Found problems: 85335
2006 National Olympiad First Round, 3
$a_1=-1$, $a_2=2$, and $a_n=\frac {a_{n-1}}{a_{n-2}}$ for $n\geq 3$. What is $a_{2006}$?
$
\textbf{(A)}\ -2
\qquad\textbf{(B)}\ -1
\qquad\textbf{(C)}\ -\frac 12
\qquad\textbf{(D)}\ \frac 12
\qquad\textbf{(E)}\ 2
$
2014 AMC 12/AHSME, 8
In the addition shown below $A$, $B$, $C$, and $D$ are distinct digits. How many different values are possible for $D$?
\[\begin{array}{lr}
&ABBCB \\
+& BCADA \\
\hline
& DBDDD
\end{array}\]
$\textbf{(A) }2\qquad\textbf{(B) }4\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9$
2024 IFYM, Sozopol, 4
Let \( n \geq 4 \) be a positive integer. Initially, each of \( n \) girls knows one piece of gossip that no one else knows, and they want to share them. For greater security, to avoid being spied, they only talk in pairs, and in a conversation, each girl shares all the gossip she knows so far with the other one. What is the minimum number of conversations needed so that every girl knows all the gossip?
2003 Tournament Of Towns, 6
An ant crawls on the outer surface of the box in a shape of rectangular parallelepiped. From ant’s point of view, the distance between two points on a surface is defined by the length of the shortest path ant need to crawl to reach one point from the other. Is it true that if ant is at vertex then from ant’s point of view the opposite vertex be the most distant point on the surface?
2025 Romania EGMO TST, P2
Let $m$ and $n$ be positive integers with $m > n \ge 2$. Set $S =\{1,2,...,m\}$, and set $T = \{a_1,a_2,...,a_n\}$ is a subset of $S$ such that every element of $S$ is not divisible by any pair of distinct elements of $T$. Prove that
$$\frac{1}{a_1}+\frac{1}{a_2}+ ...+ \frac{1}{a_n} < \frac{m+n}{m}$$
2017 Online Math Open Problems, 10
Determine the value of $-1+2+3+4-5-6-7-8-9+...+10000$, where the signs change after each perfect square.
[i]Proposed by Michael Ren
1988 Tournament Of Towns, (175) 1
Is it possible to select two natural numbers $m$ and $n$ so that the number $n$ results from a permutation of the digits of $m$, and $m+n =999 . . . 9$ ?
1996 Miklós Schweitzer, 5
Let K and D be the set of convergent and divergent series of positive terms respectively. Does there exist a bijection between K and D such that for all $\sum a_n,\sum b_n\in K$ and $\sum a_n',\sum b_n'\in D$ , $\frac{a_n}{b_n}\to 0\iff \frac{a_n'}{b_n'}\to 0$ ? Under the bijection, $\sum a_n\leftrightarrow\sum a_n'$ and $\sum b_n\leftrightarrow\sum b_n'$.
2012 Stars of Mathematics, 2
Let $\ell$ be a line in the plane, and a point $A \not \in \ell$. Also let $\alpha \in (0, \pi/2)$ be fixed. Determine the locus of the points $Q$ in the plane, for which there exists a point $P\in \ell$ such that $AQ=PQ$ and $\angle PAQ = \alpha$.
([i]Dan Schwarz[/i])
1997 Turkey MO (2nd round), 2
In a triangle $ABC$, the inner and outer bisectors of the $\angle A$ meet the line $BC$ at $D$ and $E$, respectively. Let $d$ be a common tangent of the circumcircle $(O)$ of $\triangle ABC$ and the circle with diameter $DE$ and center $F$. The projections of the tangency points onto $FO$ are denoted by $P$ and $Q$, and the length of their common chord is denoted by $m$. Prove that $PQ = m$
1995 Tuymaada Olympiad, 2
Let $x_1=a, x_2=a^{x_1}, ..., x_n=a^{x_{n-1}}$ where $a>1$. What is the maximum value of $a$ for which lim exists $\lim_{n\to \infty} x_n$ and what is this limit?
1951 Putnam, B3
Show that if $x$ is positive, then \[ \log_e (1 + 1/x) > 1 / (1 + x).\]
2017 Harvard-MIT Mathematics Tournament, 15
Start by writing the integers $1, 2, 4, 6$ on the blackboard. At each step, write the smallest positive integer $n$ that satisfies both of the following properties on the board.
[list]
[*] $n$ is larger than any integer on the board currently.
[*] $n$ cannot be written as the sum of $2$ distinct integers on the board.
[/list]
Find the $100$-th integer that you write on the board. Recall that at the beginning, there are already $4$ integers on the board.
