Found problems: 85335
2001 National High School Mathematics League, 13
$(a_n)$ is an arithmetic sequence, $(b_n)$ is a geometric sequence. If $b_1=a_1^2,b_2=a_2^2,b_3=a_3^2(a_1<a_2)$, and $\lim_{n\to\infty}(b_1+b_2+\cdots+b_n)=\sqrt2+1$, find $a_n$.
2006 China Team Selection Test, 2
Find all positive integer pairs $(a,n)$ such that $\frac{(a+1)^n-a^n}{n}$ is an integer.
2021 USMCA, 1
Let $a_1, a_2, \ldots, a_{2021}$ be a sequence, where each $a_i$ is a positive factor of $2021$. How many possible values are there for the product $a_1 a_2 \cdots a_{2021}$?
1992 India National Olympiad, 10
Determine all functions $f : \mathbb{R} - [0,1] \to \mathbb{R}$ such that \[ f(x) + f \left( \dfrac{1}{1-x} \right) = \dfrac{2(1-2x)}{x(1-x)} . \]
2005 Junior Balkan Team Selection Tests - Moldova, 7
Let $p$ be a prime number and $a$ and $n$ positive nonzero integers. Prove that
if $2^p + 3^p = a^n$ then $n=1$
2010 Albania Team Selection Test, 5
[b]a)[/b] Let's consider a finite number of big circles of a sphere that do not pass all from a point. Show that there exists such a point that is found only in two of the circles. (With big circle we understand the circles with radius equal to the radius of the sphere.)
[b]b)[/b] Using the result of part $a)$ show that, for a set of $n$ points in a plane, that are not all in a line, there exists a line that passes through only two points of the given set.
2021 239 Open Mathematical Olympiad, 3
Given are two distinct sequences of positive integers $(a_n)$ and $(b_n)$, such that their first two members are coprime and smaller than $1000$, and each of the next members is the sum of the previous two.
8-9 grade Prove that if $a_n$ is divisible by $b_n$, then $n<50$
10-11 grade Prove that if $a_n^{100}$ is divisible by $b_n$ then $n<5000$
1999 IberoAmerican, 1
Let $B$ be an integer greater than 10 such that everyone of its digits belongs to the set $\{1,3,7,9\}$. Show that $B$ has a [b]prime divisor[/b] greater than or equal to 11.
2022 JBMO Shortlist, C1
Anna and Bob, with Anna starting first, alternately color the integers of the set $S = \{1, 2, ..., 2022 \}$ red or blue. At their turn each one can color any uncolored number of $S$ they wish with any color they wish. The game ends when all numbers of $S$ get colored. Let $N$ be the number of pairs $(a, b)$, where $a$ and $b$ are elements of $S$, such that $a$, $b$ have the same color, and $b - a = 3$.
Anna wishes to maximize $N$. What is the maximum value of $N$ that she can achieve regardless of how Bob plays?
2023 Turkey EGMO TST, 6
Let $ABC$ be a scalene triangle and $l_0$ be a line that is tangent to the circumcircle of $ABC$ at point $A$. Let $l$ be a variable line which is parallel to line $l_0$. Let $l$ intersect segment $AB$ and $AC$ at the point $X$, $Y$ respectively. $BY$ and $CX$ intersects at point $T$ and the line $AT$ intersects the circumcirle of $ABC$ at $Z$. Prove that as $l$ varies, circumcircle of $XYZ$ passes through a fixed point.
2005 Greece National Olympiad, 3
We know that $k$ is a positive integer and the equation \[ x^3+y^3-2y(x^2-xy+y^2)=k^2(x-y) \quad (1) \] has one solution $(x_0,y_0)$ with
$x_0,y_0\in \mathbb{Z}-\{0\}$ and $x_0\neq y_0$. Prove that
i) the equation (1) has a finite number of solutions $(x,y)$ with $x,y\in \mathbb{Z}$ and $x\neq y$;
ii) it is possible to find $11$ addition different solutions $(X,Y)$ of the equation (1) with $X,Y\in \mathbb{Z}-\{0\}$ and $X\neq Y$ where $X,Y$ are functions of $x_0,y_0$.
1987 India National Olympiad, 8
Three congruent circles have a common point $ O$ and lie inside a given triangle. Each circle touches a pair of sides of the triangle. Prove that the incentre and the circumcentre of the triangle and the common point $ O$ are collinear.
2011 ELMO Shortlist, 1
Let $n$ be a positive integer. There are $n$ soldiers stationed on the $n$th root of unity in the complex plane. Each round, you pick a point, and all the soldiers shoot in a straight line towards that point; if their shot hits another soldier, the hit soldier dies and no longer shoots during the next round. What is the minimum number of rounds, in terms of $n$, required to eliminate all the soldiers?
[i]David Yang.[/i]
2023 AIME, 3
A plane contains $40$ lines, no $2$ of which are parallel. Suppose there are $3$ points where exactly $3$ lines intersect, $4$ points where exactly $4$ lines intersect, $5$ points where exactly $5$ lines intersect, $6$ points where exactly $6$ lines intersect, and no points where more than $6$ lines intersect. Find the number of points where exactly $2$ lines intersect.
