Found problems: 85335
Find all function pairs $(f,g)$ where each $f$ and $g$ is a function defined on the integers and with values, such that, for all integers $a$ and $b$,
\[f(a+b)=f(a)g(b)+g(a)f(b)\\
g(a+b)=g(a)g(b)-f(a)f(b).\]
Positive integers $x_1, x_2, \dots, x_n$ ($n \ge 4$) are arranged in a circle such that each $x_i$ divides the sum of the neighbors; that is \[ \frac{x_{i-1}+x_{i+1}}{x_i} = k_i \] is an integer for each $i$, where $x_0 = x_n$, $x_{n+1} = x_1$. Prove that \[ 2n \le k_1 + k_2 + \dots + k_n < 3n. \]
consider $n\geq 6$ points $x_1,x_2,\dots,x_n$ on the plane such that no three of them are colinear. We call graph with vertices $x_1,x_2,\dots,x_n$ a "road network" if it is connected, each edge is a line segment, and no two edges intersect each other at points other than the vertices. Prove that there are three road networks $G_1,G_2,G_3$ such that $G_i$ and $G_j$ don't have a common edge for $1\leq i,j\leq 3$.
Proposed by Morteza Saghafian
In a given plane, points $A$ and $B$ are $10$ units apart. How many points $C$ are there in the plane such that the perimeter of $\triangle ABC$ is $50$ units and the area of $\triangle ABC$ is $100$ square units?
$\textbf{(A) }0\qquad\textbf{(B) }2\qquad\textbf{(C) }4\qquad\textbf{(D) }8\qquad\textbf{(E) }\text{infinitely many}$
Let $a_0, a_1, . . . ,a_n$ be such that $a_n \ne 0$ and $$(1 + x + x^3)^{342} (1 + 2x + x^2 + 2x^3 + 2x^4 + x^6)^{341} =\sum^{n}_{i=0}a_ix^i.$$ Compute the number of odd terms in the sequence $a_0, a_1, . . . ,a_n$.
Find all primes $p,$ $q,$ $r$ satisfying $p^{2}+2q^{2}=r^{2}.$
Find the remainder when $2^{5^9}+5^{9^2}+9^{2^5}$ is divided by $11$.
Alex scans the list of integers between $1$ and $2020$ inclusive using the following algorithm. First, he reads off perfect squares between $1$ and $2020$ in ascending order and removes these numbers from the list. Next, he reads off numbers now at perfect square indices in ascending order, which are $2$, $6$, $12$, $...$, and removes these numbers from the list. He repeats this algorithm until he reads off $2020$, which is the nth number he has read o so far. Compute $n$.
$f(k) = k + \left[ \frac{n}{k}\right ] $,$k \in \{1,2,..., n\}$, $k_0 =\left[ \sqrt{n} \right] + 1$.
Prove that $f(k_0) < f(k)$ if $k \in \{1,2,..., n\}$
Determine all integer $n > 1$ such that
\[\gcd \left( n, \dfrac{n-m}{\gcd(n,m)} \right) = 1\]
for all integer $1 \le m < n$.
Let $p$ a prime number and $d$ a divisor of $p-1$. Find the product of elements in $\mathbb Z_p$ with order $d$. ($\mod p$).
(10 points)
There are $n$ cards on a table in a line, with a positive real written on eachcard. LTF and Sunny are playing a game where they take turns taking away the first or the last card in line. The player that has the bigger sum of all the numberson his cards wins. If LTF goes first, find all $n$ such that LTF can always prevent Sunny from winning, regardless of the numbers written on the cards.
Which of the following is satisfied by all numbers $ x$ of the form
\[ x \equal{} \frac {a_1}{3} \plus{} \frac {a_2}{3^2} \plus{} \cdots \plus{} \frac {a_{25}}{3^{25}},\]
where $ a_1$ is $ 0$ or $ 2$, $ a_2$ is $ 0$ or $ 2$,...,$ a_{25}$ is $ 0$ or $ 2$?
$ \textbf{(A)}\ 0 \le x < 1/3 \qquad \textbf{(B)}\ 1/3 \le x < 2/3 \qquad \textbf{(C)}\ 2/3 \le x < 1 \\
\textbf{(D)}\ 0 \le x < 1/3 \text{ or } 2/3 \le x < 1 \qquad \textbf{(E)}\ 1/2 \le x \le 3/4$
Let $n$ be a fixed positive integer. Ana and Banana are playing a game. First, Ana picks a subset $S$ of $\{1,2,\ldots,n\}$. Then for each $k=1,2,\ldots,n$, she tells Banana how many numbers from $k-1$ to $k+1$ she has picked (i.e. $\lvert S \cap \{k-1,k,k+1\}\rvert$). Then Banana guesses $S$; she wins if her guess is correct and she loses otherwise.
