Found problems: 85335
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
$
2024 Malaysia IMONST 2, 6
Rui Xuen has a circle $\omega$ with center $O$, and a square $ABCJ$ with vertices on $\omega$. Let $M$ be the midpoint of $AB$, and let $\Gamma$ be the circle passing through the points $J$, $O$, $M$. Suppose $\Gamma$ intersect line $AJ$ at a point $P \neq J$, and suppose $\Gamma$ intersect $\omega$ at a point $Q \neq J$. A point $R$ lies on side $BC$ so that $RC = 3RB$.
Help Rui Xuen prove that the points $P$, $Q$, $R$ are collinear.
2018 CMIMC Algebra, 1
Misha has accepted a job in the mines and will produce one ore each day. At the market, he is able to buy or sell one ore for \$3, buy or sell bundles of three wheat for \$12 each, or $\textit{sell}$ one wheat for one ore. His ultimate goal is to build a city, which requires three ore and two wheat. How many dollars must Misha begin with in order to build a city after three days of working?
2009 IMC, 4
Let $p(z)=a_0+a_1z+a_2z^2+\cdots+a_nz^n$ be a complex polynomial. Suppose that $1=c_0\ge c_1\ge \cdots \ge c_n\ge 0$ is a sequence of real numbers which form a convex sequence. (That is $2c_k\le c_{k-1}+c_{k+1}$ for every $k=1,2,\cdots ,n-1$ ) and consider the polynomial
\[ q(z)=c_0a_0+c_1a_1z+c_2a_2z^2+\cdots +c_na_nz^n \]
Prove that :
\[ \max_{|z|\le 1}q(z)\le \max_{|z|\le 1}p(z) \]
2012 NIMO Problems, 4
When flipped, coin A shows heads $\textstyle\frac{1}{3}$ of the time, coin B shows heads $\textstyle\frac{1}{2}$ of the time, and coin C shows heads $\textstyle\frac{2}{3}$ of the time. Anna selects one of the coins at random and flips it four times, yielding three heads and one tail. The probability that Anna flipped coin A can be expressed as $\textstyle\frac{p}{q}$ for relatively prime positive integers $p$ and $q$. Compute $p + q$.
[i]Proposed by Eugene Chen[/i]
2011 May Olympiad, 4
Using several white edge cubes of side $ 1$, Guille builds a large cube. Then he chooses $4$ faces of the big cube and paints them red. Finally, he takes apart the large cube and observe that the cubes with at least a face painted red is $431$. Find the number of cubes that he used to assemble the large cube.
Analyze all the possibilities.
2010 National Olympiad First Round, 32
At least two of any three students of a school with $1001$ students are friends. How many of the numbers $334,412,450,499$ can be the number of friends of the one with the highest number of friends?
$ \textbf{(A)}\ 4
\qquad\textbf{(B)}\ 3
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 1
\qquad\textbf{(E)}\ \text{None}
$