Found problems: 85335
2020 Dutch IMO TST, 3
Find all pairs $(a, b)$ of positive integers for which $a + b = \phi (a) + \phi (b) + gcd (a, b)$.
Here $ \phi (n)$ is the number of numbers $k$ from $\{1, 2,. . . , n\}$ with $gcd (n, k) = 1$.
1970 AMC 12/AHSME, 22
If the sum of the first $3n$ positive integers is $150$ more than the sum of the first $n$ positive integers, then the sum of the first $4n$ positive integers is
$\textbf{(A) }300\qquad\textbf{(B) }350\qquad\textbf{(C) }400\qquad\textbf{(D) }450\qquad \textbf{(E) }600$
1997 Israel National Olympiad, 6
In a certain country, every two cities are connected either by an airline route or by a railroad. Prove that one can always choose a type of transportation in such a way that each city can be reached from any other city with at most two transfers.
2015 Bosnia Herzegovina Team Selection Test, 4
Let $X$ be a set which consists from $8$ consecutive positive integers. Set $X$ is divided on two disjoint subsets $A$ and $B$ with equal number of elements. If sum of squares of elements from set $A$ is equal to sum of squares of elements from set $B$, prove that sum of elements of set $A$ is equal to sum of elements of set $B$.
2013 Baltic Way, 7
A positive integer is written on a blackboard. Players $A$ and $B$ play the following game: in each move one has to choose a proper divisor $m$ of the number $n$ written on the blackboard ($1<m<n$) and replaces $n$ with $n-m$. Player $A$ makes the first move, then players move alternately. The player who can't make a move loses the game. For which starting numbers is there a winning strategy for player $B$?
2017 Harvard-MIT Mathematics Tournament, 7
Determine the largest real number $c$ such that for any $2017$ real numbers $x_1, x_2, \dots, x_{2017}$, the inequality $$\sum_{i=1}^{2016}x_i(x_i+x_{i+1})\ge c\cdot x^2_{2017}$$ holds.
2014 Romania Team Selection Test, 1
Let $ABC$ be a triangle and let $X$,$Y$,$Z$ be interior points on the sides $BC$, $CA$, $AB$, respectively. Show that the magnified image of the triangle $XYZ$ under a homothety of factor $4$ from its centroid covers at least one of the vertices $A$, $B$, $C$.
LMT Team Rounds 2021+, 14
In a cone with height $3$ and base radius $4$, let $X$ be a point on the circumference of the base. Let $Y$ be a point on the surface of the cone such that the distance from $Y$ to the vertex of the cone is $2$, and $Y$ is diametrically opposite $X$ with respect to the base of the cone. The length of the shortest path across the surface of the cone from $X$ to $Y$ can be expressed as $\sqrt{a +\sqrt{b}}$, where a and b are positive integers. Find $a +b$.
2016 Sharygin Geometry Olympiad, 1
The diagonals of a parallelogram $ABCD$ meet at point $O$. The tangent to the circumcircle of triangle $BOC$ at $O$ meets ray $CB$ at point $F$. The circumcircle of triangle $FOD$ meets $BC$ for the second time at point $G$. Prove that $AG=AB$.
2011 IFYM, Sozopol, 2
Let $k>1$ and $n$ be natural numbers and
$p=\frac{((n+1)(n+2)…(n+k))}{k!}-1$.
Prove that, if $p$ is prime, then $n|k!$.
Ukrainian TYM Qualifying - geometry, VI.14
A quadrilateral whose perimeter is equal to $P$ is inscribed in a circle of radius $R$ and is circumscribed around a circle of radius $r$. Check whether the inequality $P\le \frac{r+\sqrt{r^2+4R^2}}{2}$ holds.
Try to find the corresponding inequalities for the $n$-gon ($n \ge 5$) inscribed in a circle of radius $R$ and circumscribed around a circle of radius $r$.
2024 India Regional Mathematical Olympiad, 6
Let $X$ be a set of $11$ integers. Prove that one can find a nonempty subset $\{a_1, a_2, \cdots , a_k \}$ of $X$ such that $3$ divides $k$ and $9$ divides the sum $\sum_{i=1}^{k} 4^i a_i$.
