Found problems: 85335
2004 Austria Beginners' Competition, 4
Of a rhombus $ABCD$ we know the circumradius $R$ of $\Delta ABC$ and $r$ of $\Delta BCD$. Construct the rhombus.
2018 Korea USCM, 6
Suppose a continuous function $f:[0,1]\to\mathbb{R}$ is differentiable on $(0,1)$ and $f(0)=1$, $f(1)=0$. Then, there exists $0<x_0<1$ such that
$$|f'(x_0)| \geq 2018 f(x_0)^{2018}$$
1998 IMO, 1
A convex quadrilateral $ABCD$ has perpendicular diagonals. The perpendicular bisectors of the sides $AB$ and $CD$ meet at a unique point $P$ inside $ABCD$. Prove that the quadrilateral $ABCD$ is cyclic if and only if triangles $ABP$ and $CDP$ have equal areas.
1974 IMO Longlists, 40
Three players $A,B$ and $C$ play a game with three cards and on each of these $3$ cards it is written a positive integer, all $3$ numbers are different. A game consists of shuffling the cards, giving each player a card and each player is attributed a number of points equal to the number written on the card and then they give the cards back. After a number $(\geq 2)$ of games we find out that A has $20$ points, $B$ has $10$ points and $C$ has $9$ points. We also know that in the last game B had the card with the biggest number. Who had in the first game the card with the second value (this means the middle card concerning its value).
2014 Sharygin Geometry Olympiad, 14
In a given disc, construct a subset such that its area equals the half of the disc area and its intersection with its reflection over an arbitrary diameter has the area equal to the quarter of the disc area.
1981 Spain Mathematical Olympiad, 6
Prove that the transformation product of the symmetry of center $(0, 0)$ with the symmetry of the axis, with the line of equation $x = y + 1$, can be expressed as a product of an axis symmetry the line $e$ by a translation of vector $\overrightarrow{v}$, with $e$ parallel to $\overrightarrow{v}$, .
Determine a line $e$ and a vector $\overrightarrow{v}$, that meet the indicated conditions. have to be unique $e$ and $\overrightarrow{v}$,?
2011 Kyiv Mathematical Festival, 1
Solve the equation
$mn =$ (gcd($m,n$))$^2$ + lcm($m, n$)
in positive integers, where gcd($m, n$) – greatest common divisor of $m,n$, and lcm($m, n$) – least common multiple of $m,n$.
2023 Iranian Geometry Olympiad, 4
Let $ABC$ be a triangle and $P$ be the midpoint of arc $BAC$ of circumcircle of triangle $ABC$ with orthocenter $H$. Let $Q, S$ be points such that $HAPQ$ and $SACQ$ are parallelograms. Let $T$ be the midpoint of $AQ$, and $R$ be the intersection point of the lines $SQ$ and $PB$. Prove that $AB$, $SH$ and $TR$ are concurrent.
[i]Proposed by Dominik Burek - Poland[/i]
2015 ASDAN Math Tournament, 7
The Yamaimo family is moving to a new house, so they’ve packed their belongings into boxes, which weigh $100\text{ kg}$ in total. Mr. Yamaimo realizes that $99\%$ of the weight of the boxes is due to books. Later, the family unpacks some of the books (and nothing else). Mr. Yamaimo notices that now only $95\%$ of the weight of the boxes is due to books. How much do the boxes weigh now in kilograms?
2020 Balkan MO Shortlist, N4
Let $a_1=2$ and, for every positive integer $n$, let $a_{n+1}$ be the smallest integer strictly greater than $a_n$ that has more positive divisors than $a_n$. Prove that $2a_{n+1}=3a_n$ only for finitely many indicies $n$.
[i] Proposed by Ilija Jovčevski, North Macedonia[/i]
2019 Estonia Team Selection Test, 6
It is allowed to perform the following transformations in the plane with any integers $a$:
(1) Transform every point $(x, y)$ to the corresponding point $(x + ay, y)$,
(2) Transform every point $(x, y)$ to the corresponding point $(x, y + ax)$.
Does there exist a non-square rhombus whose all vertices have integer coordinates and which can be transformed to:
a) Vertices of a square,
b) Vertices of a rectangle with unequal side lengths?
2021 Sharygin Geometry Olympiad, 16
Let circles $\Omega$ and $\omega$ touch internally at point $A$. A chord $BC$ of $\Omega$ touches $\omega$ at point $K$. Let $O$ be the center of $\omega$. Prove that the circle $BOC$ bisects segment $AK$.
