Found problems: 85335
2001 IMO Shortlist, 3
Let $ a_1 \equal{} 11^{11}, \, a_2 \equal{} 12^{12}, \, a_3 \equal{} 13^{13}$, and $ a_n \equal{} |a_{n \minus{} 1} \minus{} a_{n \minus{} 2}| \plus{} |a_{n \minus{} 2} \minus{} a_{n \minus{} 3}|, n \geq 4.$ Determine $ a_{14^{14}}$.
2008 Mathcenter Contest, 3
Set $ M= \{1,2,\cdots,2550\} $ and $\min A ,\ \max A $ represents the minimum and maximum values of the elements in the set $A$. For $ k \in \{1,2,\cdots 2006\} $define $$ x_k = \frac{1}{2008} \bigg (\sum_{A \subset M : n(A)= k} (\ min A + \max A) \, \bigg) $$. Find remainder from division $\sum_{i=1}^{2006} x_i^2$ with $2551$.
[i](passer-by)[/i]
2016 Balkan MO, 2
Let $ABCD$ be a cyclic quadrilateral with $AB<CD$. The diagonals intersect at the point $F$ and lines $AD$ and $BC$ intersect at the point $E$. Let $K$ and $L$ be the orthogonal projections of $F$ onto lines $AD$ and $BC$ respectively, and let $M$, $S$ and $T$ be the midpoints of $EF$, $CF$ and $DF$ respectively. Prove that the second intersection point of the circumcircles of triangles $MKT$ and $MLS$ lies on the segment $CD$.
[i](Greece - Silouanos Brazitikos)[/i]
2018 India IMO Training Camp, 3
Find all functions $f: \mathbb{R} \mapsto \mathbb{R}$ such that $$f(x)f\left(yf(x)-1\right)=x^2f(y)-f(x),$$for all $x,y \in \mathbb{R}$.
2017 China Team Selection Test, 5
A(x,y), B(x,y), and C(x,y) are three homogeneous real-coefficient polynomials of x and y with degree 2, 3, and 4 respectively. we know that there is a real-coefficient polinimial R(x,y) such that $B(x,y)^2-4A(x,y)C(x,y)=-R(x,y)^2$. Proof that there exist 2 polynomials F(x,y,z) and G(x,y,z) such that $F(x,y,z)^2+G(x,y,z)^2=A(x,y)z^2+B(x,y)z+C(x,y)$ if for any x, y, z real numbers $A(x,y)z^2+B(x,y)z+C(x,y)\ge 0$
2018 AMC 10, 25
For a positive integer $n$ and nonzero digits $a$, $b$, and $c$, let $A_n$ be the $n$-digit integer each of whose digits is equal to $a$; let $B_n$ be the $n$-digit integer each of whose digits is equal to $b$, and let $C_n$ be the $2n$-digit (not $n$-digit) integer each of whose digits is equal to $c$. What is the greatest possible value of $a + b + c$ for which there are at least two values of $n$ such that $C_n - B_n = A_n^2$?
$\textbf{(A)} \text{ 12} \qquad \textbf{(B)} \text{ 14} \qquad \textbf{(C)} \text{ 16} \qquad \textbf{(D)} \text{ 18} \qquad \textbf{(E)} \text{ 20}$
2002 Estonia Team Selection Test, 3
In a certain country there are $10$ cities connected by a network of one-way nonstop flights so that it is possible to fly (using one or more flights) from any city to any other. Let $n$ be the least number of flights needed to complete a trip starting from one of the cities, visiting all others and returning to the starting point. Find the greatest possible value of $n$.
2009 Federal Competition For Advanced Students, P2, 6
The quadrilateral PQRS whose vertices are the midpoints of the sides AB, BC,
CD, DA, respectively of a quadrilateral ABCD is called the midpoint quadrilateral
of ABCD.
Determine all circumscribed quadrilaterals whose mid-point quadrilaterals are
squares.
.
2024 District Olympiad, P1
Let $ABCD$ be a convex quadrilateral whose diagonals intersect at $O.$ Given that \[\overrightarrow{AB}+\overrightarrow{AD}+\overrightarrow{AO}=\overrightarrow{BC}+\overrightarrow{DC}+\overrightarrow{OC},\]prove that $ABCD$ is a parallelogram.
1952 Moscow Mathematical Olympiad, 224
a) Prove that if the square of a number begins with $0.\underbrace{\hbox{9...9}}_{\hbox{100}}$, then the number itself begins with $0.\underbrace{\hbox{9...9}}_{\hbox{100}}$,.
b) Calculate $\sqrt{0.9...9}$ ($60$ nines) to $60$ decimal places
2022 Indonesia Regional, 5
Numbers $1$ to $22$ are written on a board. A "move" is a procedure of picking two numbers $a,b$ on the board such that $b \geq a+2$, then erasing $a$ and $b$ to be replaced with $a+1$ and $b-1$. Determine the maximum possible number of moves that can be done on the board.
2022 Benelux, 4
A subset $A$ of the natural numbers $\mathbb{N} = \{0, 1, 2,\dots\}$ is called [i]good[/i] if every integer $n>0$ has at most one prime divisor $p$ such that $n-p\in A$.
(a) Show that the set $S = \{0, 1, 4, 9,\dots\}$ of perfect squares is good.
