Found problems: 85335
2023 Euler Olympiad, Round 1, 3
Leonard has a hand clock with only hour and minute hands. Determine the number of minutes in a day where the angle between the clock hands is not more than 1 degree. Both clock hands move continuously and at a constant speed.
[i]Proposed by Giorgi Arabidze, Georgia [/i]
1978 AMC 12/AHSME, 17
If $k$ is a positive number and $f$ is a function such that, for every positive number $x$, \[\left[f(x^2+1)\right]^{\sqrt{x}}=k;\] then, for every positive number $y$, \[\left[f(\frac{9+y^2}{y^2})\right]^{\sqrt{\frac{12}{y}}}\] is equal to
$\textbf{(A) }\sqrt{k}\qquad\textbf{(B) }2k\qquad\textbf{(C) }k\sqrt{k}\qquad\textbf{(D) }k^2\qquad \textbf{(E) }y\sqrt{k}$
2019 China Northern MO, 1
Find all positive intengers $x,y$, satisfying:
$$3^x+x^4=y!+2019.$$
2001 USAMO, 5
Let $S$ be a set of integers (not necessarily positive) such that
(a) there exist $a,b \in S$ with $\gcd(a,b)=\gcd(a-2,b-2)=1$;
(b) if $x$ and $y$ are elements of $S$ (possibly equal), then $x^2-y$ also belongs to $S$.
Prove that $S$ is the set of all integers.
JBMO Geometry Collection, 2019
Triangle $ABC$ is such that $AB < AC$. The perpendicular bisector of side $BC$ intersects lines $AB$ and $AC$ at points $P$ and $Q$, respectively. Let $H$ be the orthocentre of triangle $ABC$, and let $M$ and $N$ be the midpoints of segments $BC$ and $PQ$, respectively. Prove that lines $HM$ and $AN$ meet on the circumcircle of $ABC$.
2020 AMC 12/AHSME, 10
There is a unique positive integer $n$ such that \[\log_2{(\log_{16}{n})} = \log_4{(\log_4{n})}.\]
What is the sum of the digits of $n?$
$\textbf{(A) } 4 \qquad \textbf{(B) } 7 \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 11 \qquad \textbf{(E) } 13$
2016 Middle European Mathematical Olympiad, 3
A $8 \times 8$ board is given, with sides directed north-south and east-west.
It is divided into $1 \times 1$ cells in the usual manner. In each cell, there is most one [i]house[/i]. A house occupies only one cell.
A house is [i] in the shade[/i] if there is a house in each of the cells in the south, east and west sides of its cell. In particular, no house placed on the south, east or west side of the board is in the shade.
Find the maximal number of houses that can be placed on the board such that no house is in the shade.
2012 Hanoi Open Mathematics Competitions, 2
[b]Q2.[/b] Let be given a parallegogram $ABCD$ with the area of $12 \ \text{cm}^2$. The line through $A$ and the midpoint $M$ of $BC$ mects $BD$ at $N.$ Compute the area of the quadrilateral $MNDC.$
\[(A) \; 4 \text{cm}^2; \qquad (B) \; 5 \text{cm}^2; \qquad (C ) \; 6 \text{cm}^2; \qquad (D) \; 7 \text{cm}^2; \qquad (E) \; \text{None of the above.}\]
2015 IMO Shortlist, G7
Let $ABCD$ be a convex quadrilateral, and let $P$, $Q$, $R$, and $S$ be points on the sides $AB$, $BC$, $CD$, and $DA$, respectively. Let the line segment $PR$ and $QS$ meet at $O$. Suppose that each of the quadrilaterals $APOS$, $BQOP$, $CROQ$, and $DSOR$ has an incircle. Prove that the lines $AC$, $PQ$, and $RS$ are either concurrent or parallel to each other.
2005 USAMO, 5
Let $n$ be an integer greater than 1. Suppose $2n$ points are given in the plane, no three of which are collinear. Suppose $n$ of the given $2n$ points are colored blue and the other $n$ colored red. A line in the plane is called a [i]balancing line[/i] if it passes through one blue and one red point and, for each side of the line, the number of blue points on that side is equal to the number of red points on the same side.
