Found problems: 85335
2006 Paraguay Mathematical Olympiad, 1
What are the last two digits of the decimal representation of $21^{2006}$?
2024 Kyiv City MO Round 2, Problem 2
You are given a positive integer $n > 1$. What is the largest possible number of integers that can be chosen from
the set $\{1, 2, 3, \ldots, 2^n\}$ so that for any two different chosen integers $a, b$, the value $a^k + b^k$ is not divisible by $2^n$ for any positive integer $k$?
[i]Proposed by Oleksii Masalitin[/i]
2018 Malaysia National Olympiad, A2
An integer has $2018$ digits and is divisible by $7$. The first digit is $d$, while all the other digits are $2$. What is the value of $d$?
2006 Putnam, B3
Let $S$ be a finite set of points in the plane. A linear partition of $S$ is an unordered pair $\{A,B\}$ of subsets of $S$ such that $A\cup B=S,\ A\cap B=\emptyset,$ and $A$ and $B$ lie on opposite sides of some straight line disjoint from $S$ ($A$ or $B$ may be empty). Let $L_{S}$ be the number of linear partitions of $S.$ For each positive integer $n,$ find the maximum of $L_{S}$ over all sets $S$ of $n$ points.
2021 Polish Junior MO Finals, 5
Natural numbers $a$, $b$ are written in decimal using the same digits (i.e. every digit from 0 to 9 appears the same number of times in $a$ and in $b$). Prove that if $a+b=10^{1000}$ then both numbers $a$ and $b$ are divisible by $10$.
1994 National High School Mathematics League, 1
$a,b,c$ are real numbers. The sufficient and necessary condition of $\forall x\in\mathbb{R},a\sin x+b\cos x+c>0$ is
$\text{(A)}$ $a=b=0,c>0$
$\text{(B)}$ $\sqrt{a^2+b^2}=c$
$\text{(C)}$ $\sqrt{a^2+b^2}<c$
$\text{(D)}$ $\sqrt{a^2+b^2}>c$
2006 CHKMO, 1
On a planet there are $3\times2005!$ aliens and $2005$ languages. Each pair of aliens communicates with each other in exactly one language. Show that there are $3$ aliens who communicate with each other in one common language.
2020 Serbian Mathematical Olympiad, Problem 4
In a trapezoid $ABCD$ such that the internal angles are not equal to $90^{\circ}$, the diagonals $AC$ and $BD$ intersect at the point $E$. Let $P$ and $Q$ be the feet of the altitudes from $A$ and $B$ to the sides $BC$ and $AD$ respectively. Circumscribed circles of the triangles $CEQ$ and $DEP$ intersect at the point $F\neq E$. Prove that the lines $AP$, $BQ$ and $EF$ are either parallel to each other, or they meet at exactly one point.
2009 HMNT, 1
Paul starts with the number $19$. In one step, he can add $1$ to his number, divide his number by $2$, or divide his number by $3$. What is the minimum number of steps Paul needs to get to $1$?
2019 ELMO Shortlist, N1
Let $P(x)$ be a polynomial with integer coefficients such that $P(0)=1$, and let $c > 1$ be an integer. Define $x_0=0$ and $x_{i+1} = P(x_i)$ for all integers $i \ge 0$. Show that there are infinitely many positive integers $n$ such that $\gcd (x_n, n+c)=1$.
[i]Proposed by Milan Haiman and Carl Schildkraut[/i]
1976 Canada National Olympiad, 2
Suppose
\[ n(n\plus{}1)a_{n\plus{}1}\equal{}n(n\minus{}1)a_n\minus{}(n\minus{}2)a_{n\minus{}1}
\]
for every positive integer $ n\ge1$. Given that $ a_0\equal{}1,a_1\equal{}2$, find
\[ \frac{a_0}{a_1}\plus{}\frac{a_1}{a_2}\plus{}\frac{a_2}{a_3}\plus{}\dots\plus{}\frac{a_{50}}{a_{51}}.
\]
2008 AMC 8, 6
In the figure, what is the ratio of the area of the gray squares to the area of the white squares?
[asy]
size((70));
draw((10,0)--(0,10)--(-10,0)--(0,-10)--(10,0));
draw((-2.5,-7.5)--(7.5,2.5));
draw((-5,-5)--(5,5));
draw((-7.5,-2.5)--(2.5,7.5));
draw((-7.5,2.5)--(2.5,-7.5));
draw((-5,5)--(5,-5));
draw((-2.5,7.5)--(7.5,-2.5));
fill((-10,0)--(-7.5,2.5)--(-5,0)--(-7.5,-2.5)--cycle, gray);
fill((-5,0)--(0,5)--(5,0)--(0,-5)--cycle, gray);
fill((5,0)--(7.5,2.5)--(10,0)--(7.5,-2.5)--cycle, gray);
[/asy]
$ \textbf{(A)}\ 3:10 \qquad\textbf{(B)}\ 3:8 \qquad\textbf{(C)}\ 3:7 \qquad\textbf{(D)}\ 3:5 \qquad\textbf{(E)}\ 1:1 $
2021 Science ON Seniors, 4
$ABCD$ is a cyclic convex quadrilateral whose diagonals meet at $X$. The circle $(AXD)$ cuts $CD$ again at $V$ and the circle $(BXC)$ cuts $AB$ again at $U$, such that $D$ lies strictly between $C$ and $V$ and $B$ lies strictly between $A$ and $U$. Let $P\in AB\cap CD$.\\ \\
If $M$ is the intersection point of the tangents to $U$ and $V$ at $(UPV)$ and $T$ is the second intersection of circles $(UPV)$ and $(PAC)$, prove that $\angle PTM=90^o$.\\ \\
[i](Vlad Robu)[/i]
2008 Portugal MO, 3
Let $d$ be a natural number. Given two natural numbers $M$ and $N$ with $d$ digits, $M$ is a friend of $N$ if and only if the $d$ numbers obtained substituting each one of the digits of $M$ by the digit of $N$ which is on the same position are all multiples of $7$. Find all the values of $d$ for which the following condition is valid:
For any two numbers $M$ and $N$ with $d$ digits, $M$ is a friend of $N$ if and only if $N$ is a friend of $M$.