1956 Miklós Schweitzer, 9
[b]9.[/b] Show that if the trigonometric polynomial $f(\theta)= \sum_{v=1}^{n} a_v \cos v\theta$ monotonically decreases over the closed interval $[0,\pi]$, then the trigonometric polynomial $g(\theta)=\sum_{v=1}^{n}a_v \sin v\theta$ is non negative in the same interval. [b](S. 26)[/b]
2013 F = Ma, 20
A simple pendulum experiment is constructed from a point mass $m$ attached to a pivot by a massless rod of length $L$ in a constant gravitational field. The rod is released from an angle $\theta_0 < \frac{\pi}{2}$ at rest and the period of motion is found to be $T_0$. Ignore air resistance and friction.
What is the maximum value of the tension in the rod?
$\textbf{(A) } mg\\
\textbf{(B) } 2mg\\
\textbf{(C) } mL\theta_0/T_0^2\\
\textbf{(D) } mg \sin \theta_0\\
\textbf{(E) } mg(3 - 2 \cos \theta_0)$
VMEO III 2006 Shortlist, N6
Find all sets of natural numbers $(a, b, c)$ such that $$a+1|b^2+c^2\,\, , b+1|c^2+a^2\,\,, c+1|a^2+b^2.$$
KoMaL A Problems 2020/2021, A. 781
We want to construct an isosceles triangle using a compass and a straightedge. We are given two of the following four data: the length of the base of the triangle $(a),$ the length of the leg of the triangle $(b),$ the radius of the inscribed circle $(r),$ and the radius of the circumscribed circle $(R).$ In which of the six possible cases will we definitely be able to construct the triangle?
[i]Proposed by György Rubóczky, Budapest[/i]
1990 Swedish Mathematical Competition, 5
Find all monotonic positive functions $f(x)$ defined on the positive reals such that $f(xy) f\left( \frac{f(y)}{x}\right) = 1$ for all $x, y$.
2017 Saint Petersburg Mathematical Olympiad, 3
Given real numbers $x,y,z,t\in (0,\pi /2]$ such that
$$\cos^2 (x)+\cos^2 (y) +\cos^2 (z) +\cos^2 (t)=1.$$
What is the minimum possible value of $$\cot (x) +\cot (y) +\cot (z) +\cot (t)?$$
2016 Sharygin Geometry Olympiad, P19
Let $ABCDEF$ be a regular hexagon. Points $P$ and $Q$ on tangents to its circumcircle at $A$ and $D$ respectively are such that $PQ$ touches the minor arc $EF$ of this circle. Find the angle between $PB$ and $QC$.
2016 BMT Spring, 14
Consider the set of axis-aligned boxes in $R^d$ , $B(a, b) = \{x \in R^d: \forall i, a_i \le x_i \le b_i\}$ where $a, b \in R^d$. In terms of $d$, what is the maximum number $n$, such that there exists a set of $n$ points $S =\{x_1, ..., x_n\}$ such that no matter how one partition $S = P \cup Q$ with $P, Q$ disjoint and $P, Q$ can possibly be empty, there exists a box $B$ such that all the points in $P$ are contained in $B$, and all the points in $Q$ are outside $B$?
2000 Harvard-MIT Mathematics Tournament, 6
Barbara, Edward, Abhinav, and Alex took turns writing this test. Working alone, they could finish it in $10$, $9$, $11$, and $12$ days, respectively. If only one person works on the test per day, and nobody works on it unless everyone else has spent at least as many days working on it, how many days (an integer) did it take to write this test?
2008 Moldova National Olympiad, 9.6
find x and y in R
$\begin{array}{l} (\frac{1}{{\sqrt[3]{x}}} + \frac{1}{{\sqrt[3]{y}}})(\frac{1}{{\sqrt[3]{x}}} + 1)(\frac{1}{{\sqrt[3]{y}}} + 1) = 18 \\ \frac{1}{x} + \frac{1}{y} = 9 \\ \end{array}$
2005 Tournament of Towns, 1
The graphs of four functions of the form $y = x^2 + ax + b$, where a and b are real coefficients, are plotted on the coordinate plane. These graphs have exactly four points of intersection, and at each one of them, exactly two graphs intersect. Prove that the sum of the largest and the smallest $x$-coordinates of the points of intersection is equal to the sum of the other two.
[i](3 points)[/i]
2012 Harvard-MIT Mathematics Tournament, 5
A mouse lives in a circular cage with completely reflective walls. At the edge of this cage, a small flashlight with vertex on the circle whose beam forms an angle of $15^o$ is centered at an angle of $37.5^o$ away from the center. The mouse will die in the dark. What fraction of the total area of the cage can keep the mouse alive?
[img]https://cdn.artofproblemsolving.com/attachments/1/c/283276058b7b2c85a95976743c5188ee8ee008.png[/img]