Found problems: 85335
2011 Canadian Open Math Challenge, 7
In the figure, BCÂ Â is a diameter of the circle, where $BC=\sqrt{901}, BD=1$, and $DA=16$. If $EC=x$, what is the value of x?
[asy]size(2inch);
pair O,A,B,C,D,E;
B=(0,0);
O=(2,0);
C=(4,0);
D=(.333,1.333);
A=(.75,2.67);
E=(1.8,2);
draw(Arc(O,2,0,360));
draw(B--C--A--B);
label("$A$",A,N);
label("$B$",B,W);
label("$C$",C,E);
label("$D$",D,W);
label("$E$",E,N);
label("Figure not drawn to scale",(2,-2.5),S);
[/asy]
2006 Chile National Olympiad, 2
In a triangle $ \vartriangle ABC $ with sides integer numbers, it is known that the radius of the circumcircle circumscribed to $ \vartriangle ABC $ measures $ \dfrac {65} {8} $ centimeters and the area is $84$ cm². Determine the lengths of the sides of the triangle.
2015 BMT Spring, 14
Alice is at coordinate point $(0, 0)$ and wants to go to point $(11, 6)$. Similarly, Bob is at coordinate point $(5, 6)$ and wants to go to point $(16, 0)$. Both of them choose a lattice path from their current position to their target position at random (such that each lattice path has an equal probability of being chosen), where a lattice path is defined to be a path composed of unit segments with orthogonal direction (parallel to x-axis or y-axis) and of minimal length. (For instance, there are six lattice paths from $(0, 0)$ to $(2, 2)$.) If they walk with the same speed, find the probability that they meet.
2013 Iran Team Selection Test, 7
Nonnegative real numbers $p_{1},\ldots,p_{n}$ and $q_{1},\ldots,q_{n}$ are such that $p_{1}+\cdots+p_{n}=q_{1}+\cdots+q_{n}$
Among all the matrices with nonnegative entries having $p_i$ as sum of the $i$-th row's entries and $q_j$ as sum of the $j$-th column's entries, find the maximum sum of the entries on the main diagonal.
2003 AIME Problems, 3
Define a $good~word$ as a sequence of letters that consists only of the letters $A,$ $B,$ and $C$ $-$ some of these letters may not appear in the sequence $-$ and in which $A$ is never immediately followed by $B,$ $B$ is never immediately followed by $C,$ and $C$ is never immediately followed by $A.$ How many seven-letter good words are there?
2023 China Northern MO, 5
Given a finite graph $G$, let $f(G)$ be the number of triangles in graph $G$, $g(G)$ be the number of edges in graph $G$, find the minimum constant $c$, so that for each graph $G$, there is $f^ 2(G)\le c \cdot g^3(G)$.
2014 Singapore Senior Math Olympiad, 10
If $f(x)=\frac{1}{x}-\frac{4}{\sqrt{x}}+3$ where $\frac{1}{16}\le x\le 1$, find the range of $f(x)$.
$ \textbf{(A) }-2\le f(x)\le 4 \qquad\textbf{(B) }-1\le f(x)\le 3\qquad\textbf{(C) }0\le f(x)\le 3\qquad\textbf{(D) }-1\le f(x)\le 4\qquad\textbf{(E) }\text{None of the above} $
2022 LMT Fall, 8
An odd positive integer $n$ can be expressed as the sum of two or more consecutive integers in exactly $2023$ ways. Find the greatest possible nonnegative integer $k$ such that $3^k$ is a factor of the least possible value of $n$.
2007 All-Russian Olympiad Regional Round, 10.6
A point $ D$ is chosen on side $ BC$ of a triangle $ ABC$ such that the inradii of triangles $ ABD$ and $ ACD$ are equal. Consider in these triangles the excircles touching sides $ BD$ and $ CD$, respectively. Prove that their radii are also equal.
2010 Princeton University Math Competition, 4
Sterling draws 6 circles on the plane, which divide the plane into regions (including the unbounded region). What is the maximum number of resulting regions?
2021 Indonesia TST, C
In a country, there are $2018$ cities, some of which are connected by roads. Each city is connected to at least three other cities. It is possible to travel from any city to any other city using one or more roads. For each pair of cities, consider the shortest route between these two cities. What is the greatest number of roads that can be on such a shortest route?
