Found problems: 85335
Let $F_1, F_2, F_3 \cdots$ be the Fibonacci sequence, the sequence of positive integers satisfying $$F_1 =F_2=1$$ and $$F_{n+2} = F_{n+1} + F_n$$ for all $n \ge 1$.
Does there exist an $n \ge 1$ such that $F_n$ is divisible by $2014$? Prove your answer.
Find all functions $f : \mathbb{R} \to \mathbb{R}$ that satisfy
\[f(x^2 + y) + f(f(x) - y) = 2f(f(x)) + 2y^2\quad\text{ for all }x, y \in \mathbb{R}.\]
In a planet $Papella$ year has $400$ days with days coundting from $1-400$ . A holiday would be that day which is divisible by $6$ . The new gonverment decide to reform a new calendar and split in $10$ months with $40$ day each month , and they decide that day of month which is divisible by $6$ to be holiday . Show that after reform the number of holidays after one year decreased less then $ 10 $ percent .
Let $ p_1, p_2, p_3$ and $ p_4$ be four different prime numbers satisying the equations
$ 2p_1 \plus{} 3p_2 \plus{} 5p_3 \plus{} 7p_4 \equal{} 162$
$ 11p_1 \plus{} 7p_2 \plus{} 5p_3 \plus{} 4p_4 \equal{} 162$
Find all possible values of the product $ p_1p_2p_3p_4$
Find all prime numbers $p,q$ for which $pq$ divides $(5^p-2^p)(5^q-2^q)$.
Show that there are no integers $x$ and $y$ satisfying $x^2 + 5 = y^3$.
Daniel Harrer
Let $k{}$ be a positive integer. For every positive integer $n \leq 3^k$, denote $b_n$ the greatest power of $3$ that divides $C_{3^k}^n$. Compute $\sum_{n=1}^{3^k-1} \frac{1}{b_n}$.
There are $12$ members in a club. The members created some small groups, which satisfy the following:
- The small group consists of $3$ or $4$ people.
- Also, for two arbitrary members, there exists exactly one small group that has both members.
Prove that all members are in the same number of small groups.
The sequence of positive integers $a_1,a_2,a_3,\dots$ is [i]brazilian[/i] if $a_1=1$ and $a_n$ is the least integer greater than $a_{n-1}$ and $a_n$ is [b]coprime[/b] with at least half elements of the set $\{a_1,a_2,\dots, a_{n-1}\}$. Is there any odd number which does [b]not[/b] belong to the brazilian sequence?
A bakery owner turns on his doughnut machine at 8:30 AM. At 11:10 AM the machine has completed one third of the day's job. At what time will the doughnut machine complete the job?
$ \textbf{(A)}$ 1:50 PM $ \qquad
\textbf{(B)}$ 3:00 PM $ \qquad
\textbf{(C)}$ 3:30 PM $ \qquad
\textbf{(D)}$ 4:30 PM $ \qquad
\textbf{(E)}$ 5:50 PM
Prove that $4x^3-3x+1=2y^2$ has at least $31$ solutions in positive integers $x,y$ with $x\le 1980$.
[i] Variant: [/i] Prove the equation $4x^3-3x+1=2y^2$ has infinitely many solutions in positive integers x,y.
Let $p$ be an odd prime number, and let $m \ge 0$ and $N \ge 1$ be integers. Let $\Lambda$ be a free $\mathbb{Z}/p^N\mathbb{Z}$-module of rank $2m+1$, equipped with a perfect symmetric $\mathbb{Z}/p^N\mathbb{Z}$-bilinear form
\[(\, ,\,): \Lambda \times \Lambda \to \mathbb{Z}/p^N\mathbb{Z}.\]
Here ``perfect'' means that the induced map
\[\Lambda \to \text{Hom}_{\mathbb{Z}/p^N\mathbb{Z}}(\Lambda, \mathbb{Z}/p^N\mathbb{Z}), \quad x \mapsto (x,\cdot)\]
is an isomorphism. Find the cardinality of the set
\[\{x \in \Lambda: (x,x)=0\},\]
expressed in terms of $p,m,N$.
