Found problems: 85335
1996 National High School Mathematics League, 1
The sum of first $n$ items of squence $(a_n)$ : $S_n$ satisfies that $S_n=2a_n-1$, squence $(b_n)$ satisfies that $b_{k+1}=a_k+b_k$ for all $k=1,2,\cdots$. Find the sum of first $n$ items of $(b_n)$.
2004 Gheorghe Vranceanu, 2
[b]a)[/b] Let be an even number $ n\ge 4 $ and $ n $ positive real numbers $ x_1,x_2,\ldots ,x_n. $ Prove that:
$$ \min_{1\le i\le n/2} \frac{x_i}{x_{i+n/2}}\le \frac{x_1+x_2+\cdots +x_{n/2}}{x_{1+n/2}+ x_{2+n/2} +\cdots + x_n}\le \max_{1\le i\le n/2} \frac{x_i}{x_{i+n/2}}$$
[b]b)[/b] Let be $ m\ge 1 $ pairwise distinct natural numbers $ a,b,\ldots ,c. $ Show that:
$$ \frac{ab\cdots c}{a+b+\cdots +c}\ge (m-1)!\cdot\frac{2}{m+1} $$
[i]M. Tetiva[/i]
2015 Rioplatense Mathematical Olympiad, Level 3, 5
For a positive integer number $n$ we denote $d(n)$ as the greatest common divisor of the binomial coefficients $\dbinom{n+1}{n} , \dbinom{n+2}{n} ,..., \dbinom{2n}{n}$.
Find all possible values of $d(n)$
2004 239 Open Mathematical Olympiad, 7
Given an isosceles triangle $ABC$ (with $AB=BC$). A point $X$ is chosen on a side $AC$. Some circle passes through $X$, touches the side $AC$ and intersects the circumcircle of triangle $ABC$ in points $M$ and $N$ such that the segment $MN$ bisects $BX$ and intersects sides $AB$ and $BC$ in points $P$ and $Q$. Prove that the circumcircle of triangle $PBQ$ passes through the circumcentre of triangle $ABC$.
[b]proposed by Sergej Berlov[/b]
2018 Online Math Open Problems, 13
Compute the largest possible number of distinct real solutions for $x$ to the equation \[x^6+ax^5+60x^4-159x^3+240x^2+bx+c=0,\] where $a$, $b$, and $c$ are real numbers.
[i]Proposed by Tristan Shin
2005 Paraguay Mathematical Olympiad, 4
In the expression $t=\frac{8a+ 1}{b}$ where $a, b, t$ are positive integers, where $b <7$. Determine the values of $a$ and$ b$ that allow to obtain $t$ under the established conditions.
2004 Hong kong National Olympiad, 1
Let $a_{1},a_{2},...,a_{n+1}(n>1)$ are positive real numbers such that $a_{2}-a_{1}=a_{3}-a_{2}=...=a_{n+1}-a_{n}$. Prove that $\sum_{k=2}^{n}\frac{1}{a_{k}^{2}}\leq\frac{n-1}{2}.\frac{a_{1}a_{n}+a_{2}a_{n+1}}{a_{1}a_{2}a_{n}a_{n+1}}$
2017 Online Math Open Problems, 4
Steven draws a line segment between every two of the points \[A(2,2), B(-2,2), C(-2,-2), D(2,-2), E(1,0), F(0,1), G(-1,0), H(0,-1).\] How many regions does he divide the square $ABCD$ into?
[i]Proposed by Michael Ren
1985 IMO Longlists, 75
Let $ABCD$ be a rectangle, $AB = a, BC = b$. Consider the family of parallel and equidistant straight lines (the distance between two consecutive lines being $d$) that are at an the angle $\phi, 0 \leq \phi \leq 90^{\circ},$ with respect to $AB$. Let $L$ be the sum of the lengths of all the segments intersecting the rectangle. Find:
[i](a)[/i] how $L $ varies,
[i](b)[/i] a necessary and sufficient condition for $L$ to be a constant, and
[i](c)[/i] the value of this constant.
