Found problems: 85335
Find all functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy for all $x$:
$(i)$ $f(x)=-f(-x);$
$(ii)$ $f(x+1)=f(x)+1;$
$(iii)$ $f\left( \frac{1}{x}\right)=\frac{1}{x^2}f(x)$ for $x\ne 0$
In three piles there are $51, 49$, and $5$ stones, respectively. You can combine any two piles into one pile or divide a pile consisting of an even number of stones into two equal piles. Is it possible to get $105$ piles with one stone in each?
Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ with the property that \[f(x-f(y))=f(f(x))-f(y)-1\] holds for all $x,y\in\mathbb{Z}$.
$5000$ movie fans gathered at a convention. Each participant had watched at least one movie. The participants should be split into discussion groups of two kinds. In each group of the fi rst kind, the members would discuss a movie they all watched. In each group of the second kind, each member would tell about the movie that no one else in this group had watched. Prove that the chairman can always split the participants into exactly 100 groups. (A group consisting of one person is allowed; in this case this person submits a report).
Let $k \geq 1$ and $N>1$ be two integers. On a circle are placed $2N+1$ coins all showing heads. Calvin and Hobbes play the following game. Calvin starts and on his move can turn any coin from heads to tails. Hobbes on his move can turn at most one coin that is next to the coin that Calvin turned just now from tails to heads. Calvin wins if at any moment there are $k$ coins showing tails after Hobbes has made his move. Determine all values of $k$ for which Calvin wins the game.
[i]Proposed by Tejaswi Navilarekallu[/i]
Let $m>1$ be an integer. Find the smallest positive integer $n$, such that for any integers $a_1,a_2,\ldots ,a_n; b_1,b_2,\ldots ,b_n$ there exists integers $x_1,x_2,\ldots ,x_n$ satisfying the following two conditions:
i) There exists $i\in \{1,2,\ldots ,n\}$ such that $x_i$ and $m$ are coprime
ii) $\sum^n_{i=1} a_ix_i \equiv \sum^n_{i=1} b_ix_i \equiv 0 \pmod m$
Let $ABC$ be a non-degenerate triangle with incenter $I$ and circumcircle $\Gamma$. Denote by $M_a$ the midpoint of the arc $\widehat{BC}$ of $\Gamma$ not containing $A$, and define $M_b$, $M_c$ similarly. Suppose $\triangle ABC$ has inradius $4$ and circumradius $9$. Compute the maximum possible value of \[IM_a^2+IM_b^2+IM_c^2.\][i]Proposed by David Altizio[/i]
The sequence $(a_n)_{n>=1}$ satisfies that : $a_1=a_2=1$ $a_n=7a_{n-1}-a_{n-2}$ ($n>=3$) , prove that : for all positive integer n , number $a_n+2+a_{n+1}$ is a perfect square .
Prove that the integer $145^{n} + 3114\cdot 138^{n}$ is divisible by $1981$ if $n=1981$, and that it is not divisible by $1981$ if $n=1980$.
Seven distinct points are given inside a square with side length $1.$ Together with the square's vertices, they form a set of $11$ points. Consider all triangles with vertices in $M.$
a) Show that at least one of these triangles has an area not exceeding $1\slash 16.$
b) Give an example in which no four of the seven points are on a line and none of the considered triangles has an area of less than $1\slash 16.$
Prove the inequality $2\sqrt{(p-b)(p-c)}\le a$, where $a, b, c$ are the lengths of the sides, and $p$ is the semiperimeter of some triangle..
A positive integer is called [i]Tiranian[/i] if it can be written as $x^2+6xy+y^2$, where $x$ and $y$ are (not necessarily distinct) positive integers. The integer $36^{2023}$ is written as the sum of $k$ Tiranian integers. What is the smallest possible value of $k$?
[i]Proposed by Miroslav Marinov, Bulgaria[/i]
Let $f(x)$ be a rational function with complex coefficients whose denominator does not have multiple roots. Let $u_0, u_1,... , u_n$ be the complex roots of $f$ and $w_1, w_2,..., w_m$ be the roots of $f'$. Suppose that $u_0$ is a simple root of $f$. Prove that
\[ \sum_{k=1}^m \frac{1}{w_k - u_0} = 2\sum_{k = 1}^n\frac{1}{u_k - u_0}.\]
A sequence of distinct circles $\omega_1, \omega_2, \cdots$ is inscribed in the parabola $y=x^2$ so that $\omega_n$ and $\omega_{n+1}$ are tangent for all $n$. If $\omega_1$ has diameter $1$ and touches the parabola at $(0,0)$, find the diameter of $\omega_{1998}$.
