Found problems: 85335
1998 Czech and Slovak Match, 6
In a summer camp there are $n$ girls $D_1,D_2, ... ,D_n$ and $2n-1$ boys $C_1,C_2, ...,C_{2n-1}$.
The girl $D_i, i = 1,2,... ,n,$ knows only the boys $C_1,C_2, ... ,C_{2i-1}$.
Let $A(n, r)$ be the number of different ways in which $r$ girls can dance with $r$ boys forming $r$ pairs,
each girl with a boy she knows.
Prove that $A(n, r) = \binom{n}{r} \frac{r!}{(n-r)!}.$
2007 Hanoi Open Mathematics Competitions, 7
Nine points, no three of which lie on the same straight line, are located inside an equilateral triangle of side $4$. Prove that some three of these points are vertices of a triangle whose area is not greater than $\sqrt3$.
2015 Iran Team Selection Test, 6
$ABCD$ is a circumscribed and inscribed quadrilateral. $O$ is the circumcenter of the quadrilateral. $E,F$ and $S$ are the intersections of $AB,CD$ , $AD,BC$ and $AC,BD$ respectively. $E'$ and $F'$ are points on $AD$ and $AB$ such that $A\hat{E}E'=E'\hat{E}D$ and $A\hat{F}F'=F'\hat{F}B$. $X$ and $Y$ are points on $OE'$ and $OF'$ such that $\frac{XA}{XD}=\frac{EA}{ED}$ and $\frac{YA}{YB}=\frac{FA}{FB}$. $M$ is the midpoint of arc $BD$ of $(O)$ which contains $A$.
Prove that the circumcircles of triangles $OXY$ and $OAM$ are coaxal with the circle with diameter $OS$.
2020 JBMO Shortlist, 3
Find the largest integer $k$ ($k \ge 2$), for which there exists an integer $n$ ($n \ge k$) such that from any collection of $n$ consecutive positive integers one can always choose $k$ numbers, which verify the following conditions:
1. each chosen number is not divisible by $6$, by $7$, nor by $8$;
2. the positive difference of any two distinct chosen numbers is not divisible by at least one of the
numbers $6$, $7$, and $8$.
2019 CCA Math Bonanza, L3.4
Determine the maximum possible value of \[\frac{\left(x^2+5x+12\right)\left(x^2+5x-12\right)\left(x^2-5x+12\right)\left(-x^2+5x+12\right)}{x^4}\] over all non-zero real numbers $x$.
[i]2019 CCA Math Bonanza Lightning Round #3.4[/i]
2014 Rioplatense Mathematical Olympiad, Level 3, 1
Let $n \ge 3$ be a positive integer. Determine, in terms of $n$, how many triples of sets $(A,B,C)$ satisfy the conditions:
$\bullet$ $A, B$ and $C$ are pairwise disjoint , that is, $A \cap B = A \cap C= B \cap C= \emptyset$.
$\bullet$ $A \cup B \cup C= \{ 1 , 2 , ... , n \}$.
$\bullet$ The sum of the elements of $A$, the sum of the elements of $B$ and the sum of the elements of $C$ leave the same remainder when divided by $3$.
Note: One or more of the sets may be empty.
2020 Tournament Of Towns, 3
Is it possible to inscribe an $N$-gon in a circle so that all the lengths of its sides are different and all its angles (in degrees) are integer, where
a) $N = 19$,
b) $N = 20$ ?
Mikhail Malkin
2013 Junior Balkan Team Selection Tests - Moldova, 4
A train from stop $A$ to stop $B$ is traveled in $X$ minutes ($0 <X <60$). It is known that when starting from $A$, as well as when arriving at $B$, the angle formed by the hour and the minute had measure equal to $X$ degrees. Find $X $.
2019 USMCA, 17
Tommy takes a 25-question true-false test. He answers each question correctly with independent probability $\frac{1}{2}$. Tommy earns bonus points for correct streaks: the first question in a streak is worth 1 point, the second question is worth 2 points, and so on. For instance, the sequence TFFTTTFT is worth 1 + 1 + 2 + 3 + 1 = 8 points. Compute the expected value of Tommy’s score.
Kvant 2020, M2628
There are $m$ identical two-pan weighting scales. One of them is broken and it shows any outcome, at random. The other scales always show the correct outcome. Moreover, the weight of the broken scale differs from those of the other scales, which are all equal. At a move, we may choose a scale and place some of the other scales on its pans. Determine the greatest value of $m$ for which we may find the broken scale with no more than three moves.
[i]Proposed by A. Gribalko and O. Manzhina[/i]
1990 AIME Problems, 3
Let $ P_1$ be a regular $ r$-gon and $ P_2$ be a regular $ s$-gon $ (r\geq s\geq 3)$ such that each interior angle of $ P_1$ is $ \frac {59}{58}$ as large as each interior angle of $ P_2$. What's the largest possible value of $ s$?
