Found problems: 85335
2023 IFYM, Sozopol, 8
Do there exist a natural number $n$ and real numbers $a_0, a_1, \dots, a_n$, each equal to $1$ or $-1$, such that the polynomial $a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0$ is divisible by the polynomial $x^{2023} - 2x^{2022} + c$, where: \\
(a) $c = 1$ \\
(b) $c = -1$?
[i]
(For polynomials $P(x)$ and $Q(x)$ with real coefficients, we say that $P(x)$ is divisible by $Q(x)$ if there exists a polynomial $R(x)$ with real coefficients such that $P(x) = Q(x)R(x)$.)[/i]
2000 National Olympiad First Round, 28
$$\begin{array}{ rlrlrl}
f_1(x)=&x^2+x & f_2(x)=&2x^2-x & f_3(x)=&x^2 +x \\
g_1(x)=&x-2 & g_2(x)=&2x \ \ & g_3(x)=&x+2 \\
\end{array}$$
If $h(x)=x$ can be get from $f_i$ and $g_i$ by using only addition, substraction, multiplication defined on those functions where $i\in\{1,2,3\}$, then $F_i=1$. Otherwise, $F_i=0$. What is $(F_1,F_2,F_3)$ ?
$ \textbf{(A)}\ (0,0,0)
\qquad\textbf{(B)}\ (0,0,1)
\qquad\textbf{(C)}\ (0,1,0)
\qquad\textbf{(D)}\ (0,1,1)
\qquad\textbf{(E)}\ \text{None}
$
2003 Moldova Team Selection Test, 4
A square-table of dimensions $ n\times n$, where $ n\in N^*$, is filled arbitrarly with the numbers $ 1,2,...,n^2$ such that every number appears on the table exactly one time. From each row of the table
is chosen the least number and then denote by $ x$ the biggest number from the numbers chosen. From each column of the table is chosen the least number and then denote by $ y$ the biggest number from the numbers chosen. The table is called [i]balanced [/i]iff $ x \equal{} y$. How many balanced tables we can obtain?
2023 SG Originals, Q2
Find all positive integers $k$ such that there exists positive integers $a, b$ such that
\[a^2+4=(k^2-4)b^2.\]
2015 Puerto Rico Team Selection Test, 6
Find all positive integers $n$ such that $7^n + 147$ is a perfect square.
1998 Irish Math Olympiad, 4
Show that a disk of radius $ 2$ can be covered by seven (possibly overlapping) disks of radius $ 1$.
1982 All Soviet Union Mathematical Olympiad, 337
All the natural numbers from $1$ to $1982$ are gathered in an array in an arbitrary order in computer's memory. The program looks through all the sequent pairs (first and second, second and third,...) and exchanges numbers in the pair, if the number on the lower place is greater than another. Then the program repeats the process, but moves from another end of the array. The number, that stand initially on the $100$-th place reserved its place. Find that number.
2012 Online Math Open Problems, 14
Al told Bob that he was thinking of $2011$ distinct positive integers. He also told Bob the sum of those $2011$ distinct positive integers. From this information, Bob was able to determine all $2011$ integers. How many possible sums could Al have told Bob?
[i]Author: Ray Li[/i]
2017 Korea - Final Round, 5
Let there be cyclic quadrilateral $ABCD$ with $L$ as the midpoint of $AB$ and $M$ as the midpoint of $CD$. Let $AC \cap BD = E$, and let rays $AB$ and $DC$ meet again at $F$. Let $LM \cap DE = P$. Let $Q$ be the foot of the perpendicular from $P$ to $EM$. If the orthocenter of $\triangle FLM$ is $E$, prove the following equality.
$$\frac{EP^2}{EQ} = \frac{1}{2} \left( \frac{BD^2}{DF} - \frac{BC^2}{CF} \right)$$
2023 Moldova EGMO TST, 9
Solve the equation $$\left[\frac{x^2+1}{x}\right]-\left[\frac{x}{x^2+1}\right]=3.$$
2011 Iran Team Selection Test, 4
Define a finite set $A$ to be 'good' if it satisfies the following conditions:
[list][*][b](a)[/b] For every three disjoint element of $A,$ like $a,b,c$ we have $\gcd(a,b,c)=1;$
[*][b](b)[/b] For every two distinct $b,c\in A,$ there exists an $a\in A,$ distinct from $b,c$ such that $bc$ is divisible by $a.$[/list]
Find all good sets.
