}{1+b+c}+\frac{b}{1+c+a}+\frac{c}{1+a+b}\ge \frac{ab}{1+a+b}+\frac{bc}{1+b+c}+\frac{ca}{1+c+a} \]
Then prove that :
\[ \frac{a^2+b^2+c^2}{ab+bc+ca}+a+b+c+2\ge 2(\sqrt{ab}+\sqrt{bc}+\sqrt{ca}) \]
[i]Proposed by Dimitar Trenevski[/i]
Yesterday, $n\ge 4$ people sat around a round table. Each participant remembers only who his two neighbours were, but not necessarily which one sat on his left and which one sat on his right. Today, you would like the same people to sit around the same round table so that each participant has the same two neighbours as yesterday (it is possible that yesterday’s left-hand side neighbour is today’s right-hand side neighbour). You are allowed to query some of the participants: if anyone is asked, he will answer by pointing at his two neighbours from yesterday.
a) Determine the minimal number $f(n)$ of participants you have to query in order to be certain to succeed, if later questions must not depend on the outcome of the previous questions. That is, you have to choose in advance the list of people you are going to query, before effectively asking any question.
b) Determine the minimal number $g(n)$ of participants you have to query in order to be certain to succeed, if later questions may depend on the outcome of previous questions. That is, you can wait until you get the first answer to choose whom to ask the second question, and so on.
Determine all $f:\mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ such that $f(m)\geq m$ and $f(m+n) \mid f(m)+f(n)$ for all $m,n\in \mathbb{Z}^+$
Find all positive integers $l$ for which the equation
\[
a^3+b^3+ab=(lab+1)(a+b)
\]
has a solution over positive integers $a,b$.
Let \(L_3 = \{3\}\), \(L_n = \{3, 4, \ldots, h\}\) (where \(h > 3\)). For any given integer \(n \geq 3\), consider a graph \(G\) with \(n\) vertices that contains a Hamiltonian cycle \(C\) and has more than \(\frac{n^2}{4}\) edges. For which lengths \(l \in L_n\) must the graph \(G\) necessarily contain a cycle of length \(l\)?
Point $A,B,C,D,$ and $E$ are equally spaced on a minor arc of a circle. Points $E,F,G,H,I$ and $A$ are equally spaced on a minor arc of a second circle with center $C$ as shown in the figure below. The angle $\angle ABD$ exceeds $\angle AHG$ by $12^\circ$. Find the degree measure of $\angle BAG$.[asy]
pair A,B,C,D,E,F,G,H,I,O;
O=(0,0);
C=dir(90);
B=dir(70);
A=dir(50);
D=dir(110);
E=dir(130);
draw(arc(O,1,50,130));
real x=2*sin(20*pi/180);
F=x*dir(228)+C;
G=x*dir(256)+C;
H=x*dir(284)+C;
I=x*dir(312)+C;
draw(arc(C,x,200,340));
label("$A$",A,dir(0));
label("$B$",B,dir(75));
label("$C$",C,dir(90));
label("$D$",D,dir(105));
label("$E$",E,dir(180));
label("$F$",F,dir(225));
label("$G$",G,dir(260));
label("$H$",H,dir(280));
label("$I$",I,dir(315));
[/asy]
Consider a regular triangular array of $n(n+1)/2$ points.Let $f(n)$ denote the number of equilateral triangles formed by taking some $3$ points in the array as vertices.Prove that
$f(n)=\frac{(n-1)n(n+1)(n+2)}{24}$.
Let $m$ and $n$ be two positive integers for which $m<n$. $n$ distinct points $X_1,\ldots , X_n$ are in the interior of the unit disc and at least one of them is on its border. Prove that we can find $m$ distinct points $X_{i_1},\ldots , X_{i_m}$ so that the distance between their center of gravity and the center of the circle is at least $\frac{1}{1+2m(1- 1/n)}$.
Find the largest real number $\alpha$ such that, for all non-negative real numbers $x$, $y$ and $z$, the following inequality holds:
\[
(x+y+z)^3 + \alpha (x^2z + y^2x + z^2y) \geq \alpha (x^2y + y^2z + z^2x).
\]
Let $ABC$ be a right triangle, with the right angle at $A$. The altitude from $A$ meets $BC$ at $H$ and $M$ is the midpoint of the hypotenuse $[BC]$. On the legs, in the exterior of the triangle, equilateral triangles $BAP$ and $ACQ$ are constructed. If $N$ is the intersection point of the lines $AM$ and $PQ$, prove that the angles $\angle NHP$ and $\angle AHQ$ are equal.
Miguel Ochoa Sanchez and Leonard Giugiuc
Let $M$, $N$ and $P$ be three non-collinear points. Construct using straight edge and compass a triangle for which $M$ and $N$ are the midpoints of two of its sides, and $P$ is its orthocenter.
Let $a,b,c>0$ be real numbers with sum 1. Prove that \[ \frac{a^2}b + \frac{b^2}c + \frac{c^2} a \geq 3(a^2+b^2+c^2) . \]
Determine all pairs of integers $(x, y)$ satisfying the equation
\[y(x + y) = x^3- 7x^2 + 11x - 3.\]
Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$.
Prove that Sisyphus cannot reach the aim in less than
\[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \]
turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )
In an acute traingle $ABC$ with $AB< BC$ let $BH_b$ be its altitude, and let $O$ be the circumcenter. A line through $H_b$ parallel to $CO$ meets $BO$ at $X$. Prove that $X$ and the midpoints of $AB$ and $AC$ are collinear.
Angle $A$ of triangle $ABC$ is $33^o$. A straight line passing through $A$ perpendicular to $AC$ intersects straight line $BC$ at point $D$ so that $CD = 2AB$. What is angle $C$ of triangle $ABC$? (Please list all options.)
The following figure shows a $3^2 \times 3^2$ grid divided into $3^2$ subgrids of size $3 \times 3$. This grid has $81$ cells, $9$ in each subgrid.
[asy]
draw((0,0)--(9,0)--(9,9)--(0,9)--cycle, linewidth(2));
draw((0,1)--(9,1));
draw((0,2)--(9,2));
draw((0,3)--(9,3), linewidth(2));
draw((0,4)--(9,4));
draw((0,5)--(9,5));
draw((0,6)--(9,6), linewidth(2));
draw((0,7)--(9,7));
draw((0,8)--(9,8));
draw((1,0)--(1,9));
draw((2,0)--(2,9));
draw((3,0)--(3,9), linewidth(2));
draw((4,0)--(4,9));
draw((5,0)--(5,9));
draw((6,0)--(6,9), linewidth(2));
draw((7,0)--(7,9));
draw((8,0)--(8,9));
[/asy]
Now consider an $n^2 \times n^2$ grid divided into $n^2$ subgrids of size $n \times n$. Find the number of ways in which you can select $n^2$ cells from this grid such that there is exactly one cell coming from each subgrid, one from each row and one from each column.