Found problems: 85335
2019 Saint Petersburg Mathematical Olympiad, 2
In the city built are $2019$ metro stations. Some pairs of stations are connected. tunnels, and from any station through the tunnels you can reach any other. The mayor ordered to organize several metro lines: each line should include several different stations connected in series by tunnels (several lines can pass through the same tunnel), and in each station must lie at least on one line. To save money no more than $k$ lines should be made. It turned out that the order of the mayor is not feasible. What is the largest $k$ it could to happen?
1989 IMO Longlists, 5
Let $ n > 1$ be a fixed integer. Define functions $ f_0(x) \equal{} 0,$ $ f_1(x) \equal{} 1 \minus{} \cos(x),$ and for $ k > 0,$ \[ f_{k\plus{}1}(x) \equal{} f_k(x) \cdot \cos(x) \minus{} f_{k\minus{}1}(x).\] If $ F(x) \equal{} \sum^n_{r\equal{}1} f_r(x),$ prove that
[b](a)[/b] $ 0 < F(x) < 1$ for $ 0 < x < \frac{\pi}{n\plus{}1},$ and
[b](b)[/b] $ F(x) > 1$ for $ \frac{\pi}{n\plus{}1} < x < \frac{\pi}{n}.$
1979 Polish MO Finals, 4
Let $A > 1$ and $B > 1$ be real numbers and (xn) be a sequence of numbers in the interval $[1,AB]$. Prove that there exists a sequence $(y_n)$ of numbers in the interval $[1,A]$ such that
$$\frac{x_m}{x_n}\le B\frac{y_m}{y_n} \,\,\, for \,\,\, all \,\,\, m,n = 1,2,...$$
2009 Jozsef Wildt International Math Competition, W. 20
If $x \in \mathbb{R}\backslash \left \{\frac{k\pi}{2}\ |\ k\in \mathbb{Z} \right \}$, then $$\left (\sum \limits_{0\leq j<k\leq n} \sin (2(j+k)x)\right )^2 + \left (\sum \limits_{0\leq j<k\leq n} \cos (2(j+k)x)\right )^2 = \frac{\sin ^2 nx \sin ^2 (n+1)x}{\sin ^2x \sin^22x}$$
2011 Indonesia TST, 4
Given $N = 2^ap_1p_2...p_m$, $m \ge 1$, $a \in N$ with $p_1, p_2,..., p_m$ are different primes. It is known that $\sigma (N) = 3N $ where $\sigma (N)$ is the sum of all positive integers which are factors of $N$. Show that there exists a prime number $p$ such that $2^p- 1$ is also a prime, and $2^p - 1|N$.
2011 Pre-Preparation Course Examination, 2
We say that a covering of a $m\times n$ rectangle with dominos has a wall if there exists a horizontal or vertical line that splits the rectangle into two smaller rectangles and doesn't cut any of the dominos. prove that if these three conditions are satisfied:
[b]a)[/b] $mn$ is an even number
[b]b)[/b] $m\ge 5$ and $n\ge 5$
[b]c)[/b] $(m,n)\neq(6,6)$
then we can cover the rectangle with dominos in such a way that we have no walls. (20 points)
2020 Greece Team Selection Test, 4
Let $a$ and $b$ be two positive integers. Prove that the integer
\[a^2+\left\lceil\frac{4a^2}b\right\rceil\]
is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.)
[i]Russia[/i]
2022 BMT, 1
For lunch, Lamy, Botan, Nene, and Polka each choose one of three options: a hot dog, a slice of pizza, or a hamburger. Lamy and Botan choose different items, and Nene and Polka choose the same item. In how many ways could they choose their items?
2024 Mozambican National MO Selection Test, P1
A school security guard works from Monday to Saturday from $7:30 am$ to $12:00 pm$ ($7:30$ to $12:00$). He also works the night shift, from Monday to Friday from $6pm$to $10pm$ ($18:00$ to $22:00$) . He receives $75MT$ per hour, up to $40$ hours of work per week. For the remaining hours of weekly work, he receives $95MT$ per hour. So, considering that a month has four weeks, what will be this security guard's monthly salary?
2023 Balkan MO Shortlist, G2
Let $ABCD$ be a cyclic quadrilateral with circumcenter $O$ lying in the interior. Let $E$ and $F$ be the midpoints of the segments $BC$ and $AD$, respectively. Let $X$ be the point lying on the same side of the line $EF$ as the vertex $C$ such that $\triangle EXF$ and $\triangle BOA$ are similar. Prove that $XC = XD$.
