Found problems: 85335
1997 Nordic, 1
Let $A$ be a set of seven positive numbers. Determine the maximal number of triples $(x, y, z)$ of elements of $A$
satisfying $x < y$ and $x + y = z$.
1978 Swedish Mathematical Competition, 3
Two satellites are orbiting the earth in the equatorial plane at an altitude $h$ above the surface. The distance between the satellites is always $d$, the diameter of the earth. For which $h$ is there always a point on the equator at which the two satellites subtend an angle of $90^\circ$?
2000 Mexico National Olympiad, 3
Given a set $A$ of positive integers, the set $A'$ is composed from the elements of $A$ and all positive integers that can be obtained in the following way:
Write down some elements of $A$ one after another without repeating, write a sign $+ $ or $-$ before each of them, and evaluate the obtained expression. The result is included in $A'$.
For example, if $A = \{2,8,13,20\}$, numbers $8$ and $14 = 20-2+8$ are elements of $A'$.
Set $A''$ is constructed from $A'$ in the same manner.
Find the smallest possible number of elements of $A$, if $A''$ contains all the integers from $1$ to $40$.
1995 IMC, 7
Let $A$ be a $3\times 3$ real matrix such that the vectors $Au$ and $u$ are orthogonal for
every column vector $u\in \mathbb{R}^{3}$. Prove that:
a) $A^{T}=-A$.
b) there exists a vector $v \in \mathbb{R}^{3}$ such that $Au=v\times u$ for every $u\in \mathbb{R}^{3}$,
where $v \times u$ denotes the vector product in $\mathbb{R}^{3}$.
2000 Stanford Mathematics Tournament, 11
If $ a@b\equal{}\frac{a\plus{}b}{a\minus{}b}$, find $ n$ such that $ 3@n\equal{}3$.
2018 AMC 10, 24
Triangle $ABC$ with $AB=50$ and $AC=10$ has area $120$. Let $D$ be the midpoint of $\overline{AB}$, and let $E$ be the midpoint of $\overline{AC}$. The angle bisector of $\angle BAC$ intersects $\overline{DE}$ and $\overline{BC}$ at $F$ and $G$, respectively. What is the area of quadrilateral $FDBG$?
$
\textbf{(A) }60 \qquad
\textbf{(B) }65 \qquad
\textbf{(C) }70 \qquad
\textbf{(D) }75 \qquad
\textbf{(E) }80 \qquad
$
1990 IberoAmerican, 2
Let $ABC$ be a triangle. $I$ is the incenter, and the incircle is tangent to $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. $P$ is the second point of intersection of $AD$ and the incircle. If $M$ is the midpoint of $EF$, show that $P$, $I$, $M$, $D$ are concyclic.
2020 Denmark MO - Mohr Contest, 5
Alma places spies on some of the squares on a $2020\times 2020$ game board. Now Bertha secretly chooses a quadradic area consisting of $1020 \times 1020$ squares and tells Alma which spies are standing on a square in the secret quadradic area. At least how many spies must Alma have placed in order for her to determine with certainty which area Bertha has chosen?
2019-2020 Winter SDPC, 5
Let $a_1, a_2, \ldots$ be a sequence of real numbers such that $a_1=4$ and $a_2=7$ such that for all integers $n$, $\frac{1}{a_{2n-1}}, \frac{1}{a_{2n}}, \frac{1}{a_{2n+1}}$ forms an arithmetic progression, and $a_{2n}, a_{2n+1}, a_{2n+2}$ forms an arithmetic progression. Find, with proof, the prime factorization of $a_{2019}$.
1984 Tournament Of Towns, (075) T1
In convex hexagon $ABCDEF, AB$ is parallel to $CF, CD$ is parallel to $BE$ and $EF$ is parallel to $AD$. Prove that the areas of triangles $ACE$ and $BDF$ are equal .
2007 QEDMO 4th, 3
Let $ n$ be a positive integer, and let $ M\equal{}\left\{ 1,2,...,n\right\}$. Two players take turns at the following game: Each player, at his turn, has to select an element of $ M$ and remove all divisors of this element (including this element itself) from the set $ M$.
[b]a)[/b] Assume that the player who cannot move anymore (because the set $ M$ is empty when it's his move) wins. For which values of $ n$ does the first player have a winning strategy?
[b]b)[/b] Assume that the player who cannot move anymore (because the set $ M$ is empty when it's his move) loses. For which values of $ n$ does the first player have a winning strategy?
1987 Putnam, A1
Curves $A,B,C$ and $D$ are defined in the plane as follows:
\begin{align*}
A &= \left\{ (x,y): x^2-y^2 = \frac{x}{x^2+y^2} \right\}, \\
B &= \left\{ (x,y): 2xy + \frac{y}{x^2+y^2} = 3 \right\}, \\
C &= \left\{ (x,y): x^3-3xy^2+3y=1 \right\}, \\
D &= \left\{ (x,y): 3x^2 y - 3x - y^3 = 0\right\}.
\end{align*}
Prove that $A \cap B = C \cap D$.
