Found problems: 85335
Let $p$ be an odd prime number such that $p\equiv 2\pmod{3}.$ Define a permutation $\pi$ of the residue classes modulo $p$ by $\pi(x)\equiv x^3\pmod{p}.$ Show that $\pi$ is an even permutation if and only if $p\equiv 3\pmod{4}.$
For every composite positive integer $n$, define $r(n)$ to be the sum of the factors in the prime factorization of $n$. For example, $r(50)=12$ because the prime factorization of $50$ is $ 2 \cdot 5^2 $, and $ 2 + 5 + 5 = 12 $. What is the range of the function $r$, $ \{ r(n) : n \ \text{is a composite positive integer} \} $?
[b](A)[/b] the set of positive integers
[b](B)[/b] the set of composite positive integers
[b](C)[/b] the set of even positive integers
[b](D)[/b] the set of integers greater than 3
[b](E)[/b] the set of integers greater than 4
It is known that $n$ is a positive integer and $n \le 144$. Ten questions of the type “Is $n$ smaller than $a$?” are allowed. Answers are given with a delay: for $i = 1, \ldots , 9$, the $i$-th question is answered only after the $(i + 1)$-th question is asked. The answer to the tenth question is given immediately.
Find a strategy for identifying $n$.
The smaller root of the equation $ \left(x-\frac{3}{4}\right)\left(x-\frac{3}{4}\right)+\left(x-\frac{3}{4}\right)\left(x-\frac{1}{2}\right) =0$ is:
${{ \textbf{(A)}\ -\frac{3}{4}\qquad\textbf{(B)}\ \frac{1}{2}\qquad\textbf{(C)}\ \frac{5}{8}\qquad\textbf{(D)}\ \frac{3}{4} }\qquad\textbf{(E)}\ 1 } $
Consider an equilateral triangle $\triangle ABC$. The points $K$ and $L$ divide the leg $BC$ into three equal parts, the point $M$ divides the leg $AC$ in the ratio $1:2$, counting from the vertex $A$. Prove that $\angle AKM+\angle ALM=30^{\circ}$.
Proposed by V. Proizvolov
Suppose there are $4n$ line segments of unit length inside a circle of radius $n$. Furthermore, a straight line $L$ is given. Prove that there exists a straight line $L'$ that is either parallel or perpendicular to $L$ and that $L'$ cuts at least two of the given line segments.
Find the probability such that when a polynomial in $\mathbb Z_{2027}[x]$ having degree at most $2026$ is chosen uniformly at random,
$$x^{2027}-x | P^{k}(x) - x \iff 2021 | k $$
(note that $2027$ is prime).
Here $P^k(x)$ denotes $P$ composed with itself $k$ times.
[i]Proposed by Grant Yu[/i]
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that
\[f(xy)=\max\{f(x+y),f(x) f(y)\} \]
for all real numbers $x$ and $y$.
Prove that there is no function $f: \mathbb{R} \rightarrow \mathbb{R}$ with $f(0) >0$, and such that
\[f(x+y) \geq f(x) +yf(f(x)) \text{ for all } x,y \in \mathbb{R}. \]
A triangle is cut into several (not less than two) triangles. One of them is isosceles (not equilateral), and all others are equilateral. Determine the angles of the original triangle.
Let $ABCD$ be a convex quadrilateral with perimeter $\tfrac{5}{2}$ and $AC=BD=1$. Determine the maximum possible area of $ABCD$.
Find the largest and smallest value of the function $$y=\sqrt{7+5\cos x}-\cos x.$$
Determine the largest real number \(C\) such that
$$\frac{1}{x}+\frac{1}{2y}+\frac{1}{3z}\geqslant C$$
for all real numbers \(x,y,z\neq 0\) satisfying the equation
$$\frac{x}{yz}+\frac{4y}{xz}+\frac{9z}{xy}=24$$
The sequence $ \{a_{n}\}_{n \ge 1}$ is defined by \[ a_{1}=1, \; a_{2}=2, \; a_{3}=24, \; a_{n}=\frac{ 6a_{n-1}^{2}a_{n-3}-8a_{n-1}a_{n-2}^{2}}{a_{n-2}a_{n-3}}\ \ \ \ (n\ge4).\] Show that $ a_{n}$ is an integer for all $ n$, and show that $ n|a_{n}$ for every $ n\in\mathbb{N}$.
