Found problems: 85335
2022 Puerto Rico Team Selection Test, 2
Suppose $a$ is a non-zero real number such that $a +\frac{1}{a}$ is a whole number.
(a) Prove that $a^2 +\frac{1}{a^2}$ is also an integer.
(b) Prove that $a^n+\frac{1}{a^n}$ is also an integer, for any integer value positive of $n$.
2008 Romania National Olympiad, 2
a) Prove that
\[ \dfrac{1}{2} \plus{} \dfrac{1}{3} \plus{} ... \plus{} \dfrac{1}{2^{2n}} > n,
\]
for all positive integers $ n$.
b) Prove that for every positive integer $ n$ we have $ \min\left\{ k \in \mathbb{Z}, k\geq 2 \mid \dfrac{1}{2} \plus{} \dfrac{1}{3} \plus{} \cdots \plus{} \dfrac{1}{k}>n \right\} > 2^n$.
2006 Iran MO (3rd Round), 1
$n$ is a natural number. $d$ is the least natural number that for each $a$ that $gcd(a,n)=1$ we know $a^{d}\equiv1\pmod{n}$. Prove that there exist a natural number that $\mbox{ord}_{n}b=d$
2024 IMC, 1
Determine all pairs $(a,b) \in \mathbb{C} \times \mathbb{C}$ of complex numbers satisfying $|a|=|b|=1$ and $a+b+a\overline{b} \in \mathbb{R}$.
2015 MMATHS, 2
Determine, with proof, whether $22!6! + 1$ is prime.
1991 Polish MO Finals, 1
On the Cartesian plane consider the set $V$ of all vectors with integer coordinates. Determine all functions $f : V \rightarrow \mathbb{R}$ satisfying the conditions:
(i) $f(v) = 1$ for each of the four vectors $v \in V$ of unit length.
(ii) $f(v+w) = f(v)+f(w)$ for every two perpendicular vectors $v, w \in V$
(Zero vector is considered to be perpendicular to every vector).
1986 National High School Mathematics League, 10
$x,y,z$ are nonnegative real numbers, and $4^{\sqrt{5x+9y+4z}}-68\times2^{\sqrt{5x+9y+4z}}+256=0$. Then, the product of the maximum and minimum value of $x+y+z$ is________.
2010 Purple Comet Problems, 18
How many three-digit positive integers contain both even and odd digits?
1961 IMO Shortlist, 6
Consider a plane $\epsilon$ and three non-collinear points $A,B,C$ on the same side of $\epsilon$; suppose the plane determined by these three points is not parallel to $\epsilon$. In plane $\epsilon$ take three arbitrary points $A',B',C'$. Let $L,M,N$ be the midpoints of segments $AA', BB', CC'$; Let $G$ be the centroid of the triangle $LMN$. (We will not consider positions of the points $A', B', C'$ such that the points $L,M,N$ do not form a triangle.) What is the locus of point $G$ as $A', B', C'$ range independently over the plane $\epsilon$?
2020 Hong Kong TST, 6
For a sequence with some ones and zeros, we count the number of continuous runs of equal digits in it. (For example the sequence $011001010$ has $7$ continuous runs: $0,11,00,1,0,1,0$.) Find the sum of the number of all continuous runs for all possible sequences with $2019$ ones and $2019$ zeros.
2016 Poland - Second Round, 1
Point $P$ lies inside triangle of sides of length $3, 4, 5$. Show that if distances between $P$ and vertices of triangle are rational numbers then distances from $P$ to sides of triangle are rational numbers too.
2021 BMT, 12
Unit square $ABCD$ is drawn on a plane. Point $O$ is drawn outside of $ABCD$ such that lines $AO$ and $BO$ are perpendicular. Square $F ROG$ is drawn with $F$ on $AB$ such that $AF =\frac23$, $R$ is on $\overline{BO}$, and $G$ is on $\overline{AO}$. Extend segment $\overline{OF}$ past $\overline{AB}$ to intersect side $\overline{CD}$ at $E$. Compute $DE$.
2015 AMC 12/AHSME, 22
For each positive integer $n$, let $S(n)$ be the number of sequences of length $n$ consisting solely of the letters $A$ and $B$, with no more than three $A$s in a row and no more than three $B$s in a row. What is the remainder when $S(2015)$ is divided by $12$?
$\textbf{(A) }0\qquad\textbf{(B) }4\qquad\textbf{(C) }6\qquad\textbf{(D) }8\qquad\textbf{(E) }10$
2015 Romania Team Selection Tests, 2
Let $(a_n)_{n \geq 0}$ and $(b_n)_{n \geq 0}$ be sequences of real numbers such that $ a_0>\frac{1}{2}$ , $a_{n+1} \geq a_n$ and $b_{n+1}=a_n(b_n+b_{n+2})$ for all non-negative integers $n$ . Show that the sequence $(b_n)_{n \geq 0}$ is bounded .