2023 Romania National Olympiad, 2
Determine all triples $(a,b,c)$ of integers that simultaneously satisfy the following relations:
\begin{align*}
a^2 + a = b + c, \\
b^2 + b = a + c, \\
c^2 + c = a + b.
\end{align*}
1987 IMO Shortlist, 17
Prove that there exists a four-coloring of the set $M = \{1, 2, \cdots, 1987\}$ such that any arithmetic progression with $10$ terms in the set $M$ is not monochromatic.
[b][i]Alternative formulation[/i][/b]
Let $M = \{1, 2, \cdots, 1987\}$. Prove that there is a function $f : M \to \{1, 2, 3, 4\}$ that is not constant on every set of $10$ terms from $M$ that form an arithmetic progression.
[i]Proposed by Romania[/i]
2001 National Olympiad First Round, 2
Each of the football teams Istanbulspor, Yesildirek, Vefa, Karagumruk, and Adalet, played exactly one match against the other four teams. Istanbulspor defeated all teams except Yesildirek; Yesildirek defeated Istanbulspor but lost to all the other teams. Vefa defeated all except Istanbulspor. The winner of the game Karagumruk-Adalet is Karagumruk. In how many ways one can order these five teams such that each team except the last, defeated the next team?
$
\textbf{(A)}\ 5
\qquad\textbf{(B)}\ 7
\qquad\textbf{(C)}\ 8
\qquad\textbf{(D)}\ 9
\qquad\textbf{(E)}\ \text{None of the preceding}
$
2024 Thailand TST, 2
Let $a_1<a_2<a_3<\dots$ be positive integers such that $a_{k+1}$ divides $2(a_1+a_2+\dots+a_k)$ for every $k\geqslant 1$. Suppose that for infinitely many primes $p$, there exists $k$ such that $p$ divides $a_k$. Prove that for every positive integer $n$, there exists $k$ such that $n$ divides $a_k$.
2016 Junior Balkan Team Selection Tests - Romania, 3
ABCD=cyclic quadrilateral,$AC\cap BD=X$
AA'$\perp $BD,A'$\in$BD
CC'$\perp $BD,C'$\in$BD
BB'$\perp $AC,B'$\in$AC
DD'$\perp $AC,D'$\in$AC
Prove that:
a)Prove that perpendiculars from midpoints of the sides to the opposite sides are concurrent.The point is called Mathot Point
b)A',B',C',D' are concyclic
c)If O'=circumcenter of (A'B'C') prove that O'=midpoint of the line that connects the orthocente of triangle XAB and XCD
d)O' is the Mathot Point
2024 Regional Olympiad of Mexico Southeast, 1
Find all pairs of positive integers \(a, b\) such that the numbers \(a+1\), \(b+1\), \(2a+1\), \(2b+1\), \(a+3b\), and \(b+3a\) are all prime numbers.
2004 AMC 12/AHSME, 17
Let $ f$ be a function with the following properties:
(i) $f(1) \equal{} 1$, and
(ii) $ f(2n) \equal{} n\times f(n)$, for any positive integer $ n$.
What is the value of $ f(2^{100})$?
$ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2^{99} \qquad \textbf{(C)}\ 2^{100} \qquad \textbf{(D)}\ 2^{4950} \qquad \textbf{(E)}\ 2^{9999}$
Estonia Open Junior - geometry, 2008.1.3
Let $M$ be the intersection of the medians $ABC$ of the triangle and the midpoint of the side $BC$. $A$ line parallel to side $BC$ and passing through point $M$ intersects sides $AB$ and $AC$ at points $X$ and $Y$ respectively. Let the point of intersection of the lines $XC$ and $MB$ be $Q$ and let $P$ intersection point of the lines $YB$ and $MC$ be $P$ . Prove that the triangles $DPQ$ and $ABC$ are similar.
2003 India National Olympiad, 3
Show that $8x^4 - 16x^3 + 16x^2 - 8x + k = 0$ has at least one real root for all real $k$. Find the sum of the non-real roots.
2017-IMOC, A3
Solve the following system of equations:
$$\begin{cases}
x^3+y+z=1\\
x+y^3+z=1\\
x+y+z^3=1\end{cases}$$
2002 District Olympiad, 4
The cube $ABCDA' B' C' D' $has of length a. Consider the points $K \in [AB], L \in [CC' ], M \in [D'A']$.
a) Show that $\sqrt3 KL \ge KB + BC + CL$
b) Show that the perimeter of triangle $KLM$ is strictly greater than $2a\sqrt3$.