2007 Kyiv Mathematical Festival, 2
The point $D$ at the side $AB$ of triangle $ABC$ is given. Construct points $E,F$ at sides $BC, AC$ respectively such that the midpoints of $DE$ and $DF$ are collinear with $B$ and the midpoints of $DE$ and $EF$ are collinear with $C.$
2025 Junior Balkan Team Selection Tests - Romania, P4
Find all positive integers $n$ such that $2^n-n^2+1$ is a perfect square.
2020 USEMO, 1
Which positive integers can be written in the form \[\frac{\operatorname{lcm}(x, y) + \operatorname{lcm}(y, z)}{\operatorname{lcm}(x, z)}\] for positive integers $x$, $y$, $z$?
2014 Indonesia MO Shortlist, N2
Suppose that $a, b, c, k$ are natural numbers with $a, b, c \ge 3$ which fulfill the equation $abc = k^2 + 1$.
Show that at least one between $a - 1, b - 1, c -1$ is composite number.
2004 India IMO Training Camp, 2
Find all primes $p \geq 3$ with the following property: for any prime $q<p$, the number
\[
p - \Big\lfloor \frac{p}{q} \Big\rfloor q
\]
is squarefree (i.e. is not divisible by the square of a prime).
1985 AMC 8, 16
The ratio of boys to girls in Mr. Brown's math class is $ 2: 3$. If there are $ 30$ students in the class, how many more girls than boys are in the class?
\[ \textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 3 \qquad
\textbf{(C)}\ 5 \qquad
\textbf{(D)}\ 6 \qquad
\textbf{(E)}\ 10
\]
2008 Turkey Team Selection Test, 2
A graph has $ 30$ vertices, $ 105$ edges and $ 4822$ unordered edge pairs whose endpoints are disjoint. Find the maximal possible difference of degrees of two vertices in this graph.
2011 Morocco National Olympiad, 4
Let $ABC$ be a triangle. The inside bisector of the angle $\angle BAC$ cuts $[BC]$ in $L$ and the circle $(C)$ circumsbribed to the triangle $ABC$ in $D$. The perpendicular to $(AC)$ going through $D$ cuts $[AC]$ in $M$ and the circle $(C)$ in $K$. Find the value of $\frac{AM}{MC}$ knowing that $\frac{BL}{LC}=\frac{1}{2}$.
2015 Princeton University Math Competition, B1
Roy is starting a baking company and decides that he will sell cupcakes. He sells $n$ cupcakes for $(n + 20)(n + 15)$ cents. A man walks in and buys $\$10.50$ worth of cupcakes. Roy bakes cupcakes at a rate of $10$ cupcakes an hour. How many minutes will it take Roy to complete the order?
2012 HMNT, 1
Find the number of integers between $1$ and $200$ inclusive whose distinct prime divisors sum to $16$.
(For example, the sum of the distinct prime divisors of $12$ is $2 + 3 = 5$.)
In this section, the word [i]divisor [/i]is used to refer to a [i]positive divisor[/i] of an integer.
2003 Italy TST, 2
Let $B\not= A$ be a point on the tangent to circle $S_1$ through the point $A$ on the circle. A point $C$ outside the circle is chosen so that segment $AC$ intersects the circle in two distinct points. Let $S_2$ be the circle tangent to $AC$ at $C$ and to $S_1$ at some point $D$, where $D$ and $B$ are on the opposite sides of the line $AC$. Let $O$ be the circumcentre of triangle $BCD$. Show that $O$ lies on the circumcircle of triangle $ABC$.
2015 AMC 8, 5
Billy's basketball team scored the following points over the course of the first 11 games of the season:
\[42, 47, 53, 53, 58, 58, 58, 61, 64, 65, 73\]
If his team scores 40 in the 12th game, which of the following statistics will show an increase?
$
\textbf{(A) } \text{range} \qquad
\textbf{(B) } \text{median} \qquad
\textbf{(C) } \text{mean} \qquad
\textbf{(D) } \text{mode} \qquad
\textbf{(E) } \text{mid-range}
$
2016 Chile National Olympiad, 2
For a triangle $\vartriangle ABC$, determine whether or not there exists a point $P$ on the interior of $\vartriangle ABC$ in such a way that every straight line through $P$ divides the triangle $\vartriangle ABC$ in two polygons of equal area.