Found problems: 85335
2016 BMT Spring, 10
Define $T_n =\sum^{n}){i=1} i(n + 1 - i)$. Find $\lim_{n\to \infty} \frac{T_n}{n^3}$.
2007 All-Russian Olympiad, 1
Given reals numbers $a$, $b$, $c$. Prove that at least one of three equations $x^{2}+(a-b)x+(b-c)=0$, $x^{2}+(b-c)x+(c-a)=0$, $x^{2}+(c-a)x+(a-b)=0$ has a real root.
[i]O. Podlipsky[/i]
Russian TST 2019, P1
The shores of the Tvertsy River are two parallel straight lines. There are point-like villages on the shores in some order: 20 villages on the left shore and 15 villages on the right shore. We want to build a system of non-intersecting bridges, that is, segments connecting a couple of villages from different shores, so that from any village you can get to any other village only by bridges (you can't walk along the shore). In how many ways can such a bridge system be built?
1997 All-Russian Olympiad Regional Round, 11.5
Members of the State Duma formed factions in such a way that for any two fractions $A $ and $B$ (not necessarily different), $\overline{A \cup B}$ is also faction ($\overline{C}$ denotes the set of all members of the Duma, not including in $C$). Prove that for any two factions $A$ and $B$, $A \cup B$ is also a faction.
1990 IMO Longlists, 5
Given the condition that there exist exactly $1990$ triangles $ABC$ with integral side-lengths satisfying the following conditions:
(i) $\angle ABC =\frac 12 \angle BAC;$
(ii) $AC = b.$
Find the minimal value of $b.$
2003 Switzerland Team Selection Test, 9
Given integers $0 < a_1 < a_2 <... < a_{101} < 5050$, prove that one can always choose for different numbers $a_k,a_l,a_m,a_n$ such that $5050 | a_k +a_l -a_m -a_n$
2017 Baltic Way, 6
Fifteen stones are placed on a $4 \times 4$ board, one in each cell, the remaining cell being empty. Whenever two stones are on neighbouring cells (having a common side), one may jump over the other to the opposite neighbouring cell, provided this cell is empty. The stone jumped over is removed from the board.
For which initial positions of the empty cell is it possible to end up with exactly one stone on the board?
1989 French Mathematical Olympiad, Problem 4
For natural numbers $x_1,\ldots,x_k$, let $[x_k,\ldots,x_1]$ be defined recurrently as follows: $[x_2,x_1]=x_2^{x_1}$ and, for $k\ge3$, $[x_k,x_{k-1},\ldots,x_1]=x_k^{[x_{k-1},\ldots,x_1]}$.
(a) Let $3\le a_1\le a_2\le\ldots\le a_n$be integers. For a permutation $\sigma$ of the set $\{1,2,\ldots,n\}$, we set $P(\sigma)=[a_{\sigma(n)},a_{\sigma(n-1)},\ldots,a_{\sigma(1)}]$. Find the permutations $\sigma$ for which $P(\sigma)$ is minimal or maximal.
(b) Find all integers $a,b,c,d$, greater than or equal to $2$, for which $[178,9]\le[a,b,c,d]\le[198,9]$.
2020 Malaysia IMONST 2, 1
Given a trapezium with two parallel sides of lengths $m$ and $n$, where $m$, $n$ are integers, prove that it is
possible to divide the trapezium into several congruent triangles.
2010 Contests, 1
Solve in positive reals the system:
$x+y+z+w=4$
$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{w}=5-\frac{1}{xyzw}$
2011 Serbia National Math Olympiad, 1
Let $n \ge 2$ be integer. Let $a_0$, $a_1$, ... $a_n$ be sequence of positive reals such that:
$(a_{k-1}+a_k)(a_k+a_{k+1})=a_{k-1}-a_{k+1}$, for $k=1, 2, ..., n-1$.
Prove $a_n< \frac{1}{n-1}$.
1989 All Soviet Union Mathematical Olympiad, 508
A polyhedron has an even number of edges. Show that we can place an arrow on each edge so that each vertex has an even number of arrows pointing towards it (on adjacent edges).
1996 Romania National Olympiad, 3
Prove that $ \forall x\in \mathbb{R} $ , $ \cos ^7x+\cos ^7(x+\frac {2\pi}{3})+\cos ^7(x+\frac {4\pi}{3})=\frac {63}{64}\cos 3x $
2010 AMC 12/AHSME, 1
Makayla attended two meetings during her 9-hour work day. The first meeting took 45 minutes and the second meeting took twice as long. What percent of her work day was spent attending meetings?
