Found problems: 85335
A chicken breeder went to check what price per chick he had charged the previous year. He found an invoice, half erased, which read: $72$ chickens sold for $*679*$ escudos” (the digits of the units and tens of thousands were illegible). What price did each chick sell for last year?
Find all primes $p$ such that $4p^2+1$ and $6p^2+1$ are both primes.
Prove for all positive real numbers $m,n,p,q$ that
$$\frac{m}{t+n+p+q} + \frac{n}{t+p+q+m} + \frac{p}{t+q+m+n} + \frac{q}{t+m+n+p} \geq \frac{4}{5},$$
where $t=\frac{m+n+p+q}{2}.$
Let $f:[-\pi/2,\pi/2]\to\mathbb{R}$ be a twice differentiable function which satisfies \[\left(f''(x)-f(x)\right)\cdot\tan(x)+2f'(x)\geqslant 1,\]for all $x\in(-\pi/2,\pi/2)$. Prove that \[\int_{-\pi/2}^{\pi/2}f(x)\cdot \sin(x) \ dx\geqslant \pi-2.\]
Let $ABC$ be a cute triangle.$(O)$ is circumcircle of $\triangle ABC$.$D$ is on arc $BC$ not containing $A$.Line $\triangle$ moved through $H$($H$ is orthocenter of $\triangle ABC$ cuts circumcircle of $\triangle ABH$,circumcircle $\triangle ACH$ again at $M,N$ respectively.
a.Find $\triangle$ satisfy $S_{AMN}$ max
b.$d_{1},d_{2}$ are the line through $M$ perpendicular to $DB$,the line through $N$ perpendicular to $DC$ respectively.
$d_{1}$ cuts $d_{2}$ at $P$.Prove that $P$ move on a fixed circle.
The four faces of a tetrahedral die are labelled $0, 1, 2,$ and $3,$ and the die has the property that, when it is rolled, the die promptly vanishes, and a number of copies of itself appear equal to the number on the face the die landed on. For example, if it lands on the face labelled $0,$ it disappears. If it lands on the face labelled $1,$ nothing happens. If it lands on the face labelled $2$ or $3,$ there will then be $2$ or $3$ copies of the die, respectively (including the original). Suppose the die and all its copies are continually rolled, and let $p$ be the probability that they will all eventually disappear. Find $\left\lfloor \frac{10}{p} \right\rfloor$.
This questions comprises two independent parts.
(i) Let \(g:\mathbb{R}\to\mathbb{R}\) be continuous and such that \(g(0)=0\) and \(g(x)g(-x)>0\) for any \(x > 0\). Find all solutions \(f : \mathbb{R}\to\mathbb{R}\) to the functional equation
\[g(f(x+y))=g(f(x))+g(f(y)),\ x,y\in\mathbb{R}\]
(ii) Find all continuously differentiable functions \(\phi : [a, \infty) \to \mathbb{R}\), where \(a > 0\), that satisfies the equation
\[(\phi(x))^2=\int_a^x \left(\left|\phi(y)\right|\right)^2+\left(\left|\phi'(y)\right|\right)^2\mathrm{d}y -(x-a)^3,\ \forall x\geq a.\]
There are $c$ red, $p$ blue, and $b$ white balls on a table. Two players $A$ and $B$ play a game by alternately making moves. In every move, a player takes two or three balls from the table. Player $A$ begins. A player wins if after his/her move at least one of the three colors no longer exists among the balls remaining on the table. For which values of $c,p,b$ does player $A$ have a winning strategy?
The sequence of reals $a_1, a_2, a_3, \ldots$ is defined recursively by the recurrence:
$$\dfrac{a_{n+1}}{a_n} - 3 = a_n(a_n - 3)$$ Given that $a_{2021} = 2021$, find $a_1$.
Let $P(x)$ be a polynomial with integer coefficients. Prove that if there is an integer $k$ such that none of the integers $P(1),P(2), ..., P(k)$ is divisible by $k$, then $P(x)$ does not have integer roots.