2014 USA TSTST, 6
Suppose we have distinct positive integers $a, b, c, d$, and an odd prime $p$ not dividing any of them, and an integer $M$ such that if one considers the infinite sequence \begin{align*}
ca &- db \\
ca^2 &- db^2 \\
ca^3 &- db^3 \\
ca^4 &- db^4 \\
&\vdots
\end{align*} and looks at the highest power of $p$ that divides each of them, these powers are not all zero, and are all at most $M$. Prove that there exists some $T$ (which may depend on $a,b,c,d,p,M$) such that whenever $p$ divides an element of this sequence, the maximum power of $p$ that divides that element is exactly $p^T$.
2018 CMIMC Combinatorics, 10
Call a set $S \subseteq \{0,1,\dots,14\}$ $\textit{sparse}$ if $x+1 \pmod{15}$ is not in $S$ whenever $x \in S$. Find the number of sparse sets $T$ such that the sum of the elements of $T$ is a multiple of 15.
2008 China Team Selection Test, 3
Find all positive integers $ n$ having the following properties:in two-dimensional Cartesian coordinates, there exists a convex $ n$ lattice polygon whose lengths of all sides are odd numbers, and unequal to each other. (where lattice polygon is defined as polygon whose coordinates of all vertices are integers in Cartesian coordinates.)
2021 Greece JBMO TST, 2
Anna and Basilis play a game writing numbers on a board as follows:
The two players play in turns and if in the board is written the positive integer $n$, the player whose turn is chooses a prime divisor $p$ of $n$ and writes the numbers $n+p$. In the board, is written at the start number $2$ and Anna plays first. The game is won by whom who shall be first able to write a number bigger or equal to $31$.
Find who player has a winning strategy, that is who may writing the appropriate numbers may win the game no matter how the other player plays.
1980 IMO Longlists, 4
Determine all positive integers $n$ such that the following statement holds: If a convex polygon with with $2n$ sides $A_1 A_2 \ldots A_{2n}$ is inscribed in a circle and $n-1$ of its $n$ pairs of opposite sides are parallel, which means if the pairs of opposite sides
\[(A_1 A_2, A_{n+1} A_{n+2}), (A_2 A_3, A_{n+2} A_{n+3}), \ldots , (A_{n-1} A_n, A_{2n-1} A_{2n})\]
are parallel, then the sides \[ A_n A_{n+1}, A_{2n} A_1\] are parallel as well.
2008 IMS, 8
Find all natural numbers such that \[ n\sigma(n)\equiv 2\pmod {\phi( n)}\]
1967 IMO Shortlist, 1
Let $a_1,\ldots,a_8$ be reals, not all equal to zero. Let
\[ c_n = \sum^8_{k=1} a^n_k\]
for $n=1,2,3,\ldots$. Given that among the numbers of the sequence $(c_n)$, there are infinitely many equal to zero, determine all the values of $n$ for which $c_n = 0.$
2002 Tournament Of Towns, 2
$\Delta ABC$ and its mirror reflection $\Delta A^{\prime}B^{\prime}C^{\prime}$ is arbitrarily placed on the plane. Prove the midpoints of $AA^{\prime},BB^{\prime},CC^{\prime}$ are collinear.
1959 Putnam, A6
Let $m$ and $n$ be integers greater than $1$ and $a_1 ,a_2 ,\ldots, a_{m+1}$ be real numbers. Prove that there exist real $n\times n$ matrices $A_1 ,A_2,\ldots, A_m$ such that
(i) $\det(A_j) =a_j$ for $j=1,2,\ldots,m$ and
(ii) $\det(A_1 +A_2 +\ldots+A_m)=a_{m+1}.$
1988 Flanders Math Olympiad, 1
show that the polynomial $x^4+3x^3+6x^2+9x+12$ cannot be written as the product of 2 polynomials of degree 2 with integer coefficients.
1985 Greece National Olympiad, 1
Inside triangle $ABC$ consider random point $O$. Prove that: $$E_A \overrightarrow{OA}+E_B \overrightarrow{OB}+E_C\overrightarrow{OC}=\overrightarrow{O}$$
where $E_A,E_B,E_C$ the areas of triangle $BOC, COB, AOB$ respectively
Fractal Edition 2, P4
In triangle $ABC$, the points $D$, $E$, and $F$ are the feet of the perpendiculars dropped from $A$, $B$, and $C$, respectively, onto the opposite sides. The point $X_A$ is such that a circle passing through $E$ and $F$ is tangent to the circumcircle of triangle $ABC$ at $X_A$, and $X_A$ is on a different side of $EF$ as $A$. Similarly, $X_B$ and $X_C$ are defined. Prove that the lines $AX_A$, $BX_B$, and $CX_C$ are concurrent.
1997 Croatia National Olympiad, Problem 2
Let $a,b,c$ be positive reals. Prove that $$a^ab^bc^c \geq a^bb^cc^a$$