2001 Stanford Mathematics Tournament, 14
Find the prime factorization of $\textstyle\sum_{1\le i < j \le 100}ij$.
2001 Tournament Of Towns, 5
On a square board divided into $15 \times 15$ little squares there are $15$ rooks that do not attack each other. Then each rook makes one move like that of a knight. Prove that after this is done a pair of rooks will necessarily attack each other.
2020 CMIMC Geometry, Estimation
Gunmay picks $6$ points uniformly at random in the unit square. If $p$ is the probability that their convex hull is a hexagon, estimate $p$ in the form $0.abcdef$ where $a,b,c,d,e,f$ are decimal digits. (A [i]convex combination[/i] of points $x_1, x_2, \dots, x_n$ is a point of the form $\alpha_1x_1 + \alpha_2x_2 + \dots + \alpha_nx_n$ with $0 \leq \alpha_i \leq 1$ for all $i$ and $\alpha_1 + \alpha_2 + \dots + \alpha_n = 1$. [i]The convex hull[/i] of a set of points $X$ is the set of all possible convex combinations of all subsets of $X$.)
2002 Moldova National Olympiad, 2
Can a square of side $ 1024$ be partitioned into $ 31$ squares?Can a square of side $ 1023$ be partitioned into $ 30$ squares, one of which has a s side lenght not exceeding $ 1$?
2008 Bosnia And Herzegovina - Regional Olympiad, 3
Prove that equation $ p^{4}\plus{}q^{4}\equal{}r^{4}$ does not have solution in set of prime numbers.
2016 Turkey Team Selection Test, 5
Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that for all $m,n \in \mathbb{N}$ holds $f(mn)=f(m)f(n)$ and $m+n \mid f(m)+f(n)$ .
2013 Gheorghe Vranceanu, 2
Given a number $ a $ and natural number $ n\ge 3 $ having the property that $ x^n-x $ and $ x^2-x $ are integers, prove that $ x $ is integer.
2022 BMT, 20
The game Boddle uses eight cards numbered $6, 11, 12, 14, 24, 47, 54$, and $n$, where $0 \le n \le 56$. An integer D is announced, and players try to obtain two cards, which are not necessarily distinct, such that one of their differences (positive or negative) is congruent to $D$ modulo $57$. For example, if $D = 27$, then the pair $24$ and $54$ would work because $24 - 54 \equiv 27$ mod $57$. Compute $n$ such that this task is always possible for all $D$.
2023 CMIMC Team, 5
$1296$ CMU Students sit in a circle. Every pair of adjacent students rolls a standard six-sided die, and the `score' of any individual student is the sum of their two dice rolls. A 'matched pair' of students is an (unordered) pair of distinct students with the same score. What is the expected value of the number of matched pairs of students?
[i]Proposed by Dilhan Salgado[/i]
1994 Korea National Olympiad, Problem 2
Let $ \alpha,\beta,\gamma$ be the angles of a triangle. Prove that
$csc^2\frac{\alpha}{2}+csc^2\frac{\beta}{2}+csc^2\frac{\gamma}{2} \ge 12$
and find the conditions for equality.
2022 Purple Comet Problems, 16
The sum of the solutions to the equation $$x^{\log_2 x} =\frac{64}{x}$$ can be written as$ \frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
1974 Spain Mathematical Olympiad, 2
In a metallic disk, a circular sector is removed, so that with the remaining can form a conical glass of maximum volume. Calculate, in radians, the angle of the sector that is removed.
[hide=original wording]En un disco metalico se quita un sector circular, de modo que con la parte restante se pueda formar un vaso c´onico de volumen maximo. Calcular, en radianes, el angulo del sector que se quita.[/hide]
2011 IFYM, Sozopol, 4
There are $n$ points in a plane. Prove that there exist a point $O$ (not necessarily from the given $n$) such that on each side of an arbitrary line, through $O$, lie at least $\frac{n}{3}$ points (including the points on the line).
2019 India IMO Training Camp, P1
Let the points $O$ and $H$ be the circumcenter and orthocenter of an acute angled triangle $ABC.$ Let $D$ be the midpoint of $BC.$ Let $E$ be the point on the angle bisector of $\angle BAC$ such that $AE\perp HE.$ Let $F$ be the point such that $AEHF$ is a rectangle. Prove that $D,E,F$ are collinear.