(a) Determine all $n$ for which Banana will win regardless of what Ana chooses.
(b) For the values of $n$ for which Ana can win, determine the number of sets $S$ she can choose so as to do so.
In a cyclic quadrilateral $ABCD$ whose largest interior angle is $D$, lines $BC$ and $AD$ intersect at point $E$, while lines $AB$ and $CD$ intersect at point $F$. A point $P$ is taken in the interior of quadrilateral $ABCD$ for which $\angle EPD=\angle FPD=\angle BAD$. $O$ is the circumcenter of quadrilateral $ABCD$. Line $FO$ intersects the lines $AD$, $EP$, $BC$ at $X$, $Q$, $Y$, respectively. If $\angle DQX = \angle CQY$, show that $\angle AEB=90^\circ$.
We say that a positive integer is a [i]good number[/i] if the digit $2$ appears more often than the digit $3$ and that it is a [i]bad number[/i] if the digit $3$ appears more often than the digit $2$. For example, $2023$ is a good number and $123$ is neither good nor bad. Calculate the difference between the quantity of good numbers and the quantity of bad numbers for integers less than or equal to $2023$.
Let $n$ be a positive integer and let $k$ be an integer between $1$ and $n$ inclusive. There is a white board of $n \times n$. We do the following process. We draw $k$ rectangles with integer sides lenghts and sides parallel to the ones of the $n \times n$ board, and such that each rectangle covers the top-right corner of the $n \times n$ board. Then, the $k$ rectangles are painted of black. This process leaves a white figure in the board.
How many different white figures are possible to do with $k$ rectangles that can't be done with less than $k$ rectangles?
Proposed by David Torres Flores
Let $(x,y)$ be the coordinates of a point chosen uniformly at random within the unit square with vertices at $(0,0), (0,1), (1,0),$ and $(1,1).$ The probability that $|x - \tfrac{1}{2}| + |y - \tfrac{1}{2}| < \tfrac{1}{2}$ is $\tfrac{p}{q},$ where $p$ and $q$ are relatively prime integers. Find $p + q.$
Given a positive integer $a,$ prove that $n!$ is divisible by $n^2 + n + a$ for infinitely many positive integers $n.{}$
[i]Proposed by Andrei Bâra[/i]
The set consists of equal three-cell corners ( $L$ -triminoes), the middle cells of which are marked with paint. A rectangular board has been covered with these triminoes in a single layer so that all triminoes were entirely on the board. Then the triminoes were removed leaving the paint marks where the marked cells were. Is it always possible to know the location of the triminoes on the board using only those paint marks?
Alexandr Gribalko
Concider two sequences $x_n=an+b$, $y_n=cn+d$ where $a,b,c,d$ are natural numbers and $gcd(a,b)=gcd(c,d)=1$, prove that there exist infinite $n$ such that $x_n$, $y_n$ are both square-free.
[i]Proposed by Siavash Rahimi Shateranloo, Matin Yadollahi[/i] [b]Rated 3[/b]
Determine the largest integer $c$ for which the following statement holds: there exists at least one triple $(x,y,z)$ of integers such that
\begin{align*} x^2 + 4(y + z) = y^2 + 4(z + x) = z^2 + 4(x + y) = c \end{align*}
and all triples $(x,y,z)$ of real numbers, satisfying the equations, are such that $x,y,z$ are integers.
[i]Marek Maruin, Slovakia [/i]
Let $XY$ be a segment, which is a diameter of a semi-circle. Let $Z$ be a point on $XY$ and 9 rays from $Z$ are drawn that divide $\angle XZY=180^{\circ}$ into $10$ equal angles. These rays meet the semi-circle at $A_1, A_2, \ldots, A_9$ in this order in the direction from $X$ to $Y$. Prove that the sum of the areas of triangles $ZA_2A_3$ and $ZA_7A_8$ equals the area of the quadrilateral $A_2A_3A_7A_8$.
Solve the equation $\frac{1}{\sin x}+\frac{1}{\cos x}=\frac 1p$ where $p$ is a real parameter.
Discuss for which values of $p$ the equation has at least one real solution and determine the number of solutions in $[0, 2\pi)$ for a given $p.$
For any $k \in \mathbb{Z},$ define $$F_k=X^4+2(1-k)X^2+(1+k)^2.$$ Find all values $k \in \mathbb{Z}$ such that $F_k$ is irreducible over $\mathbb{Z}$ and reducible over $\mathbb{Z}_p,$ for any prime $p.$
[i]Marius Vladoiu[/i]