2004 Postal Coaching, 19
Suppose a circle passes through the feet of the symmedians of a non-isosceles triangle $ABC$ , and is tangent to one of the sides. Show that $a^2 +b^2, b^2 + c^2 , c^2 + a^2$ are in geometric progression when taken in some order
MOAA Gunga Bowls, 2021.5
Joshua rolls two dice and records the product of the numbers face up. The probability that this product is composite can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
[i]Proposed by Nathan Xiong[/i]
1989 China Team Selection Test, 3
$1989$ equal circles are arbitrarily placed on the table without overlap. What is the least number of colors are needed such that all the circles can be painted with any two tangential circles colored differently.
2013 IMC, 2
Let $\displaystyle{f:{\cal R} \to {\cal R}}$ be a twice differentiable function. Suppose $\displaystyle{f\left( 0 \right) = 0}$. Prove there exists $\displaystyle{\xi \in \left( { - \frac{\pi }{2},\frac{\pi }{2}} \right)}$ such that \[\displaystyle{f''\left( \xi \right) = f\left( \xi \right)\left( {1 + 2{{\tan }^2}\xi } \right)}.\]
[i]Proposed by Karen Keryan, Yerevan State University, Yerevan, Armenia.[/i]
2011 Iran MO (3rd Round), 3
In triangle $ABC$, $X$ and $Y$ are the tangency points of incircle (with center $I$) with sides $AB$ and $AC$ respectively. A tangent line to the circumcircle of triangle $ABC$ (with center $O$) at point $A$, intersects the extension of $BC$ at $D$. If $D,X$ and $Y$ are collinear then prove that $D,I$ and $O$ are also collinear.
[i]proposed by Amirhossein Zabeti[/i]
2023 JBMO Shortlist, G1
Let $ABC$ be a triangle with circumcentre $O$ and circumcircle $\Omega$. $\Gamma$ is the circle passing through $O,B$ and tangent to $AB$ at $B$. Let $\Gamma$ intersect $\Omega$ a second time at $P \neq B$. The circle passing through $P,C$ and tangent to $AC$ at $C$ intersects with $\Gamma$ at $M$. Prove that $|MP|=|MC|$.
2015 Balkan MO Shortlist, A2
Let $a,b,c$ be sidelengths of a triangle and $r,R,s$ be the inradius, the circumradius and the semiperimeter respectively of the same triangle. Prove that:
$$\frac{1}{a + b} + \frac{1}{a + c} + \frac{1}{b + c}
\leq \frac{r}{16Rs}+\frac{s}{16Rr} + \frac{11}{8s}$$
(Albania)
2017 HMNT, 2
[b]H[/b]orizontal parallel segments $AB=10$ and $CD=15$ are the bases of trapezoid $ABCD$. Circle $\gamma$ of radius $6$ has center within the trapezoid and is tangent to sides $AB$, $BC$, and $DA$. If side $CD$ cuts out an arc of $\gamma$ measuring $120^{\circ}$, find the area of $ABCD$.
2012 Kyiv Mathematical Festival, 3
Let $O$ be the circumcenter of triangle $ABC$: Points $D$ and $E$ are chosen at sides $AB$ and $AC$ respectively such that $\angle ADO = \angle AEO = 60^o$ and $BDEC$ is inscribed quadrangle. Prove or disprove that $ABC$ is isosceles triangle.
2023 AMC 8, 19
An equilateral triangle is placed inside a larger equilateral triangle so that the region between them can be divided into three congruent trapezoids, as shown below. The side length of the inner triangle is $\tfrac23$ the side length of the larger triangle. What is the ratio of the area of one trapezoid to the area of the inner triangle?
