Found problems: 85335
1995 Baltic Way, 11
In how many ways can the set of integers $\{1,2,\ldots ,1995\}$ be partitioned into three non-empty sets so that none of these sets contains any pair of consecutive integers?
1986 IMO Longlists, 56
Let $A_1A_2A_3A_4A_5A_6$ be a hexagon inscribed into a circle with center $O$. Consider the circular arc with endpoints $A_1,A_6$ not containing $A_2$. For any point $M$ of that arc denote by $h_i$ the distance from $M$ to the line $A_iA_{i+1} \ (1 \leq i \leq 5)$. Construct $M$ such that the sum $h_1 + \cdots + h_5$ is maximal.
1996 Turkey MO (2nd round), 2
Prove that $\prod\limits_{k=0}^{n-1}{({{2}^{n}}-{{2}^{k}})}$ is divisible by $n!$ for all positive integers $n$.
2016 BMT Spring, 5
Find
$$\frac{\tan 1^o}{1 + \tan 1^o }+\frac{\tan 2^o}{1 + \tan 2^o } + ... + \frac{\tan 89^o}{1 + \tan 89^o}$$
2020 USMCA, 22
Kelvin the Frog places $40$ rooks on a uniformly random subset of $40$ squares of a $20 \times 20$ chessboard. Then, Alex the Kat chooses two of the $40$ rooks uniformly at random. What is the probability that Alex's two rooks attack each other? Two rooks attack each other if they are on the same row or column, and no piece stands between them.
2014 NIMO Summer Contest, 6
Suppose $x$ is a random real number between $1$ and $4$, and $y$ is a random real number between $1$ and $9$. If the expected value of \[ \left\lceil \log_2 x \right\rceil - \left\lfloor \log_3 y \right\rfloor \] can be expressed as $\frac mn$ where $m$ and $n$ are relatively prime positive integers, compute $100m + n$.
[i]Proposed by Lewis Chen[/i]
1999 All-Russian Olympiad, 2
There are several cities in a country. Some pairs of the cities are connected by a two-way airline of one of the $N$ companies, so that each company serves exactly one airline from each city, and one can travel between any two cities, possibly with transfers. During a financial crisis, $N-1$ airlines have been canceled, all from different companies. Prove that it is still possible to travel between any two cities.
1991 Arnold's Trivium, 68
Find
\[\inf\iint_{x^2+y^2\le1}\left(\frac{\partial u}{\partial x}\right)^2+\left(\frac{\partial u}{\partial y}\right)^2dxdy\]
1969 IMO Longlists, 34
$(HUN 1)$ Let $a$ and $b$ be arbitrary integers. Prove that if $k$ is an integer not divisible by $3$, then $(a + b)^{2k}+ a^{2k} +b^{2k}$ is divisible by $a^2 +ab+ b^2$
2020 Caucasus Mathematical Olympiad, 1
Determine if there exists a finite set $A$ of positive integers satisfying the following condition: for each $a\in{A}$ at least one of two numbers $2a$ and
$\frac{a}{3}$ belongs to $A$.
2009 IMO Shortlist, 6
Let $k$ be a positive integer. Show that if there exists a sequence $a_0,a_1,\ldots$ of integers satisfying the condition \[a_n=\frac{a_{n-1}+n^k}{n}\text{ for all } n\geq 1,\] then $k-2$ is divisible by $3$.
[i]Proposed by Okan Tekman, Turkey[/i]
2017 Ukraine Team Selection Test, 12
Let $m_1,m_2,...,m_{2013} > 1$ be 2013 pairwise relatively prime positive integers and $A_1,A_2,...,A_{2013}$ be 2013 (possibly empty) sets with $A_i\subseteq \{1,2,...,m_i-1\}$ for $i=1,2,...,2013$. Prove that there is a positive integer $N$ such that
\[ N \le \left( 2\left\lvert A_1 \right\rvert + 1 \right)\left( 2\left\lvert A_2 \right\rvert + 1 \right)\cdots\left( 2\left\lvert A_{2013} \right\rvert + 1 \right) \]
and for each $i = 1, 2, ..., 2013$, there does [i]not[/i] exist $a \in A_i$ such that $m_i$ divides $N-a$.
[i]Proposed by Victor Wang[/i]
2004 239 Open Mathematical Olympiad, 6
Do there exist a set $A\subset [0,1]$ such that
$(a)$ $A$ is a finite union of segments of total length $\frac{1}{2}$,
$(b)$ The symmetric difference of $A$ and $B:=A/2\cup(A/2+1/2)$ is a union of segments of the total length less than $\frac{1}{10000}$?
2019 Polish Junior MO First Round, 7
A cube $ABCDA'B'C'D'$ is given with an edge of length $2$ and vertices marked as in the figure. The point $K$ is center of the edge $AB$. The plane containing the points $B',D', K$ intersects the edge $AD$ at point $L$. Calculate the volume of the pyramid with apex $A$ and base the quadrilateral $D'B'KL$.