For the positive integer $n$, define $f(n)=\min\limits_{m\in\Bbb Z}\left|\sqrt2-\frac mn\right|$. Let $\{n_i\}$ be a strictly increasing sequence of positive integers. $C$ is a constant such that $f(n_i)<\dfrac C{n_i^2}$ for all $i\in\{1,2,\ldots\}$. Show that there exists a real number $q>1$ such that $n_i\geqslant q^{i-1}$ for all $i\in\{1,2,\ldots \}$.
The incircle of the triangle $ ABC $ touches the sides of $ AB, BC, CA $ at points $ C_0, A_0, B_0 $, respectively. Let the point $ M $ be the midpoint of the segment connecting the vertex $ C_0 $ with the intersection point of the altitudes of the triangle $ A_0B_0C_0 $, point $ N $ be the midpoint of the arc $ ACB $ of the circumscribed circle of the triangle $ ABC $. Prove that line $ MN $ passes through the center of incircle of triangle $ ABC $.
Among all fractions (whose numerator and denominator are positive integers) strictly between $\tfrac{6}{17}$ and $\tfrac{9}{25}$, which one has the smallest denominator?
Each side of the large square in the figure is trisected (divided into three equal parts). The corners of an inscribed square are at these trisection points, as shown. The ratio of the area of the inscribed square to the area of the large square is
[asy]
draw((0,0)--(3,0)--(3,3)--(0,3)--cycle);
draw((1,0)--(1,0.2)); draw((2,0)--(2,0.2));
draw((3,1)--(2.8,1)); draw((3,2)--(2.8,2));
draw((1,3)--(1,2.8)); draw((2,3)--(2,2.8));
draw((0,1)--(0.2,1)); draw((0,2)--(0.2,2));
draw((2,0)--(3,2)--(1,3)--(0,1)--cycle);
[/asy]
$\textbf{(A)}\ \dfrac{\sqrt{3}}{3} \qquad \textbf{(B)}\ \dfrac{5}{9} \qquad \textbf{(C)}\ \dfrac{2}{3} \qquad \textbf{(D)}\ \dfrac{\sqrt{5}}{3} \qquad \textbf{(E)}\ \dfrac{7}{9}$
Let $a_1, a_2... $ be an infinite sequence of real numbers such that $a_{n+1}=\sqrt{{a_n}^2+a_n-1}$. Prove that $a_1 \notin (-2,1)$
[i]Proposed by Oleg Mushkarov and Nikolai Nikolov
[/i]
In triangle $ABC$ with circumcircle $\Omega$ and incenter $I$, point $M$ bisects arc $BAC$ and line $\overline{AI}$ meets $\Omega$ at $N\ne A$. The excircle opposite to $A$ touches $\overline{BC}$ at point $E$. Point $Q\ne I$ on the circumcircle of $\triangle MIN$ is such that $\overline{QI}\parallel\overline{BC}$. Prove that the lines $\overline{AE}$ and $\overline{QN}$ meet on $\Omega$.
Solve $(x-4)(x-5)(x-6)(x-7)=1680$
Let $n$ be a posititve integer. On a $n \times n$ grid there are $n^2$ unit squares and on these we color the sides with blue such that every unit square has exactly one side with blue.
[b]a)[/b] Find the maximun number of blue unit sides we can have on the $n \times n$ grid.
[b]b)[/b] Find the minimun number of blue unit sides we can have on the $n \times n$ grid.
In a simple graph, there exist two vertices $A,B$ such that there are exactly $100$ shortest paths from $A$ to $B$. Find the minimum number of edges in the graph.
[i]CSJL[/i]
Oleksii and Solomiya play the following game on a square $6n\times 6n$, where $n$ is a positive integer. Oleksii in his turn places a piece of type $F$, consisting of three cells, on the board. Solomia, in turn, after each move of Oleksii, places the numbers $0, 1, 2$ in the cells of the figure that Oleksii has just placed, using each of the numbers exactly once. If two of Oleksii's pieces intersect at any moment (have a common square), he immediately loses.
Once the square is completely filled with numbers, the game stops. In this case, if the sum of the numbers in each row and each column is divisible by $3$, Solomiya wins, and otherwise Oleksii wins. Who can win this game if the figure of type $F$ is:
a) a rectangle ;
b) a corner of three cells?
[i]Proposed by Oleksii Masalitin[/i]
* Assume that the number of a tree’s leaves is a multiple of $15$. Neglecting the shade of the trunk and branches prove that one can rip off the tree $7/15$ of its leaves so that not less than $8/15$ of its shade remains.