$ \textbf{(A)}\ 15 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 25 \qquad \textbf{(D)}\ 30 \qquad \textbf{(E)}\ 35$
1992 Austrian-Polish Competition, 7
Consider triangles $ABC$ in space.
(a) What condition must the angles $\angle A, \angle B , \angle C$ of $\triangle ABC$ fulfill in order that there is a point $P$ in space such that $\angle APB, \angle BPC, \angle CPA$ are right angles?
(b) Let $d$ be the longest of the edges $PA,PB,PC$ and let $h$ be the longest altitude of $\triangle ABC$. Show that $\frac{1}{3}\sqrt6 h \le d \le h$.
2006 Italy TST, 3
Find all functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that for all integers $m,n$,
\[f(m - n + f(n)) = f(m) + f(n).\]
Durer Math Competition CD Finals - geometry, 2019.C5
$A, B, C, D$ are four distinct points such that triangles $ABC$ and $CBD$ are both equilateral. Find as many circles as you can, which are equidistant from the four points. How can these circles be constructed?
[i]Remark: The distance between a point $P$ and a circle c is measured as follows: we join $P$ and the centre of the circle with a straight line, and measure how much we need to travel along thisline (starting from $P$) to hit the perimeter of the circle. If $P$ is an internal point of the circle, the distance is the length of the shorter such segment. The distance between a circle and itscentre is the radius of the circle.[/i]
2005 APMO, 5
In a triangle $ABC$, points $M$ and $N$ are on sides $AB$ and $AC$, respectively, such that $MB = BC = CN$. Let $R$ and $r$ denote the circumradius and the inradius of the triangle $ABC$, respectively. Express the ratio $MN/BC$ in terms of $R$ and $r$.
1940 Moscow Mathematical Olympiad, 059
Consider all positive integers written in a row: $123456789101112131415...$ Find the $206788$-th digit from the left.
1977 IMO Longlists, 11
Let $n$ and $z$ be integers greater than $1$ and $(n,z)=1$. Prove:
(a) At least one of the numbers $z_i=1+z+z^2+\cdots +z^i,\ i=0,1,\ldots ,n-1,$ is divisible by $n$.
(b) If $(z-1,n)=1$, then at least one of the numbers $z_i$ is divisible by $n$.
1986 AMC 12/AHSME, 17
A drawer in a darkened room contains $100$ red socks, $80$ green socks, $60$ blue socks and $40$ black socks. A youngster selects socks one at a time from the drawer but is unable to see the color of the socks drawn. What is the smallest number of socks that must be selected to guarantee that the selection contains at least $10$ pairs? (A pair of socks is two socks of the same color. No sock may be counted in more than one pair.)
$ \textbf{(A)}\ 21\qquad\textbf{(B)}\ 23\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 50$
1955 AMC 12/AHSME, 23
In checking the petty cash a clerk counts $ q$ quarters, $ d$ dimes, $ n$ nickels, and $ c$ cents. Later he discovers that $ x$ of the nickels were counted as quarters and $ x$ of the dimes were counted as cents. To correct the total obtained the clerk must:
$ \textbf{(A)}\ \text{make no correction} \qquad
\textbf{(B)}\ \text{subtract 11 cents} \qquad
\textbf{(C)}\ \text{subtract 11}x\text{ cents} \\
\textbf{(D)}\ \text{add 11}x\text{ cents} \qquad
\textbf{(E)}\ \text{add }x\text{ cents}$
2014 AMC 12/AHSME, 12
Two circles intersect at points $A$ and $B$. The minor arcs $AB$ measure $30^\circ$ on one circle and $60^\circ$ on the other circle. What is the ratio of the area of the larger circle to the area of the smaller circle?
$\textbf{(A) }2\qquad
\textbf{(B) }1+\sqrt3\qquad
\textbf{(C) }3\qquad
\textbf{(D) }2+\sqrt3\qquad
\textbf{(E) }4\qquad$
2021-2022 OMMC, 21
For some real number $a$, define two parabolas on the coordinate plane with equations $x = y^2 + a$ and $y = x^2 + a$. Suppose there are $3$ lines, each tangent to both parabolas, that form an equilateral triangle with positive area $s$. If $s^2 = \tfrac pq$ for coprime positive integers $p$, $q$, find $p + q$.
[i]Proposed by Justin Lee[/i]
2007 Korea National Olympiad, 1
For all positive reals $ a$, $ b$, and $ c$, what is the value of positive constant $ k$ satisfies the following inequality?
$ \frac{a}{c\plus{}kb}\plus{}\frac{b}{a\plus{}kc}\plus{}\frac{c}{b\plus{}ka}\geq\frac{1}{2007}$ .