Found problems: 85335
2013 NIMO Problems, 5
For every integer $n \ge 1$, the function $f_n : \left\{ 0, 1, \cdots, n \right\} \to \mathbb R$ is defined recursively by $f_n(0) = 0$, $f_n(1) = 1$ and \[ (n-k) f_n(k-1) + kf_n(k+1) = nf_n(k) \] for each $1 \le k < n$. Let $S_N = f_{N+1}(1) + f_{N+2}(2) + \cdots + f_{2N} (N)$. Find the remainder when $\left\lfloor S_{2013} \right\rfloor$ is divided by $2011$. (Here $\left\lfloor x \right\rfloor$ is the greatest integer not exceeding $x$.)
[i]Proposed by Lewis Chen[/i]
2002 Indonesia MO, 7
Let $ABCD$ be a rhombus where $\angle DAB = 60^\circ$, and $P$ be the intersection between $AC$ and $BD$. Let $Q,R,S$ be three points on the boundary of $ABCD$ such that $PQRS$ is a rhombus. Prove that exactly one of $Q,R,S$ lies on one of $A,B,C,D$.
1940 Putnam, A2
Let $A,B$ be two fixed points on the curve $y=f(x)$, $f$ is continuous with continuous derivative and the arc $\widehat{AB}$ is concave to the chord $AB$. If $P$ is a point on the arc $\widehat{AB}$ for which $AP+PB$ is maximal, prove that $PA$ and $PB$ are equally inclined to the tangent to the curve $y=f(x)$ at $P$.
2021 JHMT HS, 9
Right triangle $ABC$ has a right angle at $A.$ Points $D$ and $E$ respectively lie on $\overline{AC}$ and $\overline{BC}$ so that $\angle BDA \cong \angle CDE.$ If the lengths $DE,$ $DA,$ $DC,$ and $DB,$ in this order, form an arithmetic sequence of distinct positive integers, then the set of all possible areas of $\triangle ABC$ is a subset of the positive integers. Compute the smallest element in this set that is greater than $1000.$
Kvant 2023, M2762
The sum of $n > 2$ nonzero real numbers (not necessarily distinct) equals zero. For each of the $2^n - 1$ ways to choose one or more of these numbers, their sums are written in non-increasing order in a row. The first number in the row is $S$. Find the smallest possible value of the second number.
2017 Iberoamerican, 2
Let $ABC$ be an acute angled triangle and $\Gamma$ its circumcircle. Led $D$ be a point on segment $BC$, different from $B$ and $C$, and let $M$ be the midpoint of $AD$. The line perpendicular to $AB$ that passes through $D$ intersects $AB$ in $E$ and $\Gamma$ in $F$, with point $D$ between $E$ and $F$. Lines $FC$ and $EM$ intersect at point $X$. If $\angle DAE = \angle AFE$, show that line $AX$ is tangent to $\Gamma$.
2017 Caucasus Mathematical Olympiad, 6
Given real numbers $a$, $b$, $c$ satisfy inequality $\left| \frac{a^2+b^2-c^2}{ab} \right|<2$. Prove that they also satisfy equalities $\left| \frac{b^2+c^2-a^2}{bc} \right|<2$ and $\left| \frac{c^2+a^2-b^2}{ca} \right| <2$.
2009 IMO Shortlist, 3
Let $f$ be a non-constant function from the set of positive integers into the set of positive integer, such that $a-b$ divides $f(a)-f(b)$ for all distinct positive integers $a$, $b$. Prove that there exist infinitely many primes $p$ such that $p$ divides $f(c)$ for some positive integer $c$.
[i]Proposed by Juhan Aru, Estonia[/i]
Kyiv City MO Seniors 2003+ geometry, 2012.10.4
The triangle $ABC$ with $AB> AC$ is inscribed in a circle, the angle bisector of $\angle BAC$ intersects the side $BC$ of the triangle at the point $K$, and the circumscribed circle at the point $M$. The midline of $\Delta ABC$, which is parallel to the side $AB$, intersects $AM$ at the point $O$, the line $CO$ intersects the line $AB$ at the point $N$. Prove that a circle can be circumscribed around the quadrilateral $BNKM$.
(Nagel Igor)
1985 All Soviet Union Mathematical Olympiad, 415
All the points situated more close than $1$ cm to ALL the vertices of the regular pentagon with $1$ cm side, are deleted from that pentagon. Find the area of the remained figure.
2007 IMO Shortlist, 7
Given an acute triangle $ ABC$ with $ \angle B > \angle C$. Point $ I$ is the incenter, and $ R$ the circumradius. Point $ D$ is the foot of the altitude from vertex $ A$. Point $ K$ lies on line $ AD$ such that $ AK \equal{} 2R$, and $ D$ separates $ A$ and $ K$. Lines $ DI$ and $ KI$ meet sides $ AC$ and $ BC$ at $ E,F$ respectively. Let $ IE \equal{} IF$.
Prove that $ \angle B\leq 3\angle C$.