[asy]
size(5cm);
fill((0,0)--(2/3,1.155/3)--(4-(4-2)/3,1.155/3)--(4,0)--cycle,lightgray*0.5+mediumgray*0.5);
draw((0,0)--(4,0)--(2,2*sqrt(3))--cycle);
//center: 2,1.155
draw((2/3,1.155/3)--(4-(4-2)/3,1.155/3)--(2,2*sqrt(3)-0.7697)--cycle);
dot((0,0)^^(4,0)^^(2,2*sqrt(3))^^(2/3,1.155/3)^^(4-(4-2)/3,1.155/3)^^(2,2*sqrt(3)-0.7697));
draw((0,0)--(2/3,1.155/3));
draw((4,0)--(4-(4-2)/3,1.155/3));
draw((2,2*sqrt(3))--(2,2*sqrt(3)-0.7697));
[/asy]
$\textbf{(A) } 1:3\qquad\textbf{(B) } 3:8\qquad\textbf{(C) } 5:12\qquad\textbf{(D) } 7:16\qquad\textbf{(E) } 4:9$
1966 Swedish Mathematical Competition, 5
Let $f(r)$ be the number of lattice points inside the circle radius $r$, center the origin.
Show that $\lim_{r\to \infty} \frac{f(r)}{r^2}$ exists and find it. If the limit is $k$, put $g(r) = f(r) - kr^2$.
Is it true that $\lim_{r\to \infty} \frac{g(r)}{r^h} = 0$ for any $h < 2$?
2017 Moscow Mathematical Olympiad, 6
There are $36$ gangsters bands.And there are war between some bands. Every gangster can belongs to several bands and every 2 gangsters belongs to different set of bands. Gangster can not be in feuding bands. Also for every gangster is true, that every band, where this gangster is not in, is in war with some band, where this gangster is in. What is maximum number of gangsters in city?
2016 HMNT, 16-18
16. Create a cube $C_1$ with edge length $1$. Take the centers of the faces and connect them to form an octahedron $O_1$. Take the centers of the octahedron’s faces and connect them to form a new cube $C_2$. Continue this process infinitely. Find the sum of all the surface areas of the cubes and octahedrons.
17. Let $p(x) = x^2 - x + 1$. Let $\alpha$ be a root of $p(p(p(p(x)))$. Find the value of
$$(p(\alpha) - 1)p(\alpha)p(p(\alpha))p(p(p(\alpha))$$
18. An $8$ by $8$ grid of numbers obeys the following pattern:
1) The first row and first column consist of all $1$s.
2) The entry in the $i$th row and $j$th column equals the sum of the numbers in the $(i - 1)$ by $(j - 1)$ sub-grid with row less than i and column less than $j$.
What is the number in the 8th row and 8th column?
2025 Francophone Mathematical Olympiad, 4
Charlotte writes the integers $1,2,3,\ldots,2025$ on the board. Charlotte has two operations available: the GCD operation and the LCM operation.
[list]
[*]The GCD operation consists of choosing two integers $a$ and $b$ written on the board, erasing them, and writing the integer $\operatorname{gcd}(a, b)$.
[*]The LCM operation consists of choosing two integers $a$ and $b$ written on the board, erasing them, and writing the integer $\operatorname{lcm}(a, b)$.
[/list]
An integer $N$ is called a [i]winning number[/i] if there exists a sequence of operations such that, at the end, the only integer left on the board is $N$. Find all winning integers among $\{1,2,3,\ldots,2025\}$ and, for each of them, determine the minimum number of GCD operations Charlotte must use.
[b]Note:[/b] The number $\operatorname{gcd}(a, b)$ denotes the [i]greatest common divisor[/i] of $a$ and $b$, while the number $\operatorname{lcm}(a, b)$ denotes the [i]least common multiple[/i] of $a$ and $b$.
1987 Putnam, B6
Let $F$ be the field of $p^2$ elements, where $p$ is an odd prime. Suppose $S$ is a set of $(p^2-1)/2$ distinct nonzero elements of $F$ with the property that for each $a\neq 0$ in $F$, exactly one of $a$ and $-a$ is in $S$. Let $N$ be the number of elements in the intersection $S \cap \{2a: a \in S\}$. Prove that $N$ is even.
LMT Team Rounds 2021+, 5
Find the sum $$\sum^{2020}_{n=1} \gcd (n^3 -2n^2 +2021,n^2 -3n +3).$$