Prove that there exist at least two balancing lines.
2020 Junior Balkan Team Selection Tests - Moldova, 12
Find all numbers $n \in \mathbb{N}^*$ for which there exists a finite set of natural numbers $A=(a_1, a_2,...a_n)$ so that for any $k$ $(1\leq k \leq n)$ the number $a_k$ is the number of all multiples of $k$ in set $A$.
2015 HMNT, 8
Find $ \textbf{any} $ quadruple of positive integers $(a,b,c,d)$ satisfying $a^3+b^4+c^5=d^{11}$ and $abc<10^5$.
2021 Saudi Arabia Training Tests, 37
Given $n \ge 2$ distinct positive integers $a_1, a_2, ..., a_n$ none of which is a perfect cube. Find the maximal possible number of perfect cubes among their pairwise products.
2019-IMOC, C4
Determine the largest $k$ such that for all competitive graph with $2019$ points, if the difference between in-degree and out-degree of any point is less than or equal to $k$, then this graph is strongly connected.
2016 Harvard-MIT Mathematics Tournament, 3
Let $A$ denote the set of all integers $n$ such that $1 \le n \le 10000$, and moreover the sum of the decimal digits of $n$ is $2$. Find the sum of the squares of the elements of $A$.
1986 French Mathematical Olympiad, Problem 1
Let $ABCD$ be a tetrahedron.
(a) Prove that the midpoints of the edges $AB,AC,BD$, and $CD$ lie in a plane.
(b) Find the point in that plane, whose sum of distances from the lines $AD$ and $BC$ is minimal.
1957 AMC 12/AHSME, 1
The number of distinct lines representing the altitudes, medians, and interior angle bisectors of a triangle that is isosceles, but not equilateral, is:
$ \textbf{(A)}\ 9\qquad
\textbf{(B)}\ 7\qquad
\textbf{(C)}\ 6\qquad
\textbf{(D)}\ 5\qquad
\textbf{(E)}\ 3$
2000 Baltic Way, 15
Let $n$ be a positive integer not divisible by $2$ or $3$. Prove that for all integers $k$, the number $(k+1)^n-k^n-1$ is divisible by $k^2+k+1$.
2015 Kazakhstan National Olympiad, 2
Solve in positive integers
$x^yy^x=(x+y)^z$
2009 Canadian Mathematical Olympiad Qualification Repechage, 5
Determine all positive integers $n$ for which $n(n + 9)$ is a perfect square.
1955 Moscow Mathematical Olympiad, 308
* Two circles are tangent to each other externally, and to a third one from the inside. Two common tangents to the first two circles are drawn, one outer and one inner. Prove that the inner tangent divides in halves the arc intercepted by the outer tangent on the third circle.
2007 Bosnia Herzegovina Team Selection Test, 4
Let $P(x)$ be a polynomial such that $P(x)=x^3-2x^2+bx+c$. Roots of $P(x)$ belong to interval $(0,1)$. Prove that $8b+9c \leq 8$. When does equality hold?
1995 Tournament Of Towns, (459) 4
Some points with integer coordinates in the plane are marked. It is known that no four of them lie on a circle. Show that there exists a circle of radius 1995 without any marked points inside.
(AV Shapovelov)
1999 Mongolian Mathematical Olympiad, Problem 6
Two circles in the plane intersect at $C$ and $D$. A chord $AB$ of the first circle and a chord $EF$ of the second circle pass through a point on the common chord $CD$. Show that the points $A,B,E,F$ lie on a circle.
2019 Jozsef Wildt International Math Competition, W. 59
In the any $[ABCD]$ tetrahedron we denote with $\alpha$, $\beta$, $\gamma$ the measures, in radians, of the angles of the three pairs of opposite edges and with $r$, $R$ the lengths of the rays of the sphere inscribed and respectively circumscribed the tetrahedron. Demonstrate inequality$$\left(\frac{3r}{R}\right)^3\leq \sin \frac{\alpha +\beta +\gamma}{3}$$(A refinement of inequality $R \geq 3r$).