1959 AMC 12/AHSME, 40
In triangle $ABC$, $BD$ is a median. $CF$ intersects $BD$ at $E$ so that $\overline{BE}=\overline{ED}$. Point $F$ is on $AB$. Then, if $\overline{BF}=5$, $\overline{BA}$ equals:
$ \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 20\qquad\textbf{(E)}\ \text{none of these} $
1996 Moldova Team Selection Test, 5
Find all polynomials $P(X)$ of fourth degree with real coefficients that verify the properties:
[b]a)[/b] $P(-x)=P(x), \forall x\in\mathbb{R};$
[b]b)[/b] $P(x)\geq0, \forall x\in\mathbb{R};$
[b]c)[/b] $P(0)=1;$
[b]d)[/b] $P(X)$ has exactly two local minimums $x_1$ and $x_2$ such that $|x_1-x_2|=2.$
1940 Putnam, B5
Suppose that the rational numbers $a, b$ and $c$ are the roots of the equation $x^3+ax^2 + bx + c = 0$. Find all such rational numbers $a, b$ and $c$. Justify your answer
2016 Korea Summer Program Practice Test, 3
Let $p > 10^9$ be a prime number such that $4p + 1$ is also prime.
Prove that the decimal expansion of $\frac{1}{4p+1}$ contains all the digits $0,1, \ldots, 9$.
2011 Kyiv Mathematical Festival, 3
Quadrilateral can be cut into two isosceles triangles in two different ways.
a) Can this quadrilateral be nonconvex?
b) If given quadrilateral is convex, is it necessarily a rhomb?
2013 District Olympiad, 4
Let $n\in {{\mathbb{N}}^{*}}$. Prove that $2\sqrt{{{2}^{n}}}\cos \left( n\arccos \frac{\sqrt{2}}{4} \right)$ is an odd integer.
1988 Romania Team Selection Test, 10
Let $ p > 2$ be a prime number. Find the least positive number $ a$ which can be represented as
\[ a \equal{} (X \minus{} 1)f(X) \plus{} (X^{p \minus{} 1} \plus{} X^{p \minus{} 2} \plus{} \cdots \plus{} X \plus{} 1)g(X),
\]
where $ f(X)$ and $ g(X)$ are integer polynomials.
[i]Mircea Becheanu[/i].
2012 Balkan MO, 3
Let $n$ be a positive integer. Let $P_n=\{2^n,2^{n-1}\cdot 3, 2^{n-2}\cdot 3^2, \dots, 3^n \}.$ For each subset $X$ of $P_n$, we write $S_X$ for the sum of all elements of $X$, with the convention that $S_{\emptyset}=0$ where $\emptyset$ is the empty set. Suppose that $y$ is a real number with $0 \leq y \leq 3^{n+1}-2^{n+1}.$
Prove that there is a subset $Y$ of $P_n$ such that $0 \leq y-S_Y < 2^n$
2014 India IMO Training Camp, 2
Let $n$ be a natural number.A triangulation of a convex n-gon is a division of the polygon into $n-2$ triangles by drawing $n-3$ diagonals no two of which intersect at an interior point of the polygon.Let $f(n)$ denote the number of triangulations of a regular n-gon such that each of the triangles formed is isosceles.Determine $f(n)$ in terms of $n$.
LMT Accuracy Rounds, 2022 S5
A bag contains $5$ identical blue marbles and $5$ identical green marbles. In how many ways can $5$ marbles from the bag be arranged in a row if each blue marble must be adjacent to at least $1$ green marble?
1989 Nordic, 3
Let $S$ be the set of all points $t$ in the closed interval $[-1, 1]$ such that for the sequence $x_0, x_1, x_2, ...$ defined by the equations $x_0 = t, x_{n+1} = 2x_n^2-1$, there exists a positive integer $N$ such that $x_n = 1$ for all $n \ge N$. Show that the set $S$ has infinitely many elements.
1980 IMO, 11
A triangle $(ABC)$ and a point $D$ in its plane satisfy the relations
\[\frac{BC}{AD}=\frac{CA}{BD}=\frac{AB}{CD}=\sqrt{3}.\]
Prove that $(ABC)$ is equilateral and $D$ is its center.
2022 Taiwan TST Round 1, N
Find all positive integers $n$ with the following property: the $k$ positive divisors of $n$ have a permutation $(d_1,d_2,\ldots,d_k)$ such that for $i=1,2,\ldots,k$, the number $d_1+d_2+\cdots+d_i$ is a perfect square.
1983 Tournament Of Towns, (043) A5
$k$ vertices of a regular $n$-gon $P$ are coloured. A colouring is called almost uniform if for every positive integer $m$ the following condition is satisfied:
If $M_1$ is a set of m consecutive vertices of $P$ and $M_2$ is another such set then the number of coloured vertices of $M_1$ differs from the number of coloured vertices of $M_2$ at most by $1$.
Prove that for all positive integers $k$ and $n$ ($k \le n$) an almost uniform colouring exists and that it is unique within a rotation.
(M Kontsevich, Moscow)