(b) Find an infinite good set disjoint from $S$. (Two sets are [i]disjoint[/i] if they have no common elements.)
2019 USAJMO, 1
There are $a+b$ bowls arranged in a row, numbered $1$ through $a+b$, where $a$ and $b$ are given positive integers. Initially, each of the first $a$ bowls contains an apple, and each of the last $b$ bowls contains a pear.
A legal move consists of moving an apple from bowl $i$ to bowl $i+1$ and a pear from bowl $j$ to bowl $j-1$, provided that the difference $i-j$ is even. We permit multiple fruits in the same bowl at the same time. The goal is to end up with the first $b$ bowls each containing a pear and the last $a$ bowls each containing an apple. Show that this is possible if and only if the product $ab$ is even.
2011 Cono Sur Olympiad, 2
The numbers $1$ through $4^{n}$ are written on a board. In each step, Pedro erases two numbers $a$ and $b$ from the board, and writes instead the number $\frac{ab}{\sqrt{2a^2+2b^2}}$. Pedro repeats this procedure until only one number remains. Prove that this number is less than $\frac{1}{n}$, no matter what numbers Pedro chose in each step.
1974 AMC 12/AHSME, 27
If $ f(x)\equal{}3x\plus{}2$ for all real $ x$, then the statement:
"$ |f(x)\plus{}4| < a$ whenever $ |x\plus{}2| < b$ and $ a>0$ and $ b>0$"
is true when
$ \textbf{(A)}\ b \le a/3
\qquad \textbf{(B)}\ b > a/3
\qquad \textbf{(C)}\ a \le b/3
\qquad \textbf{(D)}\ a > b/3 \\
\qquad \textbf{(E)}\ \text{The statement is never true.}$
1989 IMO Longlists, 98
Let $ A$ be an $ n \times n$ matrix whose elements are non-negative real numbers. Assume that $ A$ is a non-singular matrix and all elements of $ A^{\minus{}1}$ are non-negative real numbers. Prove that every row and every column of $ A$ has exactly one non-zero element.
the 11th XMO, 4
We define a beehive of order $n$ as follows:
a beehive of order 1 is one hexagon
To construct a beehive of order $n$, take a beehive of order $n-1$ and draw a layer of hexagons in the exterior of these hexagons. See diagram for examples of $n=2,3$
Initially some hexagons are infected by a virus. If a hexagon has been infected, it will always be infected. Otherwise, it will be infected if at least 5 out of the 6 neighbours are infected.
Let $f(n)$ be the minimum number of infected hexagons in the beginning so that after a finite time, all hexagons become infected. Find $f(n)$.
2019 Estonia Team Selection Test, 4
Let us call a real number $r$ [i]interesting[/i], if $r = a + b\sqrt2$ for some integers a and b. Let $A(x)$ and $B(x)$ be polynomial functions with interesting coefficients for which the constant term of $B(x)$ is $1$, and $Q(x)$ be a polynomial function with real coefficients such that $A(x) = B(x) \cdot Q(x)$. Prove that the coefficients of $Q(x)$ are interesting.
PEN D Problems, 21
Determine the last three digits of \[2003^{2002^{2001}}.\]
2002 All-Russian Olympiad, 4
A hydra consists of several heads and several necks, where each neck joins two heads. When a hydra's head $A$ is hit by a sword, all the necks from head $A$ disappear, but new necks grow up to connect head $A$ to all the heads which weren't connected to $A$. Heracle defeats a hydra by cutting it into two parts which are no joined. Find the minimum $N$ for which Heracle can defeat any hydra with $100$ necks by no more than $N$ hits.
2013 Saudi Arabia GMO TST, 2
Let $f(X) = a_nX^n + a_{n-1}X^{n-1} + ...+ a_1X + p$ be a polynomial of integer coefficients where $p$ is a prime number. Assume that $p >\sum_{i=1}^n |a_i|$. Prove that $f(X)$ is irreducible.
2021 Romania National Olympiad, 2
Determine all non-trivial finite rings with am unit element in which the sum of all elements is invertible.
[i]Mihai Opincariu[/i]
1988 Dutch Mathematical Olympiad, 4
Given is an isosceles triangle $ABC$ with $AB = 2$ and $AC = BC = 3$. We consider squares where $A, B$ and $C$ lie on the sides of the square (so not on the extension of such a side). Determine the maximum and minimum value of the area of such a square. Justify the answer.
2019 Azerbaijan Senior NMO, 1
Solve the following equation $$\sqrt{\frac{x^2}3-ax+a^2}+\sqrt{\frac{x^2}3-bx+b^2}=\sqrt{a^2-ab+b^2}$$ where $a;b\in\mathbb{R^+}$
2022 Bulgaria EGMO TST, 1
The finite set $M$ of real numbers is such that among any three of its elements there are two whose sum is in $M$.
What is the maximum possible cardinality of $M$?
[hide=Remark about the other problems] Problem 2 is UK National Round 2022 P2, Problem 3 is UK National Round 2022 P4, Problem 4 is Balkan MO 2021 Shortlist N2 (the one with Bertrand), Problem 5 is IMO Shortlist 2021 A1 and Problem 6 is USAMO 2002/1. Hence neither of these will be posted here. [/hide]