2022 China Team Selection Test, 5
Let $n$ be a positive integer, $x_1,x_2,\ldots,x_{2n}$ be non-negative real numbers with sum $4$. Prove that there exist integer $p$ and $q$, with $0 \le q \le n-1$, such that
\[ \sum_{i=1}^q x_{p+2i-1} \le 1 \mbox{ and } \sum_{i=q+1}^{n-1} x_{p+2i} \le 1, \]
where the indices are take modulo $2n$.
[i]Note:[/i] If $q=0$, then $\sum_{i=1}^q x_{p+2i-1}=0$; if $q=n-1$, then $\sum_{i=q+1}^{n-1} x_{p+2i}=0$.
2023 HMNT, 27
Compute the number of ways to color the vertices of a regular heptagon red, green, or blue (with rotations and reflections distinct) such that no isosceles triangle whose vertices are vertices of the heptagon has all three vertices the same color.
2013 USAMTS Problems, 2
Let $ABCD$ be a quadrilateral with $\overline{AB}\parallel\overline{CD}$, $AB=16$, $CD=12$, and $BC<AD$. A circle with diameter $12$ is inside of $ABCD$ and tangent to all four sides. Find $BC$.
1998 Baltic Way, 18
Determine all positive integers $n$ for which there exists a set $S$ with the following properties:
(i) $S$ consists of $n$ positive integers, all smaller than $2^{n-1}$;
(ii) for any two distinct subsets $A$ and $B$ of $S$, the sum of the elements of $A$ is different from the sum of the elements of $B$.
2022 Princeton University Math Competition, A1 / B3
Circle $\Gamma$ is centered at $(0, 0)$ in the plane with radius $2022\sqrt3$. Circle $\Omega$ is centered on the $x$-axis, passes through the point $A = (6066, 0)$, and intersects $\Gamma$ orthogonally at the point $P = (x, y)$ with $y > 0$. If the length of the minor arc $AP$ on $\Omega$ can be expressed as $\frac{m\pi}{n}$ forrelatively prime positive integers $m, n$, find $m + n$.
(Two circles intersect orthogonally at a point $P$ if the tangent lines at $P$ form a right angle.)
2014 NIMO Problems, 8
Let $a$, $b$, $c$, $d$ be complex numbers satisfying
\begin{align*}
5 &= a+b+c+d \\
125 &= (5-a)^4 + (5-b)^4 + (5-c)^4 + (5-d)^4 \\
1205 &= (a+b)^4 + (b+c)^4 + (c+d)^4 + (d+a)^4 + (a+c)^4 + (b+d)^4 \\
25 &= a^4+b^4+c^4+d^4
\end{align*}
Compute $abcd$.
[i]Proposed by Evan Chen[/i]
PEN S Problems, 36
For every natural number $n$, denote $Q(n)$ the sum of the digits in the decimal representation of $n$. Prove that there are infinitely many natural numbers $k$ with $Q(3^{k})>Q(3^{k+1})$.
1967 IMO Shortlist, 4
Does there exist an integer such that its cube is equal to $3n^2 + 3n + 7,$ where $n$ is an integer.
2005 Czech-Polish-Slovak Match, 4
We distribute $n\ge1$ labelled balls among nine persons $A,B,C, \dots , I$. How many ways are there to do this so that $A$ gets the same number of balls as $B,C,D$ and $E$ together?
1976 IMO Longlists, 15
Let $ABC$ and $A'B'C'$ be any two coplanar triangles. Let $L$ be a point such that $AL || BC, A'L || B'C'$ , and $M,N$ similarly defined. The line $BC$ meets $B'C'$ at $P$, and similarly defined are $Q$ and $R$. Prove that $PL, QM, RN$ are concurrent.
2012 Morocco TST, 1
Find all positive integers $n, k$ such that $(n-1)!=n^{k}-1$.
2005 Postal Coaching, 25
Find all pairs of cubic equations $x^3 +ax^2 +bx +c =0$ and $x^3 +bx^2 + ax +c = 0$ where $a,b,c$ are integers, such that each equation has three integer roots and both the equations have exactly one common root.
2018 PUMaC Number Theory A, 1
Find the number of positive integers $n < 2018$ such that $25^n + 9^n$ is divisible by $13$.
2011 Oral Moscow Geometry Olympiad, 1
$AD$ and $BE$ are the altitudes of the triangle $ABC$. It turned out that the point $C'$, symmetric to the vertex $C$ wrt to the midpoint of the segment $DE$, lies on the side $AB$. Prove that $AB$ is tangent to the circle circumscribed around the triangle $DEC'$.
2013 Korea National Olympiad, 2
Let $ a, b, c>0 $ such that $ ab+bc+ca=3 $. Prove that
\[ \sum_{cyc} { \frac{ (a+b)^{3} }{ {(2(a+b)(a^2 + b^2))}^{\frac{1}{3}}} \ge 12 }\]