Found problems: 85335
1992 Hungary-Israel Binational, 5
We examine the following two sequences: The Fibonacci sequence: $F_{0}= 0, F_{1}= 1, F_{n}= F_{n-1}+F_{n-2 }$ for $n \geq 2$; The Lucas sequence: $L_{0}= 2, L_{1}= 1, L_{n}= L_{n-1}+L_{n-2}$ for $n \geq 2$. It is known that for all $n \geq 0$
\[F_{n}=\frac{\alpha^{n}-\beta^{n}}{\sqrt{5}},L_{n}=\alpha^{n}+\beta^{n}, \]
where $\alpha=\frac{1+\sqrt{5}}{2},\beta=\frac{1-\sqrt{5}}{2}$. These formulae can be used without proof.
Show that $L_{2n+1}+(-1)^{n+1}(n \geq 1)$ can be written as a product of three (not necessarily distinct) Fibonacci numbers.
2004 Croatia National Olympiad, Problem 1
Parts of a pentagon have areas $x,y,z$ as shown in the picture. Given the area $x$, find the areas $y$ and $z$ and the area of the entire pentagon.
[img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvOS9mLzM5NjNjNDcwY2ZmMzgzY2QwYWM0YzI1NmYzOWU2MWY1NTczZmYxLnBuZw==&rn=U2NyZWVuIFNob3QgMjAyMS0wNC0wOCBhdCA0LjMwLjU1IFBNLnBuZw[/img]
2018 Saudi Arabia GMO TST, 2
Let $p$ be a prime number of the form $9k + 1$. Show that there exists an integer n such that $p | n^3 - 3n + 1$.
2014 Canada National Olympiad, 4
The quadrilateral $ABCD$ is inscribed in a circle. The point $P$ lies in the interior of $ABCD$, and $\angle P AB = \angle P BC = \angle P CD = \angle P DA$. The lines $AD$ and $BC$ meet at $Q$, and the lines $AB$ and $CD$ meet at $R$. Prove that the lines $P Q$ and $P R$ form the same angle as the diagonals of $ABCD$.
2001 Mongolian Mathematical Olympiad, Problem 1
Suppose that a sequence $x_1,x_2,\ldots,x_{2001}$ of positive real numbers satisfies
$$3x^2_{n+1}=7x_nx_{n+1}-3x_{n+1}-2x^2_n+x_n\enspace\text{ and }\enspace x_{37}=x_{2001}.$$Find the maximum possible value of $x_1$.
India EGMO 2025 TST, 3
Let $\Delta ABC$ be an acute angled scalene triangle with circumcircle $\omega$. Let $O$ and $H$ be the circumcenter and orthocenter of $\Delta ABC,$ respectively. Let $E,F$ and $Q$ be points on segments $AB,AC$ and $\omega$, respectively, such that
$$\angle BHE=\angle CHF=\angle AQH=90^\circ.$$
Prove that $OQ$ and $AH$ intersect on the circumcircle of $\Delta AEF$.
[i] Proposed by Antareep Nath [/i]
2024 Harvard-MIT Mathematics Tournament, 10
A [i]peacock [/i] is a ten-digit positive integer that uses each digit exactly once. Compute the number of peacocks that are exactly twice another peacock.
1990 French Mathematical Olympiad, Problem 3
(a) Find all triples of integers $(a,b,c)$ for which $\frac14=\frac1{a^2}+\frac1{b^2}+\frac1{c^2}$.
(b) Determine all positive integers $n$ for which there exist positive integers $x_1,x_2,\ldots,x_n$ such that $1=\frac1{x_1^2}+\frac1{x_2^2}+\ldots+\frac1{x_n^2}$.
2012 QEDMO 11th, 1
Find all $x, y, z \in N_0$ with $(2^x + 1) (2^y-1) = 2^z-1$.
2020 JBMO TST of France, 3
Let n be a nonzero natural number. We say about a function f ∶ R ⟶ R that is n-positive
if, for any real numbers $x_1, x_2,...,x_n$
with the property that $x_1+x_2+...+x_n = 0$,
the inequality $f(x_1)+f(x_2)+...+f(x_n)=>0$ is true
a) Is it true that any 2020-positive function is also 1010-positive?
b) Is it true that any 1010-positive function is 2020-positive?
1977 AMC 12/AHSME, 28
Let $g(x)=x^5+x^4+x^3+x^2+x+1$. What is the remainder when the polynomial $g(x^{12})$ is divided by the polynomial $g(x)$?
$\textbf{(A) }6\qquad\textbf{(B) }5-x\qquad\textbf{(C) }4-x+x^2\qquad$
$\textbf{(D) }3-x+x^2-x^3\qquad \textbf{(E) }2-x+x^2-x^3+x^4$
2021 Junior Balkan Team Selection Tests - Moldova, 1
Find all values of the real parameter $a$, for which the equation $(x -6\sqrt{x} + 8)\cdot \sqrt{x- a} = 0$ has exactly two distinct real solutions.