2021 Durer Math Competition Finals, 3
Let $A$ and $B$ different points of a circle $k$ centered at $O$ in such a way such that $AB$ is not a diagonal of $k$. Furthermore, let $X$ be an arbitrary inner point of the segment $AB$. Let $k_1$ be the circle that passes through the points $A$ and $X$, and $A$ is the only common point of $k$ and $k_1$. Similarly, let $k_2$ be the circle that passes through the points $B$ and $X$, and $B$ is the only common point of $k$ and $k_2$. Let $M$ be the second intersection point of $k_1$ and $k_2$. Let $Q$ denote the center of circumscribed circle of the triangle $AOB$. Let $O_1$ and $O_2$ be the centers of $k_1$ and $k_2$. Show that the points $M,O,O_1,O_2,Q$ are on a circle.
2013 Harvard-MIT Mathematics Tournament, 3
Let $A_1A_2A_3A_4A_5A_6$ be a convex hexagon such that $A_iA_{i+2} \parallel A_{i+3}A_{i+5}$ for $i = 1, 2, 3$ (we take $A_{i+6} = A_i$ for each $i$). Segment $A_iA_{i+2}$ intersects segment $A_{i+1}A_{i+3}$ at $B_i$, for $1 \le i \le 6$, as shown. Furthermore, suppose that $\vartriangle A_1A_3A_5 \cong \vartriangle A_4A_6A_2$. Given that $[A_1B_5B_6] = 1$, $[A_2B_6B_1] = 4$, and $[A_3B_1B_2] = 9$ (by $[XY Z]$ we mean the area of $ \vartriangle XY Z$), determine the area of hexagon $B_1B_2B_3B_4B_5B_6$.
[img]https://cdn.artofproblemsolving.com/attachments/d/0/1a8997c9eb7dea5223b6805dacd79c10a2cd33.png[/img]
1986 Tournament Of Towns, (130) 6
Squares of an $8 \times 8$ chessboard are each allocated a number between $1$ and $32$ , with each number being used twice. Prove that it is possible to choose $32$ such squares, each allocated a different number, so that there is at least one such square on each row or column .
(A . Andjans, Riga
1997 Baltic Way, 8
If we add $1996$ to $1997$, we first add the unit digits $6$ and $7$. Obtaining $13$, we write down $3$ and “carry” $1$ to the next column. Thus we make a carry. Continuing, we see that we are to make three carries in total.
Does there exist a positive integer $k$ such that adding $1996\cdot k$ to $1997\cdot k$ no carry arises during the whole calculation?
Mathley 2014-15, 3
Given a regular $2013$-sided polygon, how many isosceles triangles are there whose vertices are vertices vertex of given polygon and haave an angle greater than $120^o$?
Nguyen Tien Lam, High School for Natural Science,Hanoi National University.
1955 AMC 12/AHSME, 18
The discriminant of the equation $ x^2\plus{}2x\sqrt{3}\plus{}3\equal{}0$ is zero. Hence, its roots are:
$ \textbf{(A)}\ \text{real and equal} \qquad
\textbf{(B)}\ \text{rational and equal} \qquad
\textbf{(C)}\ \text{rational and unequal} \\
\textbf{(D)}\ \text{irrational and unequal} \qquad
\textbf{(E)}\ \text{imaginary}$
2010 Romanian Master of Mathematics, 1
For a finite non empty set of primes $P$, let $m(P)$ denote the largest possible number of consecutive positive integers, each of which is divisible by at least one member of $P$.
(i) Show that $|P|\le m(P)$, with equality if and only if $\min(P)>|P|$.
(ii) Show that $m(P)<(|P|+1)(2^{|P|}-1)$.
(The number $|P|$ is the size of set $P$)
[i]Dan Schwarz, Romania[/i]
2005 Taiwan National Olympiad, 2
$x,y,z,a,b,c$ are positive integers that satisfy $xy \equiv a \pmod z$, $yz \equiv b \pmod x$, $zx \equiv c \pmod y$. Prove that
$\min{\{x,y,z\}} \le ab+bc+ca$.
1983 IMO Longlists, 45
Let two glasses, numbered $1$ and $2$, contain an equal quantity of liquid, milk in glass $1$ and coffee in glass $2$. One does the following: Take one spoon of mixture from glass $1$ and pour it into glass $2$, and then take the same spoon of the new mixture from glass $2$ and pour it back into the first glass. What happens after this operation is repeated $n$ times, and what as $n$ tends to infinity?
MBMT Guts Rounds, 2015.23
A positive integer is called [i]oneic[/i] if it consists of only $1$'s. For example, the smallest three oneic numbers are $1$, $11$, and $111$. Find the number of $1$'s in the smallest oneic number that is divisible by $63$.
2018 Bosnia And Herzegovina - Regional Olympiad, 2
Find all positive integers $n$ such that number $n^4-4n^3+22n^2-36n+18$ is perfect square of positive integer