Found problems: 85335
2006 Turkey MO (2nd round), 2
There are $2006$ students and $14$ teachers in a school. Each student knows at least one teacher (knowing is a symmetric relation). Suppose that, for each pair of a student and a teacher who know each other, the ratio of the number of the students whom the teacher knows to that of the teachers whom the student knows is at least $t.$ Find the maximum possible value of $t.$
2013 Bosnia Herzegovina Team Selection Test, 2
The sequence $a_n$ is defined by $a_0=a_1=1$ and $a_{n+1}=14a_n-a_{n-1}-4$,for all positive integers $n$.
Prove that all terms of this sequence are perfect squares.
1989 Putnam, B5
Label the vertices of a trapezoid $T$ inscribed in the unit circle as $A,B,C,D$ counterclockwise with $AB\parallel CD$. Let $s_1,s_2,$ and $d$ denote the lengths of $AB$, $CD$, and $OE$, where $E$ is the intersection of the diagonals of $T$, and $O$ is the center of the circle. Determine the least upper bound of $\frac{s_1-s_2}d$ over all $T$ for which $d\ne0$, and describe all cases, if any, in which equality is attained.
1981 Spain Mathematical Olympiad, 7
In a tennis ball factory there are $4$ machines $m_1 , m_2 , m_3 , m_4$, which produce, respectively, $10\%$, $20\%$, $30\%$ and $40\%$ of the balls that come out of the factory. The machine $m_1$ introduces defects in $1\%$ of the balls it manufactures, the machine $m_2$ in $2\%$, $m_3$ in $4\%$ and $m_4$ in $15\%$. Of the balls manufactured In one day, one is chosen at random and it turns out to be defective. What is the probability that Has this ball been made by the machine $ m_3$ ?
2017 Simon Marais Mathematical Competition, B4
[hide=Note][i]The following problem is open in the sense that no solution is currently known. Progress on the problem may be awarded points. An example of progress on the problem is a non-trivial bound on the sequence defined below.[/i][/hide]
For each integer $n\ge2$, consider a regular polygon with $2n$ sides, all of length $1$. Let $C(n)$ denote the number of ways to tile this polygon using quadrilaterals whose sides all have length $1$.
Determine the limit inferior and the limit superior of the sequence defined by
$$\frac1{n^2}\log_2C(n).$$
2019 Iran Team Selection Test, 1
$S$ is a subset of Natural numbers which has infinite members.
$$S’=\left\{x^y+y^x: \, x,y\in S, \, x\neq y\right\}$$
Prove the set of prime divisors of $S’$ has also infinite members
[i]Proposed by Yahya Motevassel[/i]
2020 LIMIT Category 1, 1
Find all polynomial $P(x)$ with degree $\leq n$and non negative coefficients such that $$P(x)P(\frac{1}{x})\leq P(1)^2$$ for all positive $x$. Here $n$ is a natuaral number
2022 IOQM India, 4
Consider the set of all 6-digit numbers consisting of only three digits, $a,b,c$ where $a,b,c$ are distinct. Suppose the sum of all these numbers is $593999406$. What is the largest remainder when the three digit number $abc$ is divided by $100$?
1959 Polish MO Finals, 6
Given a triangle in which the sides $ a $, $ b $, $ c $ form an arithmetic progression and the angles also form an arithmetic progression. Find the ratios of the sides of this triangle.
2004 China Team Selection Test, 1
Given sequence $ \{ c_n \}$ satisfying the conditions that $ c_0\equal{}1$, $ c_1\equal{}0$, $ c_2\equal{}2005$, and $ c_{n\plus{}2}\equal{}\minus{}3c_n \minus{} 4c_{n\minus{}1} \plus{}2008$, ($ n\equal{}1,2,3, \cdots$). Let $ \{ a_n \}$ be another sequence such that $ a_n\equal{}5(c_{n\plus{}1} \minus{} c_n) \cdot (502 \minus{} c_{n\minus{}1} \minus{} c_{n\minus{}2}) \plus{} 4^n \times 2004 \times 501$, ($ n\equal{}2,3, \cdots$).
Is $ a_n$ a perfect square for every $ n > 2$?
2018 India PRMO, 22
A positive integer $k$ is said to be [i]good [/i] if there exists a partition of $ \{1, 2, 3,..., 20\}$ into disjoint proper subsets such that the sum of the numbers in each subset of the partition is $k$. How many [i]good [/i] numbers are there?
2020 USOMO, 5
A finite set $S$ of points in the coordinate plane is called [i]overdetermined[/i] if $|S|\ge 2$ and there exists a nonzero polynomial $P(t)$, with real coefficients and of degree at most $|S|-2$, satisfying $P(x)=y$ for every point $(x,y)\in S$.