2015 Peru IMO TST, 9
Let $A$ be a finite set of functions $f: \Bbb{R}\to \Bbb{R.}$ It is known that: [list] [*] If $f, g\in A$ then $f (g (x)) \in A.$ [*] For all $f \in A$ there exists $g \in A$ such that $f (f (x) + y) = 2x + g (g (y) - x),$ for all $x, y\in \Bbb{R}.$ [/list] Let $i:\Bbb{R}\to \Bbb{R}$ be the identity function, ie, $i (x) = x$ for all $x\in \Bbb{R}.$ Prove that $i \in A.$
MathLinks Contest 1st, 2
In a triangle $\vartriangle ABC$, $\angle B = 2\angle C$. Let $P$ and $Q$ be points on the perpendicular bisector of segment $BC$ such that rays $AP$ and $AQ$ trisect $\angle A$. Prove that $PQ$ is smaller than $AB$ if and only if $\angle B$ is obtuse.
2003 Federal Math Competition of S&M, Problem 2
Let $ f : [0, 1] \to\ R $ be a function such that :-
$1.)$ $f(x) \ge 0$ for all $x$ in $[0,1]$ .
$2.)$ $f(1) = 1$ .
$3.)$ If $x_1 , x_2$ are in $[0,1]$ such that $x_1 + x_2 \le 1$ , then $f(x_1) + f(x_2) \le f(x_1 + x_2)$ .
Show that $f(x) \le 2x $ for all $x$ in $ [0,1] $.
1973 AMC 12/AHSME, 1
A chord which is the perpendicular bisector of a radius of length 12 in a circle, has length
$ \textbf{(A)}\ 3\sqrt3 \qquad \textbf{(B)}\ 27 \qquad \textbf{(C)}\ 6\sqrt3 \qquad \textbf{(D)}\ 12\sqrt3 \qquad
\textbf{(E)}\ \text{ none of these}$
2019 Nigeria Senior MO Round 2, 4
Let $h(t)$ and $f(t)$ be polynomials such that $h(t)=t^2$ and $f_n(t)=h(h(h(h(h...h(t))))))-1$ where $h(t)$ occurs $n$ times. Prove that $f_n(t)$ is a factor of $f_N(t)$ whenever $n$ is a factor of $N$
1996 Moldova Team Selection Test, 12
Suppose that in a certain society, each pair of persons can be classified as either [i]amicable [/i]or [i]hostile[/i]. We shall say that each member of an amicable pair is a [i]friend[/i] of the other, and each member of a hostile pair is a [i]foe[/i] of the other. Suppose that the society has $\, n \,$ persons and $\, q \,$ amicable pairs, and that for every set of three persons, at least one pair is hostile. Prove that there is at least one member of the society whose foes include $\, q(1 - 4q/n^2) \,$ or fewer amicable pairs.
1984 Poland - Second Round, 5
Calculate the lower bound of the areas of convex hexagons whose vertices all have integer coordinates.
1984 IMO Shortlist, 4
Let $ d$ be the sum of the lengths of all the diagonals of a plane convex polygon with $ n$ vertices (where $ n>3$). Let $ p$ be its perimeter. Prove that:
\[ n\minus{}3<{2d\over p}<\Bigl[{n\over2}\Bigr]\cdot\Bigl[{n\plus{}1\over 2}\Bigr]\minus{}2,\]
where $ [x]$ denotes the greatest integer not exceeding $ x$.
1992 Tournament Of Towns, (323) 4
A circle is divided into $7$ arcs. The sum of the angles subtending any two neighbouring arcs is no more than $103^o$. Find the maximal number $A$ such that any of the $7$ arcs is subtended by no less than $A^o$. Prove that this value $A$ is really maximal.
(A. Tolpygo, Kiev)
2019 Vietnam TST, P5
Given a scalene triangle $ABC$ inscribed in the circle $(O)$. Let $(I)$ be its incircle and $BI,CI$ cut $AC,AB$ at $E,F$ respectively. A circle passes through $E$ and touches $OB$ at $B$ cuts $(O)$ again at $M$. Similarly, a circle passes through $F$ and touches $OC$ at $C$ cuts $(O)$ again at $N$. $ME,NF$ cut $(O)$ again at $P,Q$. Let $K$ be the intersection of $EF$ and $BC$ and let $PQ$ cuts $BC$ and $EF$ at $G,H$, respectively. Show that the median correspond to $G$ of the triangle $GHK$ is perpendicular to $IO$.
2005 Sharygin Geometry Olympiad, 4
At what smallest $n$ is there a convex $n$-gon for which the sines of all angles are equal and the lengths of all sides are different?
1971 IMO Longlists, 54
A set $M$ is formed of $\binom{2n}{n}$ men, $n=1,2,\ldots$. Prove that we can choose a subset $P$ of the set $M$ consisting of $n+1$ men such that one of the following conditions is satisfied:
$(1)$ every member of the set $P$ knows every other member of the set $P$;
$(2)$ no member of the set $P$ knows any other member of the set $P$.
2022 Harvard-MIT Mathematics Tournament, 10
On a board the following six vectors are written: $$(1, 0, 0), (-1, 0, 0), (0, 1, 0), (0, -1, 0), (0, 0, 1), (0, 0, -1).$$ Given two vectors $v$ and $w$ on the board, a move consists of erasing $v$ and $w$ and replacing them with $\frac{1}{\sqrt2} (v + w)$ and $\frac{1}{\sqrt2} (v - w)$. After some number of moves, the sum of the six vectors on the board is $u$. Find, with proof, the maximum possible length of $u$.