Found problems: 85335
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
2006 Baltic Way, 12
Let $ABC$ be a triangle, let $B_{1}$ be the midpoint of the side $AB$ and $C_{1}$ the midpoint of the side $AC$. Let $P$ be the point of intersection, other than $A$, of the circumscribed circles around the triangles $ABC_{1}$ and $AB_{1}C$. Let $P_{1}$ be the point of intersection, other than $A$, of the line $AP$ with the circumscribed circle around the triangle $AB_{1}C_{1}$. Prove that $2AP=3AP_{1}$.
2020 Grand Duchy of Lithuania, 3
The tangents of the circumcircle $\Omega$ of the triangle $ABC$ at points $B$ and $C$ intersect at point $P$. The perpendiculars drawn from point $P$ to lines $AB$ and $AC$ intersect at points$ D$ and $E$ respectively. Prove that the altitudes of the triangle $ADE$ intersect at the midpoint of the segment $BC$.
1997 AMC 8, 23
There are positive integers that have these properties:
* the sum of the squares of their digits is 50, and
* each digit is larger than the one to its left.
The product of the digits of the largest integer with both properties is
$\textbf{(A)}\ 7 \qquad \textbf{(B)}\ 25 \qquad \textbf{(C)}\ 36 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 60$
2020 MMATHS, 1
A positive integer $n$ is called an untouchable number if there is no positive integer $m$ for which the sum of the factors of $m$ (including $m$ itself) is $n + m$. Find the sum of all of the untouchable numbers between $1$ and $10$ (inclusive)
2009 AMC 12/AHSME, 1
Kim's flight took off from Newark at 10:34 AM and landed in Miami at 1:18 PM. Both cities are in the same time zone. If her flight took $ h$ hours and $ m$ minutes, with $ 0<m<60$, what is $ h\plus{}m$?
$ \textbf{(A)}\ 46 \qquad
\textbf{(B)}\ 47 \qquad
\textbf{(C)}\ 50 \qquad
\textbf{(D)}\ 53 \qquad
\textbf{(E)}\ 54$