Consider a group $G$ with at least $2$ elements and the property that each nontrivial element has infinite order. Let $H$ be a cyclic subgroup of $G$ such that the set $\{xH\mid x\in G\}$ has $2$ elements. \\
$\textbf{(a)}$ Prove that $G$ is cyclic. \\
$\textbf{(b)}$ Does the conclusion from $\textbf{(a)}$ stand true if $G$ contains nontrivial elements of finite order?
Prove that there exists a positive $ c$ such that for every positive integer $ N$ among any $ N$ positive integers not exceeding $ 2N$ there are two numbers whose greatest common divisor is greater than $ cN$.
In the diagram, if points $ A$, $ B$ and $ C$ are points of tangency, then $ x$ equals:
[asy]unitsize(5cm);
defaultpen(linewidth(.8pt)+fontsize(8pt));
dotfactor=3;
pair A=(-3*sqrt(3)/32,9/32), B=(3*sqrt(3)/32, 9/32), C=(0,9/16);
pair O=(0,3/8);
draw((-2/3,9/16)--(2/3,9/16));
draw((-2/3,1/2)--(-sqrt(3)/6,1/2)--(0,0)--(sqrt(3)/6,1/2)--(2/3,1/2));
draw(Circle(O,3/16));
draw((-2/3,0)--(2/3,0));
label("$A$",A,SW);
label("$B$",B,SE);
label("$C$",C,N);
label("$\frac{3}{8}$",O);
draw(O+.07*dir(60)--O+3/16*dir(60),EndArrow(3));
draw(O+.07*dir(240)--O+3/16*dir(240),EndArrow(3));
label("$\frac{1}{2}$",(.5,.25));
draw((.5,.33)--(.5,.5),EndArrow(3));
draw((.5,.17)--(.5,0),EndArrow(3));
label("$x$",midpoint((.5,.5)--(.5,9/16)));
draw((.5,5/8)--(.5,9/16),EndArrow(3));
label("$60^{\circ}$",(0.01,0.12));
dot(A);
dot(B);
dot(C);[/asy]$ \textbf{(A)}\ \frac {3}{16}" \qquad \textbf{(B)}\ \frac {1}{8}" \qquad \textbf{(C)}\ \frac {1}{32}" \qquad \textbf{(D)}\ \frac {3}{32}" \qquad \textbf{(E)}\ \frac {1}{16}"$
Write the numbers from $1$ to $16$ in the cells of a of a $4 \times 4$ square so that:
1. Each cell contains exactly one number;
2. Each number is written exactly once;
3. For any two cells that are symmetrical with respect to any of the perpendicular bisectors of sides of the
original $4 \times 4$ square, the sum of numbers in them is a prime number
The figure below shows examples of such pairs of cells, sums of numbers in which have to be prime.
[img]https://i.ibb.co/fqX05dY/Kyiv-MO-2024-Round-1-8-2.png[/img]
[i]Proposed by Mykhailo Shtandenko[/i]
We say that a quadruple $(A,B,C,D)$ is [i]dobarulho[/i] when $A,B,C$ are non-zero algorisms and $D$ is a positive integer such that:
$1.$ $A \leq 8$
$2.$ $D>1$
$3.$ $D$ divides the six numbers $\overline{ABC}$, $\overline{BCA}$, $\overline{CAB}$, $\overline{(A+1)CB}$, $\overline{CB(A+1)}$, $\overline{B(A+1)C}$.
Find all such quadruples.
On an infinite white checkered sheet, a square $Q$ of size $12$ × $12$ is selected. Petya wants to paint some (not necessarily all!) cells of the square with seven colors of the rainbow (each cell is just one color) so that no two of the $288$ three-cell rectangles whose centers lie in $Q$ are the same color. Will he succeed in doing this?