2022 Federal Competition For Advanced Students, P2, 1
Find all functions $f : Z_{>0} \to Z_{>0}$ with $a - f(b) | af(a) - bf(b)$ for all $a, b \in Z_{>0}$.
[i](Theresia Eisenkoelbl)[/i]
2021 Belarusian National Olympiad, 11.4
State consists of $2021$ cities, between some of them there are direct flights. Each pair of cities has not more than one flight, every flight belongs to one of $2021$ companies. Call a group of cities [i]incomplete[/i], if at least one company doesn't have any flights between cities of the group.
Find the maximum positive integer $m$, so that one can always find an incomplete group of $m$ cities.
2023-24 IOQM India, 23
In the coordinate plane, a point is called a $\text{lattice point}$ if both of its coordinates are integers. Let $A$ be the point $(12,84)$. Find the number of right angled triangles $ABC$ in the coordinate plane $B$ and $C$ are lattice points, having a right angle at vertex $A$ and whose incenter is at the origin $(0,0)$.
2003 AMC 12-AHSME, 1
Which of the following is the same as
\[ \frac{2\minus{}4\plus{}6\minus{}8\plus{}10\minus{}12\plus{}14}{3\minus{}6\plus{}9\minus{}12\plus{}15\minus{}18\plus{}21}?
\]$ \textbf{(A)}\ \minus{}1 \qquad
\textbf{(B)}\ \minus{}\frac23 \qquad
\textbf{(C)}\ \frac23 \qquad
\textbf{(D)}\ 1 \qquad
\textbf{(E)}\ \frac{14}{3}$
2012 HMNT, 2
Let $Q(x) = x^2 + 2x + 3$, and suppose that $P(x)$ is a polynomial such that $$P(Q(x)) = x^6 + 6x^5 + 18x^4 + 32x^3 + 35x^2 + 22x + 8.$$ Compute $P(2)$.
2018 Greece Team Selection Test, 3
Find all functions $f:\mathbb{Z}_{>0}\mapsto\mathbb{Z}_{>0}$ such that
$$xf(x)+(f(y))^2+2xf(y)$$
is perfect square for all positive integers $x,y$.
**This problem was proposed by me for the BMO 2017 and it was shortlisted. We then used it in our TST.
2019 Miklós Schweitzer, 2
Let $R$ be a noncommutative finite ring with multiplicative identity element $1$. Show that if the subring generated by $I \cup \{1\}$ is $R$ for each nonzero ideal $I$ then $R$ is simple.
2009 Kazakhstan National Olympiad, 6
Is there exist four points on plane, such that distance between any two of them is integer odd number?
May be it is geometry or number theory or combinatoric, I don't know, so it here :blush:
2004 Irish Math Olympiad, 1
Determine all pairs of prime numbers $(p, q)$, with $2 \leq p, q < 100$, such that $p+6, p+10, q+4, q+10$ and $p+q+1$ are all prime numbers.
1997 Turkey Team Selection Test, 2
The sequences $(a_{n})$, $(b_{n})$ are defined by $a_{1} = \alpha$, $b_{1} = \beta$, $a_{n+1} = \alpha a_{n} - \beta b_{n}$, $b_{n+1} = \beta a_{n} + \alpha b_{n}$ for all $n > 0.$ How many pairs $(\alpha, \beta)$ of real numbers are there such that $a_{1997} = b_{1}$ and $b_{1997} = a_{1}$?
2006 Sharygin Geometry Olympiad, 24
a) Two perpendicular rays are drawn through a fixed point $P$ inside a given circle, intersecting the circle at points $A$ and $B$. Find the geometric locus of the projections of $P$ on the lines $AB$.
b) Three pairwise perpendicular rays passing through the fixed point $P$ inside a given sphere intersect the sphere at points $A, B, C$. Find the geometrical locus of the projections $P$ on the $ABC$ plane
2019 USEMO, 1
Let $ABCD$ be a cyclic quadrilateral. A circle centered at $O$ passes through $B$ and $D$ and meets lines $BA$ and $BC$ again at points $E$ and $F$ (distinct from $A,B,C$). Let $H$ denote the orthocenter of triangle $DEF.$ Prove that if lines $AC,$ $DO,$ $EF$ are concurrent, then triangle $ABC$ and $EHF$ are similar.
[i]Robin Son[/i]
2014 IMAR Test, 4
Let $n$ be a positive integer. A Steiner tree associated with a finite set $S$ of points in the Euclidean $n$-space is a finite collection $T$ of straight-line segments in that space such that any two points in $S$ are joined by a unique path in $T$ , and its length is the sum of the segment lengths. Show that there exists a Steiner tree of length $1+(2^{n-1}-1)\sqrt{3}$ associated with the vertex set of a unit $n$-cube.
2012 AMC 12/AHSME, 6
The sums of three whole numbers taken in pairs are $12$, $17$, and $19$. What is the middle number?
$ \textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 8 $