2021 LMT Spring, A2
The function $f(x)$ has the property that $f(x) = -\frac{1}{f(x-1)}.$ Given that $f(0)=-\frac{1}{21},$ find the value of $f(2021).$
[i]Proposed by Ada Tsui[/i]
2006 International Zhautykov Olympiad, 2
Let $ a,b,c,d$ be real numbers with sum 0. Prove the inequality:
\[ (ab \plus{} ac \plus{} ad \plus{} bc \plus{} bd \plus{} cd)^2 \plus{} 12\geq 6(abc \plus{} abd \plus{} acd \plus{} bcd).
\]
2016 Dutch IMO TST, 1
Prove that for all positive reals $a, b,c$ we have: $a +\sqrt{ab}+ \sqrt[3]{abc}\le \frac43 (a + b + c)$
2019 PUMaC Geometry A, 5
Let $\Gamma$ be a circle with center $A$, radius $1$ and diameter $BX$. Let $\Omega$ be a circle with center $C$, radius $1$ and diameter $DY $, where $X$ and $Y$ are on the same side of $AC$. $\Gamma$ meets $\Omega$ at two points, one of which is $Z$. The lines tangent to $\Gamma$ and $\Omega$ that pass through $Z$ cut out a sector of the plane containing no part of either circle and with angle $60^\circ$. If $\angle XY C = \angle CAB$ and $\angle XCD = 90^\circ$, then the length of $XY$ can be written in the form $\tfrac{\sqrt a+\sqrt b}{c}$ for integers $a, b, c$ where $\gcd(a, b, c) = 1$. Find $a + b + c$.
2017 Harvard-MIT Mathematics Tournament, 10
Compute the number of possible words $w=w_1w_2\dots w_{100}$ satisfying:
$\bullet$ $w$ has exactly $50$ $A$'s and $50$ $B$'s (and no other letter).
$\bullet$ For $i=1,2,\dots,100$, the number of $A$'s among $w_1, w_2, \dots, w_i$ is at most the number of $B$'s among $w_1, w_2, \dots, w_i$.
$\bullet$ For all $i=44,45,\dots,57$, if $w_i$ is a $B$, then $w_{i+1}$ must be a $B$.
1985 IMO Longlists, 80
Let $E = \{1, 2, \dots , 16\}$ and let $M$ be the collection of all $4 \times 4$ matrices whose entries are distinct members of $E$. If a matrix $A = (a_{ij} )_{4\times4}$ is chosen randomly from $M$, compute the probability $p(k)$ of $\max_i \min_j a_{ij} = k$ for $k \in E$. Furthermore, determine $l \in E$ such that $p(l) = \max \{p(k) | k \in E \}.$
2019-IMOC, A5
Find all functions $f : \mathbb N \mapsto \mathbb N$ such that the following identity
$$f^{x+1}(y)+f^{y+1}(x)=2f(x+y)$$
holds for all $x,y \in \mathbb N$
2004 Turkey Team Selection Test, 3
Let $n$ be a positive integer. Determine integers, $n+1 \leq r \leq 3n+2$ such that for all integers $a_1,a_2,\dots,a_m,b_1,b_2,\dots,b_m$ satisfying the equations
\[
a_1b_1^k+a_2b_2^k+\dots + a_mb_m^k=0 \]
for every $1 \leq k \leq n$, the condition
\[
r \mid
a_1b_1^r+a_2b_2^r+\dots + a_mb_m^r=0 \]
also holds.
2002 BAMO, 2
In the illustration, a regular hexagon and a regular octagon have been tiled with rhombuses.
In each case, the sides of the rhombuses are the same length as the sides of the regular polygon.
(a) Tile a regular decagon ($10$-gon) into rhombuses in this manner.
(b) Tile a regular dodecagon ($12$-gon) into rhombuses in this manner.
(c) How many rhombuses are in a tiling by rhombuses of a $2002$-gon?
Justify your answer.
[img]https://cdn.artofproblemsolving.com/attachments/8/a/8413e4e2712609eba07786e34ba2ce4aa72888.png[/img]
MBMT Guts Rounds, 2015.24
In cyclic quadrilateral $ABCD$, $\angle DBC = 90^\circ$ and $\angle CAB = 30^\circ$. The diagonals of $ABCD$ meet at $E$. If $\frac{BE}{ED} = 2$ and $CD = 60$, compute $AD$. (Note: a cyclic quadrilateral is a quadrilateral that can be inscribed in a circle.)
2007 National Olympiad First Round, 13
Let $ABCD$ be an circumscribed quadrilateral such that $m(\widehat{A})=m(\widehat{B})=120^\circ$, $m(\widehat{C})=30^\circ$, and $|BC|=2$. What is $|AD|$?
$
\textbf{(A)}\ \sqrt 3 - 1
\qquad\textbf{(B)}\ \sqrt 2 - 3
\qquad\textbf{(C)}\ \sqrt 6 - \sqrt 2
\qquad\textbf{(D)}\ 2 - \sqrt 2
\qquad\textbf{(E)}\ 3 - \sqrt 3
$