Found problems: 85335
1999 AMC 12/AHSME, 21
A circle is circumscribed about a triangle with sides $ 20$, $ 21$, and $ 29$, thus dividing the interior of the circle into four regions. Let $ A$, $ B$, and $ C$ be the areas of the non-triangular regions, with $ C$ being the largest. Then
$ \textbf{(A)}\ A \plus{} B \equal{} C\qquad
\textbf{(B)}\ A \plus{} B \plus{} 210 \equal{} C\qquad
\textbf{(C)}\ A^2 \plus{} B^2 \equal{} C^2\qquad \\
\textbf{(D)}\ 20A \plus{} 21B \equal{} 29C\qquad
\textbf{(E)}\ \frac{1}{A^2} \plus{} \frac{1}{B^2} \equal{} \frac{1}{C^2}$
1993 AIME Problems, 13
Jenny and Kenny are walking in the same direction, Kenny at 3 feet per second and Jenny at 1 foot per second, on parallel paths that are 200 feet apart. A tall circular building 100 feet in diameter is centered midway between the paths. At the instant when the building first blocks the line of sight between Jenny and Kenny, they are 200 feet apart. Let $t$ be the amount of time, in seconds, before Jenny and Kenny can see each other again. If $t$ is written as a fraction in lowest terms, what is the sum of the numerator and denominator?
2001 Rioplatense Mathematical Olympiad, Level 3, 1
Find all integer numbers $a, b, m$ and $n$, such that the following two equalities are verified:
$a^2+b^2=5mn$ and $m^2+n^2=5ab$
VI Soros Olympiad 1999 - 2000 (Russia), 11.4
For prime numbers $p$ and $q$, natural numbers $n$, $k$, $r$, the equality $p^{2k}+q^{2n}=r^2$ holds. Prove that the number $r$ is prime.
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 $x,y \in \mathbb N$