[i]Author: Davoud Vakili, Iran[/i]
2016 CMIMC, 3
Let $\varepsilon$ denote the empty string. Given a pair of strings $(A,B)\in\{0,1,2\}^*\times\{0,1\}^*$, we are allowed the following operations:
\[\begin{cases}
(A,1)\to(A0,\varepsilon)\\
(A,10)\to(A00,\varepsilon)\\
(A,0B)\to(A0,B)\\
(A,11B)\to(A01,B)\\
(A,100B)\to(A0012,1B)\\
(A,101B)\to(A00122,10B)
\end{cases}\]
We perform these operations on $(A,B)$ until we can no longer perform any of them. We then iteratively delete any instance of $20$ in $A$ and replace any instance of $21$ with $1$ until there are no such substrings remaining. Among all binary strings $X$ of size $9$, how many different possible outcomes are there for this process performed on $(\varepsilon,X)$?
2003 AIME Problems, 14
The decimal representation of $m/n$, where $m$ and $n$ are relatively prime positive integers and $m < n$, contains the digits 2, 5, and 1 consecutively, and in that order. Find the smallest value of $n$ for which this is possible.
1982 Brazil National Olympiad, 3
$S$ is a $(k+1) \times (k+1)$ array of lattice points. How many squares have their vertices in $S$?
2002 National Olympiad First Round, 12
What is the least possible value of $ab + bc + ac$ such that $a^2 + b^2 + c^2 = 1$ where $a,b,c$ are real numbers?
$
\textbf{a)}\ -1
\qquad\textbf{b)}\ -\dfrac 12
\qquad\textbf{c)}\ -\dfrac 13
\qquad\textbf{d)}\ -\dfrac{1}{2\sqrt 2}
\qquad\textbf{e)}\ 0
$
2014 Online Math Open Problems, 25
If
\[
\sum_{n=1}^{\infty}\frac{\frac11 + \frac12 + \dots + \frac 1n}{\binom{n+100}{100}} = \frac pq
\]
for relatively prime positive integers $p,q$, find $p+q$.
[i]Proposed by Michael Kural[/i]
1991 Tournament Of Towns, (307) 4
A sequence $a_n$ is determined by the rules $a_0 = 9$ and for any nonnegative $k$,
$$a_{k+1}=3a_k^4+4a_k^3.$$
Prove that $a_{10}$ contains more than $1000$ nines in decimal notation.
(Yao)
2014 Contests, 3
The diagram below shows a rectangle with side lengths $36$ and $48$. Each of the sides is trisected and edges are added between the trisection points as shown. Then the shaded corner regions are removed, leaving the octagon which is not shaded in the diagram. Find the perimeter of this octagon.
[asy]
size(4cm);
dotfactor=3.5;
pair A,B,C,D,E,F,G,H,W,X,Y,Z;
A=(0,12);
B=(0,24);
C=(16,36);
D=(32,36);
E=(48,24);
F=(48,12);
G=(32,0);
H=(16,0);
W=origin;
X=(0,36);
Y=(48,36);
Z=(48,0);
filldraw(W--A--H--cycle^^B--X--C--cycle^^D--Y--E--cycle^^F--Z--G--cycle,rgb(.76,.76,.76));
draw(W--X--Y--Z--cycle,linewidth(1.2));
dot(A);
dot(B);
dot(C);
dot(D);
dot(E);
dot(F);
dot(G);
dot(H);
[/asy]
2017 CMIMC Computer Science, 2
We are given the following function $f$, which takes a list of integers and outputs another list of integers. (Note that here the list is zero-indexed.)
\begin{tabular}{l}
1: \textbf{FUNCTION} $f(A)$ \\
2: $\quad$ \textbf{FOR} $i=1,\ldots, \operatorname{length}(A)-1$: \\
3: $\quad\quad$ $A[i]\leftarrow A[A[i]]$ \\
4: $\quad\quad$ $A[0]\leftarrow A[0]-1$ \\
5: $\quad$ \textbf{RETURN} $A$
\end{tabular}
Suppose the list $B$ is equal to $[0,1,2,8,2,0,1,7,0]$. In how many entries do $B$ and $f(B)$ differ?
Kvant 2021, M2670
There are 100 points on the plane so that any 10 of them are vertices of a convex polygon. Does it follow from this that all these points are the vertices of a convex 100-gon?
[i]From the folklore[/i]
2018 IFYM, Sozopol, 3
The number 1 is a solution of the equation
$(x + a)(x + b)(x + c)(x + d) = 16$,
where $a, b, c, d$ are positive real numbers. Find the largest value of $abcd$.
2021 Kazakhstan National Olympiad, 5
Find all functions $f : \mathbb{R^{+}}\to \mathbb{R^{+}}$ such that $$f(x)^2=f(xy)+f(x+f(y))-1$$ for all $x, y\in \mathbb{R^{+}}$
2013 IMO Shortlist, A1
Let $n$ be a positive integer and let $a_1, \ldots, a_{n-1} $ be arbitrary real numbers. Define the sequences $u_0, \ldots, u_n $ and $v_0, \ldots, v_n $ inductively by $u_0 = u_1 = v_0 = v_1 = 1$, and $u_{k+1} = u_k + a_k u_{k-1}$, $v_{k+1} = v_k + a_{n-k} v_{k-1}$ for $k=1, \ldots, n-1.$
Prove that $u_n = v_n.$
2019 Durer Math Competition Finals, 9
A cube has been divided into $27$ equal-sized sub-cubes. We take a line that passes through the interiors of as many sub-cubes as possible. How many does it pass through?
2008 Moldova National Olympiad, 11.2
Let $ (a_{n})_{n\ge 1} $ be a sequence such that: $ a_{1}=1; a_{n+1}=\frac{n}{a_{n}+1}.$ Find $ [a_{2008}] $