Let $p \ge 5$ be a prime number. For a positive integer $k$, let $R(k)$ be the remainder when $k$ is divided by $p$, with $0 \le R(k) \le p-1$. Determine all positive integers $a < p$ such that, for every $m = 1, 2, \cdots, p-1$, $$ m + R(ma) > a. $$
Three $n\times n$ squares form the figure $\Phi$ on the checkered plane as shown on the picture. (Neighboring squares are tpuching along the segment of length $n-1$.)
Find all $n > 1$ for which the figure $\Phi$ can be covered with tiles $1\times 3$ and $3\times 1$ without overlapping.[img]https://pp.userapi.com/c850332/v850332712/115884/DKxvALE-sAc.jpg[/img]
For a positive integer $n$ let $S(n)$ be the sum of digits in the decimal representation of $n$. Any positive integer obtained by removing several (at least one) digits from the right-hand end of the decimal representation of $n$ is called a [i]stump[/i] of $n$. Let $T(n)$ be the sum of all stumps of $n$. Prove that $n=S(n)+9T(n)$.
Let $ABC$ be an acute triangle, let $H$ and $O$ be its orthocentre and circumcentre, respectively,
and let $S$ and $T$ be the feet of the altitudes from $B$ to $AC$ and from $C$ to $AB$, respectively.
Let $M$ be the midpoint of the segment $ST$, and let $N$ be the midpoint of the segment $AH$. The line
through $O$, parallel to $BC$, crosses the sides $AC$ and $AB$ at $F$ and $G$, respectively. The line $NG$
meets the circle $BGO$ again at $K$, and the line $NF$ meets the circle $CFO$ again at $L$. Prove that
the triangles $BCM$ and $KLN$ are similar.
Given a cyclic $ABCD$ with $AB=AD$. Points $M$ and $N$ are marked on the sides $CD$ and $BC$, respectively, so that $DM+BN=MN$. Prove that the circumcenter of the triangle $AMN$ belongs to the segment $AC$.
N.Sedrakian
Does there exist an infinite sequence of positive integers $a_1, a_2, a_3, . . .$ such that $a_m$ and $a_n$ are coprime if and only if $|m - n| = 1$?
The incircle of a triangle $ABC$ with $AB\ne BC$ touches $BC$ at $A_1$ and $AC$ at $B_1$. The segments $AA_1$ and $BB_1$ meet the incircle at $A_2$ and $B_2$, respectively. Prove that the lines $AB,A_1B_1,A_2B_2$ are concurrent.
Consider all possible functions defined for $x = 1, 2, ..., M$ and taking values $y = 1, 2, ..., n$. We denote the set of such functions by $T.$ By $T_0$ we denote the subset of $T$ consisting of functions whose value changes exactly by $ 1$ (in one direction or another) when the argument changes by $1$. Prove that if $M\ge 2n-4$, then among the functions from of the set $T$, there is a function that coincides at least at one point with any function from $T_0$. Specify at least one such function. Prove that if $M <2n-4$, then there is no such function.
$A,B,C,D$ are points on a circle in that order. Prove that $|AB-CD|+|AD-BC| \ge 2|AC-BD|$.
Which of the following numbers is [b]not[/b] a perfect square?
$\textbf{(A) }1^{2016}\qquad\textbf{(B) }2^{2017}\qquad\textbf{(C) }3^{2018}\qquad\textbf{(D) }4^{2019}\qquad \textbf{(E) }5^{2020}$
If $x, y$ and $z$ are real numbers such that $x^2 + y^2 + z^2 = 2$, prove that $x + y + z \le xyz + 2$.
Find all triplets $(a, k, m)$ of positive integers that satisfy the equation $k + a^k = m + 2a^m$.
Find all primes $1 < p < 100$ such that the equation $x^2-6y^2=p$ has an integer solution $(x, y)$.
Let be a natural number $ n\ge 2, $ a group $ G $ and two elements of it $ e_1,e_2 $ such that $ e_2e_1x=xe_2e_1, $ for any element $ x $ of $ G. $
Prove that $ \left( e_1xe_2 \right)^n =e_1x^ne_2, $ for any element $ x $ of $ G, $ if and only if $ e_2e_1=\left( e_2e_1\right)^n. $
[i]Ion Bursuc[/i]
In one galaxy, there exist more than one million stars. Let $M$ be the set of the distances between any $2$ of them. Prove that, in every moment, $M$ has at least $79$ members. (Suppose each star as a point.)