$(CZS 2)$ Let $p$ be a prime odd number. Is it possible to find $p-1$ natural numbers $n + 1, n + 2, . . . , n + p -1$ such that the sum of the squares of these numbers is divisible by the sum of these numbers?
2. Let $x_1, . . . , x_n$ be $n$ integers, and let $p$ be a positive integer, with $p < n$. Put
$$S_1 = x_1 + x_2 + . . . + x_p$$
$$T_1 = x_{p+1} + x_{p+2} + . . . + x_n$$
$$S_2 = x_2 + x_3 + . . . + x_{p+1}$$
$$T_2 = x_{p+2} + x_{p+3} + . . . + x_n + x_1$$
$$...$$
$$S_n=x_n+x_1+...+x_{p-1}$$
$$T_n=x_p+x_{p+1}+...+x_{n-1}$$
For $a = 0, 1, 2, 3$, and $b = 0, 1, 2, 3$, let $m(a, b)$ be the number of numbers $i$, $1 \leq i \leq n$, such that $S_i$ leaves remainder $a$ on division by $4$ and $T_i$ leaves remainder $b$ on division by $4$. Show that $m(1, 3)$ and $m(3, 1)$ leave the same remainder when divided by $4$ if, and only if, $m(2, 2)$ is even.
Let $ABC$ be a scalene triangle, and let $I$ be its incenter. A point $D$ is chosen on line $BC$, such that the circumcircle of triangle $BID$ intersects $AB$ at $E\neq B$, and the circumcircle of triangle $CID$ intersects $AC$ at $F\neq C$. Circumcircle of triangle $EDF$ intersects $AB$ and $AC$ at $M$ and $N$, respectively. Lines $FD$ and $IC$ intersect at $Q$, and lines $ED$ and $BI$ intersect at $P$. Prove that $EN\parallel MF\parallel PQ$.
Let $ABC$ be a triangle with $AB > AC$. Let $D$ be a point on the side $AB$ such that $DB = DC$ and let $M$ be the midpoint of $AC$. The line parallel to $BC$ passing through $D$ intersects the line $BM$ in $K$. Show that $\angle KCD = \angle DAC.$
Prove that the system \begin{align*}
x^6+x^3+x^3y+y & = 147^{157} \\
x^3+x^3y+y^2+y+z^9 & = 157^{147}
\end{align*} has no solutions in integers $x$, $y$, and $z$.
Determine the smallest positive integer $n$ for which the following statement holds true: From any $n$ consecutive integers one can select a non-empty set of consecutive integers such that their sum is divisible by $2019$.
[i]Proposed by Kartal Nagy, Hungary[/i]
Let $A$ be the closed region bounded by the following three lines in the $xy$ plane: $x=1, y=0$ and $y=t(2x-t)$, where $0<t<1$. Prove that the area of any triangle inside the region $A$, with two vertices $P(t,t^2)$ and $Q(1,0)$, does not exceed $\frac{1}{4}.$
A board game spinner is divided into three regions labeled $A$, $B$ and $C$. The probability of the arrow stopping on region $A$ is $\frac{1}{3}$ and on region $B$ is $\frac{1}{2}$. The probability of the arrow stopping on region $C$ is:
$\text{(A)}\ \frac{1}{12} \qquad \text{(B)}\ \frac{1}{6} \qquad \text{(C)}\ \frac{1}{5} \qquad \text{(D)}\ \frac{1}{3} \qquad \text{(E)}\ \frac{2}{5}$
What is the last two digits of the number $(11 + 12 + 13 + ... + 2006)^2$?
Let $n$ be a positive integer. In an $n\times n$ garden, a fountain is to be built with $1\times 1$ platforms covering the entire garden. Ana places all the platforms at a different height. Afterwards, Beto places water sources in some of the platforms. The water in each platform can flow to other platforms sharing a side only if they have a lower height. Beto wins if he fills all platforms with water.
Find the least number of water sources that Beto needs to win no matter how Ana places the platforms.
In triangle $ABC$, let $I$ be the incentre. Let $H$ be the orthocentre of triangle $BIC$. Show that $AH$ is parallel to $BC$ if and only if $H$ lies on the circle with diameter $AI$.