Found problems: 85335
2012 IMC, 3
Given an integer $n>1$, let $S_n$ be the group of permutations of the numbers $1,\;2,\;3,\;\ldots,\;n$. Two players, A and B, play the following game. Taking turns, they select elements (one element at a time) from the group $S_n$. It is forbidden to select an element that has already been selected. The game ends when the selected elements generate the whole group $S_n$. The player who made the last move loses the game. The first move is made by A. Which player has a winning strategy?
[i]Proposed by Fedor Petrov, St. Petersburg State University.[/i]
2021 Durer Math Competition Finals, 14
How many functions $f : \{1, 2, . . . , 16\} \to \{1, 2, . . . , 16\}$ have the property that $f(f(x))-4x$ is divisible by $17$ for all integers $1 \le x \le 16$?
2007 Tournament Of Towns, 4
Several diagonals (possibly intersecting each other) are drawn in a convex $n$-gon in such a way that no three diagonals intersect in one point. If the $n$-gon is cut into triangles, what is the maximum possible number of these triangles?
2021 BMT, Tie 2
Let $\vartriangle A_0B_0C_0$ be an equilateral triangle with area $1$, and let $A_1$, $B_1$, $C_1$ be the midpoints of $\overline{A_0B_0}$, $\overline{B_0C_0}$, and $\overline{C_0A_0}$, respectively. Furthermore, set $A_2$, $B_2$, $C_2$ as the midpoints of segments $\overline{A_0A_1}$, $\overline{B_0B_1}$, and $\overline{C_0C_1}$ respectively. For $n \ge 1$, $A_{2n+1}$ is recursively defined as the midpoint of $A_{2n}A_{2n-1}$, and $A_{2n+2}$ is recursively defined as the midpoint of $\overline{A_{2n+1}A_{2n-1}}$. Recursively define $B_n$ and $C_n$ the same way. Compute the value of $\lim_{n \to \infty }[A_nB_nC_n]$, where $[A_nB_nC_n]$ denotes the area of triangle $\vartriangle A_nB_nC_n$.
1988 Romania Team Selection Test, 12
The four vertices of a square are the centers of four circles such that the sum of theirs areas equals the square's area. Take an arbitrary point in the interior of each circle. Prove that the four arbitrary points are the vertices of a convex quadrilateral.
[i]Laurentiu Panaitopol[/i]
Cono Sur Shortlist - geometry, 2003.G1
Let $O$ be the circumcenter of the isosceles triangle $ABC$ ($AB = AC$). Let $P$ be a point of the segment $AO$ and $Q$ the symmetric of $P$ with respect to the midpoint of $AB$. If $OQ$ cuts $AB$ at $K$ and the circle that passes through $A, K$ and $O$ cuts $AC$ in $L$, show that $\angle ALP = \angle CLO$.
2004 USAMTS Problems, 4
The interior angles of a convex polygon form an arithmetic progression with a common difference of $4^\circ$. Determine the number of sides of the polygon if its largest interior angle is $172^\circ.$
1967 IMO Shortlist, 6
Given a segment $AB$ of the length 1, define the set $M$ of points in the
following way: it contains two points $A,B,$ and also all points obtained from $A,B$ by iterating the following rule: With every pair of points $X,Y$ the set $M$ contains also the point $Z$ of the segment $XY$ for which $YZ = 3XZ.$
2004 Purple Comet Problems, 20
A $70$ foot pole stands vertically in a plane supported by three $490$ foot wires, all attached to the top of the pole, pulled taut, and anchored to three equally spaced points in the plane. How many feet apart are any two of those anchor points?
STEMS 2021 Math Cat B, Q4
Let $n$ be a fixed positive integer.
- Show that there exist real polynomials $p_1, p_2, p_3, \cdots, p_k \in \mathbb{R}[x_1, \cdots, x_n]$ such that
\[(x_1 + x_2 + \cdots + x_n)^2 + p_1(x_1, \cdots, x_n)^2 + p_2(x_1, \cdots, x_n)^2 + \cdots + p_k(x_1, \cdots, x_n)^2 = n(x_1^2 + x_2^2 + \cdots + x_n^2)\]
- Find the least natural number $k$, depending on $n$, such that the above polynomials $p_1, p_2, \cdots, p_k$ exist.