1998 Croatia National Olympiad, Problem 3
Points $E$ and $F$ are chosen on the sides $AB$ and $BC$ respectively of a square $ABCD$ such that $BE=BF$. Let $BN$ be an altitude of the triangle $BCE$. Prove that the triangle $DNF$ is right-angled.
2009 USAMTS Problems, 3
Prove that if $a$ and $b$ are positive integers such that $a^2 + b^2$ is a multiple of $7^{2009}$, then $ab$ is a multiple of $7^{2010}$.
2011 Harvard-MIT Mathematics Tournament, 3
Find all integers $x$ such that $2x^2+x-6$ is a positive integral power of a prime positive integer.
1989 IMO Longlists, 64
A natural number is written in each square of an $ m \times n$ chess board. The allowed move is to add an integer $ k$ to each of two adjacent numbers in such a way that non-negative numbers are obtained. (Two squares are adjacent if they have a common side.) Find a necessary and sufficient condition for it to be possible for all the numbers to be zero after finitely many operations.
2012 NIMO Summer Contest, 13
For the NEMO, Kevin needs to compute the product
\[
9 \times 99 \times 999 \times \cdots \times 999999999.
\]
Kevin takes exactly $ab$ seconds to multiply an $a$-digit integer by a $b$-digit integer. Compute the minimum number of seconds necessary for Kevin to evaluate the expression together by performing eight such multiplications.
[i]Proposed by Evan Chen[/i]
2021 Hong Kong TST, 1
Find all real triples $(a,b,c)$ satisfying
\[(2^{2a}+1)(2^{2b}+2)(2^{2c}+8)=2^{a+b+c+5}.\]
2023 Singapore Junior Math Olympiad, 3
Define a domino to be a $1\times 2$ rectangular block. A $2023\times 2023$ square grid is filled with non-overlapping dominoes, leaving a single $1\times 1$ gap. John then repeatedly slides dominoes into the gap; each domino is moved at most once. What is the maximum number of times that John could have moved a domino? (Example: In the $3\times 3$ grid shown below, John could move 2 dominoes: $D$, followed by $A$.)
[asy]
unitsize(18);
draw((0,0)--(3,0)--(3,3)--(0,3)--(0,0)--cycle);
draw((0,1)--(3,1));
draw((2,0)--(2,3));
draw((1,1)--(1,3));
label("A",(0.5,2));
label("B",(1.5,2));
label("C",(2.5,2));
label("D",(1,0.5));
[/asy]
2019 ASDAN Math Tournament, 2
Let $P_1,P_2,\dots,P_{720}$ denote the integers whose digits are a permutation of $123456$, arranged in ascending order (so $P_1=123456$, $P_2=123465$, and $P_{720}=654321$). What is $P_{144}$?
2015 Purple Comet Problems, 4
Janet played a song on her clarinet in 3 minutes and 20 seconds. Janet’s goal is to play the song at a 25%
faster rate. Find the number of seconds it will take Janet to play the song when she reaches her goal.
2020 Serbian Mathematical Olympiad, Problem 2
We are given a polyhedron with at least $5$ vertices, such that exactly $3$ edges meet in each of the vertices. Prove that we can assign a rational number to every vertex of the given polyhedron such that the following conditions are met:
$(i)$ At least one of the numbers assigned to the vertices is equal to $2020$.
$(ii)$ For every polygonal face, the product of the numbers assigned to the vertices of that face is equal to $1$.
2000 Baltic Way, 11
A sequence of positive integers $a_1,a_2,\ldots $ is such that for each $m$ and $n$ the following holds: if $m$ is a divisor of $n$ and $m<n$, then $a_m$ is a divisor of $a_n$ and $a_m<a_n$. Find the least possible value of $a_{2000}$.
2011 IFYM, Sozopol, 3
Let $g_1$ and $g_2$ be some lines, which intersect in point $A$. A circle $k_1$ is tangent to $g_1$ at point $A$ and intersects $g_2$ for a second time in $C$. A circle $k_2$ is tangent to $g_2$ at point $A$ and intersects $g_1$ for a second time in $D$. The circles $k_1$ and $k_2$ intersect for a second time in point $B$. Prove that, if $\frac{AC}{AD}=\sqrt{2}$, then $\frac{BC}{BD}=2$.
2001 USA Team Selection Test, 5
In triangle $ABC$, $\angle B = 2\angle C$. Let $P$ and $Q$ be points on the perpendicular bisector of segment $BC$ such that rays $AP$ and $AQ$ trisect $\angle A$. Prove that $PQ < AB$ if and only if $\angle B$ is obtuse.