For each integer $n\ge 2$, find the largest integer $k$ (in terms of $n$) such that there exists a set of $n$ distinct points that is [i]not[/i] overdetermined, but has $k$ overdetermined subsets.
[i]Proposed by Carl Schildkraut[/i]
2019 Lusophon Mathematical Olympiad, 2
Prove that for every $n$ nonzero integer , there are infinite triples of nonzero integers $a, b$ and $c$ that satisfy the conditions:
1. $a + b + c = n$
2. $ax^2 + bx + c = 0$ has rational roots.
2006 Harvard-MIT Mathematics Tournament, 8
Triangle $ABC$ has a right angle at $B$. Point $D$ lies on side $BC$ such that $3\angle BAD = \angle BAC$. Given $AC=2$ and $CD=1$, compute $BD$.
2016 NIMO Summer Contest, 14
Find the smallest positive integer $n$ such that $n^2+4$ has at least four distinct prime factors.
[i]Proposed by Michael Tang[/i]
2016 IFYM, Sozopol, 6
$a,b,m,k\in \mathbb{Z}$, $a,b,m>2,k>1$, for which $k^n a+b$ is an $m$-th power of a natural number for $\forall n\in \mathbb{N}$. Prove that $b$ is an $m$-th power of a non-negative integer.
LMT Team Rounds 2010-20, A19
Euhan and Minjune are playing a game. They choose a number $N$ so that they can only say integers up to $N$. Euhan starts by saying the $1$, and each player takes turns saying either $n+1$ or $4n$ (if possible), where $n$ is the last number said. The player who says $N$ wins. What is the smallest number larger than $2019$ for which Minjune has a winning strategy?
[i]Proposed by Janabel Xia[/i]
Kvant 2020, M2596
The circle $\omega{}$ is inscribed in the quadrilateral $ABCD$. Prove that the diameter of the circle $\omega{}$ does not exceed the length of the segment connecting the midpoints of the sides $BC$ and $AD$.
[i]Proposed by O. Yuzhakov[/i]
2010 Turkey Junior National Olympiad, 1
A circle that passes through the vertex $A$ of a rectangle $ABCD$ intersects the side $AB$ at a second point $E$ different from $B.$ A line passing through $B$ is tangent to this circle at a point $T,$ and the circle with center $B$ and passing through $T$ intersects the side $BC$ at the point $F.$ Show that if $\angle CDF= \angle BFE,$ then $\angle EDF=\angle CDF.$
1989 Tournament Of Towns, (236) 4
The numbers $2^{1989}$ and $5^{1989}$ are written out one after the other (in decimal notation). How many digits are written altogether?
(G. Galperin)
2009 India IMO Training Camp, 8
Let $ n$ be a natural number $ \ge 2$ which divides $ 3^n\plus{}4^n$.Prove That $ 7\mid n$.
1989 AMC 12/AHSME, 16
A lattice point is a point in the plane with integer coordinates. How many lattice points are on the line segment whose endpoints are (3,17) and (48,281)? (Include both endpoints of the segment in your count.)
$\textbf{(A)}\ 2 \qquad
\textbf{(B)}\ 4 \qquad
\textbf{(C)}\ 6 \qquad
\textbf{(D)}\ 16 \qquad
\textbf{(E)}\ 46$
2016 China Team Selection Test, 2
Find the smallest positive number $\lambda $ , such that for any complex numbers ${z_1},{z_2},{z_3}\in\{z\in C\big| |z|<1\}$ ,if $z_1+z_2+z_3=0$, then $$\left|z_1z_2 +z_2z_3+z_3z_1\right|^2+\left|z_1z_2z_3\right|^2 <\lambda .$$
2012 France Team Selection Test, 2
Determine all non-constant polynomials $X^n+a_{n-1}X^{n-1}+\cdots +a_1X+a_0$ with integer coefficients for which the roots are exactly the numbers $a_0,a_1,\ldots ,a_{n-1}$ (with multiplicity).
Kyiv City MO Seniors 2003+ geometry, 2015.11.4
In the acute-angled triangle $ ABC $, the sides $ AB $ and $BC$ have different lengths, and the extension of the median $ BM $ intersects the circumscribed circle at the point $ N $. On this circle we note such a point $ D $ that $ \angle BDH = 90 {} ^ \circ $, where $ H $ is the point of intersection of the altitudes of the triangle $ ABC $. The point $K$ is chosen so that $ ANCK $ is a parallelogram. Prove that the lines $ AC $, $ KH $ and $ BD $ intersect at one point.
(Igor Nagel)