[img]https://cdn.artofproblemsolving.com/attachments/7/9/721989193ffd830fd7ad43bdde7e177c942c76.png[/img]
2004 Tournament Of Towns, 4
We have a circle and a line which does not intersect the circle. Using only compass and straightedge, construct a square whose two adjacent vertices are on the circle, and two other vertices are on the given line (it is known that such a square exists).
2004 IMO Shortlist, 5
Let $A_1A_2A_3\ldots A_n$ be a regular $n$-gon. Let $B_1$ and $B_{n-1}$ be the midpoints of its sides $A_1A_2$ and $A_{n-1}A_n$. Also, for every $i\in\left\{2,3,4,\ldots ,n-2\right\}$. Let $S$ be the point of intersection of the lines $A_1A_{i+1}$ and $A_nA_i$, and let $B_i$ be the point of intersection of the angle bisector bisector of the angle $\measuredangle A_iSA_{i+1}$ with the segment $A_iA_{i+1}$.
Prove that $\sum_{i=1}^{n-1} \measuredangle A_1B_iA_n=180^{\circ}$.
[i]Proposed by Dusan Dukic, Serbia and Montenegro[/i]
2013 National Olympiad First Round, 32
How many $10$-digit positive integers containing only the numbers $1,2,3$ can be written such that the first and the last digits are same, and no two consecutive digits are same?
$
\textbf{(A)}\ 768
\qquad\textbf{(B)}\ 642
\qquad\textbf{(C)}\ 564
\qquad\textbf{(D)}\ 510
\qquad\textbf{(E)}\ 456
$
2017 AMC 8, 24
Mrs. Sanders has three grandchildren, who call her regularly. One calls her every three days, one calls her every four days, and one calls her every five days. All three called her on December 31, 2016. On how many days during the next year did she not receive a phone call from any of her grandchildren?
$\textbf{(A) }78\qquad\textbf{(B) }80\qquad\textbf{(C) }144\qquad\textbf{(D) }146\qquad\textbf{(E) }152$
1951 AMC 12/AHSME, 45
If you are given $ \log 8 \approx .9031$ and $ \log 9 \approx .9542$, then the only logarithm that cannot be found without the use of tables is:
$ \textbf{(A)}\ \log 17 \qquad\textbf{(B)}\ \log \frac {5}{4} \qquad\textbf{(C)}\ \log 15 \qquad\textbf{(D)}\ \log 600 \qquad\textbf{(E)}\ \log .4$
2016 PUMaC Number Theory B, 8
Compute the number of positive integers $n$ between $2017$ and $2017^2$ such that $n^n \equiv 1$ (mod $2017$). ($2017$ is prime.)
1986 All Soviet Union Mathematical Olympiad, 428
A line is drawn through the $A$ vertex of triangle $ABC$ with $|AB|\ne|AC|$. Prove that the line can not contain more than one point $M$ such, that $M$ is not a triangle vertex, and $\angle ABM = \angle ACM$. What lines do not contain such a point $M$ at all?
2008 Alexandru Myller, 2
Let $ A,B,S $ be three $ 3\times 3 $ complex matrices with $ B=S^{-1}AS , $ and $ S $ nonsingular. Show:
$$ \text{tr} \left( B^2\right) +2\text{tr}(C(B)) = \left(\text{tr} (A)\right)^2 , $$
where $ C(B) $ is the cofactor of $ B. $
[i]Mihai Haivas[/i]
1999 Korea Junior Math Olympiad, 7
$A_0B, A_0C$ rays that satisfy $\angle BA_0C=14^{\circ}$. You are to place points $A_1, A_2, ...$ by the following rules.
[b]Rules[/b]
(1) On the first move, place $A_1$ on any point on $A_0B$(except $A_0$).
(2) On the $n>1$th move, place $A_n$ on $A_0B$ iff $A_{n-1}$ is on $A_0C$, and place $A_n$ on $A_0C$ iff $A_{n-1}$ is one $A_0B$. $A_n$ must be place on the point that satisfies $A_{n-2}A_n{n-1}=A_{n-1}A_n$.
All the points must be placed in different locations. What is the maximum number of points that can be placed?
2017 Poland - Second Round, 2
In an acute triangle $ABC$ the bisector of $\angle BAC$ crosses $BC$ at $D$. Points $P$ and $Q$ are orthogonal projections of $D$ on lines $AB$ and $AC$. Prove that $[APQ]=[BCQP]$ if and only if the circumcenter of $ABC$ lies on $PQ$.
1996 Moscow Mathematical Olympiad, 4
Consider an equilateral triangle $\triangle ABC$. The points $K$ and $L$ divide the leg $BC$ into three equal parts, the point $M$ divides the leg $AC$ in the ratio $1:2$, counting from the vertex $A$. Prove that $\angle AKM+\angle ALM=30^{\circ}$.
Proposed by V. Proizvolov