(Two three-celled rectangles are painted the same if one of them can be moved and possibly rotated so that each cell of it is overlaid on the cell of the second rectangle having the same color.)
(Bogdanov. I)
The sides of an equilateral triangle $ABC$ are divided into $n$ equal parts $(n \geq 2) .$ For each point on a side, we draw the lines parallel to other sides of the triangle $ABC,$ e.g. for $n=3$ we have the following diagram:
[asy]
unitsize(150);
defaultpen(linewidth(0.7));
int n = 3; /* # of vertical lines, including AB */
pair A = (0,0), B = dir(-30), C = dir(30);
draw(A--B--C--cycle,linewidth(2)); dot(A,UnFill(0)); dot(B,UnFill(0)); dot(C,UnFill(0));
label("$A$",A,W); label("$C$",C,NE); label("$B$",B,SE);
for(int i = 1; i < n; ++i) {
draw((i*A+(n-i)*B)/n--(i*A+(n-i)*C)/n);
draw((i*B+(n-i)*A)/n--(i*B+(n-i)*C)/n);
draw((i*C+(n-i)*A)/n--(i*C+(n-i)*B)/n);
}
[/asy]
For each $n \geq 2,$ find the number of existing parallelograms.
Solve: \[ \begin{cases}x_{2}x_{3}x_{4}\cdots x_{n}=a_{1}x_{1}\\ x_{1}x_{3}x_{4}\cdots x_{n}=a_{2}x_{2}\\x_{1}x_{2}x_{4}\cdots x_{n}=a_{3}x_{3}\\ \ldots\\x_{1}x_{2}x_{3}\cdots x_{n-1}=a_{n-1}x_{n-1} \end{cases} \]
Let $ ABC$ be a triangle, and $ I$ its incenter. Let the incircle of $ ABC$ touch side $ BC$ at $ D$, and let lines $ BI$ and $ CI$ meet the circle with diameter $ AI$ at points $ P$ and $ Q$, respectively. Given $ BI \equal{} 6, CI \equal{} 5, DI \equal{} 3$, determine the value of $ \left( DP / DQ \right)^2$.
In the triangle $ABC$, it is known that $AC <AB$. Let $\ell$ be tangent to the circumcircle of triangle $ABC$ drawn at point $A$. A circle with center $A$ and radius $AC$ intersects segment $AB$ at point $D$, and line $\ell$ at points $E$ and $F$. Prove that one of the lines $DE$ and $DF$ passes through the center inscribed circle of triangle $ABC$.
For each $1\leq i\leq 9$ and $T\in\mathbb N$, define $d_i(T)$ to be the total number of times the digit $i$ appears when all the multiples of $1829$ between $1$ and $T$ inclusive are written out in base $10$.
Show that there are infinitely many $T\in\mathbb N$ such that there are precisely two distinct values among $d_1(T)$, $d_2(T)$, $\dots$, $d_9(T)$.
You are given a positive integer $k$ and not necessarily distinct positive integers $a_1, a_2 , a_3 , \ldots,
a_k$. It turned out that for any coloring of all positive integers from $1$ to $2021$ in one of the $k$ colors so that there are exactly $a_1$ numbers of the first color, $a_2$ numbers of the second color, $\ldots$, and $a_k$ numbers of the $k$-th color, there is always a number $x \in \{1, 2, \ldots, 2021\}$, such that the total number of numbers colored in the same color as $x$ is exactly $x$. What are the possible values of $k$?
[i]Proposed by Arsenii Nikolaiev[/i]
Suppose that $S$ is a finite set of at least five points in the plane; some are coloured red, the others are coloured blue. No subset of three or more similarly coloured points is collinear. Show that there is a triangle
(i) whose vertices are all the same colour, and
(ii) at least one side of the triangle does not contain a point of the opposite colour.