1970 Putnam, A4
Given a sequence $(x_n )$ such that $\lim_{n\to \infty} x_n - x_{n-2}=0,$ prove that
$$\lim_{n\to \infty} \frac{x_n -x_{n-1}}{n}=0.$$
PEN M Problems, 25
Let $\{a_{n}\}_{n \ge 1}$ be a sequence of positive integers such that \[0 < a_{n+1}-a_{n}\le 2001 \;\; \text{for all}\;\; n \in \mathbb{N}.\] Show that there are infinitely many pairs $(p, q)$ of positive integers such that $p>q$ and $a_{q}\; \vert \; a_{p}$.
1997 All-Russian Olympiad Regional Round, 10.7
Points $O_1$ and $O_2$ are the centers of the circumscribed and inscribed circles of an isosceles triangle $ABC$ ($AB = BC$). The circumcircles of triangles $ABC$ and $O_1O_2A$ intersect at points $A$ and $D$. Prove that line $BD$ is tangent to the circumcircle of the triangle $O_1O_2A$.
2009 India IMO Training Camp, 10
For a certain triangle all of its altitudes are integers whose sum is less than 20. If its Inradius is also an integer Find all possible values of area of the triangle.
2017 CMIMC Number Theory, 6
Find the largest positive integer $N$ satisfying the following properties:
[list]
[*]$N$ is divisible by $7$;
[*]Swapping the $i^{\text{th}}$ and $j^{\text{th}}$ digits of $N$ (for any $i$ and $j$ with $i\neq j$) gives an integer which is $\textit{not}$ divisible by $7$.
[/list]
2023 CCA Math Bonanza, T8
What is the smallest positive integer (in base 10) that has a digit sum of 23 in base 20, and a digit sum of 20 in base 23? (The digit sums are in base 10.)
[i]Team #8[/i]
2006 JBMO ShortLists, 5
Determine all pairs $ (m,n)$ of natural numbers for which $ m^2\equal{}nk\plus{}2$ where $ k\equal{}\overline{n1}$.
EDIT. [color=#FF0000]It has been discovered the correct statement is with $ k\equal{}\overline{1n}$.[/color]
1990 IMO Shortlist, 28
Prove that on the coordinate plane it is impossible to draw a closed broken line such that
[i](i)[/i] the coordinates of each vertex are rational;
[i](ii)[/i] the length each of its edges is 1;
[i](iii)[/i] the line has an odd number of vertices.
2005 Kyiv Mathematical Festival, 2
Find the rightmost nonzero digit of $ \frac{100!}{5^{20}}$ (here $ n!\equal{}1\cdot2\cdot3\cdot\ldots\cdot
n$).
2014 USA Team Selection Test, 3
Let $n$ be an even positive integer, and let $G$ be an $n$-vertex graph with exactly $\tfrac{n^2}{4}$ edges, where there are no loops or multiple edges (each unordered pair of distinct vertices is joined by either 0 or 1 edge). An unordered pair of distinct vertices $\{x,y\}$ is said to be [i]amicable[/i] if they have a common neighbor (there is a vertex $z$ such that $xz$ and $yz$ are both edges). Prove that $G$ has at least $2\textstyle\binom{n/2}{2}$ pairs of vertices which are amicable.
[i]Zoltán Füredi (suggested by Po-Shen Loh)[/i]
2024 LMT Fall, 18
Find the number of ways to split the numbers from $1$ to $12$ into $4$ non-intersecting sets of size $3$ such that each set has sum divisible by $3$.
1972 Czech and Slovak Olympiad III A, 6
Two different points $A,S$ are given in the plane. Furthermore, positive numbers $d,\omega$ are given, $\omega<180^\circ.$ Let $X$ be a point and $X'$ its image under the rotation by the angle $\omega$ (in counter-clockwise direction) with respect to the origin $S.$ Construct all points $X$ such that $XX'=d$ and $A$ is a point of the segment $XX'.$ Discuss conditions of solvability (in terms of $d,\omega,SA$).
1978 Austrian-Polish Competition, 4
Let $c\neq 1$ be a positive rational number. Show that it is possible to partition $\mathbb{N}$, the set of positive integers, into two disjoint nonempty subsets $A,B$ so that $x/y\neq c$ holds whenever $x$ and $y$ lie both in $A$ or both in $B$.
2000 Balkan MO, 2
Let $ABC$ be an acute-angled triangle and $D$ the midpoint of $BC$. Let $E$ be a point on segment $AD$ and $M$ its projection on $BC$. If $N$ and $P$ are the projections of $M$ on $AB$ and $AC$ then the interior angule bisectors of $\angle NMP$ and $\angle NEP$ are parallel.
2004 Iran Team Selection Test, 2
Suppose that $ p$ is a prime number. Prove that the equation $ x^2\minus{}py^2\equal{}\minus{}1$ has a solution if and only if $ p\equiv1\pmod 4$.