Found problems: 85335
1998 National High School Mathematics League, 3
For positive integers $a,n$, define $F_n(a)=q+r$, where $a=qn+r$ ($q,r$ are nonnegative integers, $0\leq q<n$). Find the largest integer $A$, there are positive integers $n_1,n_2,n_3,n_4,n_5,n_6$, for all positive integer $a\leq A$, $F_{n_6}(F_{n_5}(F_{n_4}(F_{n_3}(F_{n_2}(F_{n_1}(a))))))=1$.
2001 China Team Selection Test, 3
For a positive integer \( n \geq 6 \), find the smallest integer \( S(n) \) such that any graph with \( n \) vertices and at least \( S(n) \) edges must contain at least two disjoint cycles (cycles with no common vertices).
2010 Germany Team Selection Test, 1
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]
2007 Junior Balkan MO, 4
Prove that if $ p$ is a prime number, then $ 7p+3^{p}-4$ is not a perfect square.
2015 Saudi Arabia JBMO TST, 2
Let $A$ and $B$ be the number of odd positive integers $n<1000$ for which the number formed by the last three digits of $n^{2015}$ is greater and smaller than $n$, respectively. Prove that $A=B$.
2017 AIME Problems, 12
Call a set $S$ [i]product-free[/i] if there do not exist $a, b, c \in S$ (not necessarily distinct) such that $a b = c$. For example, the empty set and the set $\{16, 20\}$ are product-free, whereas the sets $\{4, 16\}$ and $\{2, 8, 16\}$ are not product-free. Find the number of product-free subsets of the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$.
2020 Yasinsky Geometry Olympiad, 3
Point $M$ is the midpoint of the side $CD$ of the trapezoid $ABCD$, point $K$ is the foot of the perpendicular drawn from point $M$ to the side $AB$. Give that $3BK \le AK$. Prove that $BC + AD\ge 2BM$.
2018 Dutch IMO TST, 1
A set of lines in the plan is called [i]nice [/i]i f every line in the set intersects an odd number of other lines in the set.
Determine the smallest integer $k \ge 0$ having the following property:
for each $2018$ distinct lines $\ell_1, \ell_2, ..., \ell_{2018}$ in the plane, there exist lines $\ell_{2018+1},\ell_{2018+2}, . . . , \ell_{2018+k}$ such that the lines $\ell_1, \ell_2, ..., \ell_{2018+k}$ are distinct and form a [i]nice [/i] set.
2008 iTest Tournament of Champions, 1
Let \[X = \cos\frac{2\pi}7 + \cos\frac{4\pi}7 + \cos\frac{6\pi}7 + \cdots + \cos\frac{2006\pi}7 + \cos\frac{2008\pi}7.\] Compute $\Big|\lfloor 2008 X\rfloor\Big|$.
1990 IMO Longlists, 28
Let $ABC$ be an arbitrary acute triangle. Circle $\Gamma$ satisfies the following conditions:
(i) Circle $\Gamma$ intersects all three sides of triangle $ABC.$
(ii) In the convex hexagon formed by above six intersections, the three pairs of opposite sides are parallel respectively. (The hexagon maybe degenerate, that is, two or more vertices are coincide. In this case, "opposite sides are parallel" is defined through limit opinion.)
Find the locus of the center of circle $\Gamma$, and explain how to construct the locus.
1977 IMO Longlists, 9
Let $ABCD$ be a regular tetrahedron and $\mathbf{Z}$ an isometry mapping $A,B,C,D$ into $B,C,D,A$, respectively. Find the set $M$ of all points $X$ of the face $ABC$ whose distance from $\mathbf{Z}(X)$ is equal to a given number $t$. Find necessary and sufficient conditions for the set $M$ to be nonempty.
2003 Putnam, 1
Do there exist polynomials $a(x)$, $b(x)$, $c(y)$, $d(y)$ such that \[1 + xy + x^2y^2= a(x)c(y) + b(x)d(y)\] holds identically?
2024 Ukraine National Mathematical Olympiad, Problem 7
You are given $2024$ yellow and $2024$ blue points on the plane, and no three of the points are on the same line. We call a pair of nonnegative integers $(a, b)$ [i]good[/i] if there exists a half-plane with exactly $a$ yellow and $b$ blue points. Find the smallest possible number of good pairs. The points that lie on the line that is the boundary of the half-plane are considered to be outside the half-plane.
[i]Proposed by Anton Trygub[/i]
2022 AMC 12/AHSME, 8
The infinite product
$$\sqrt[3]{10}\cdot\sqrt[3]{\sqrt[3]{10}}\cdot\sqrt[3]{\sqrt[3]{\sqrt[3]{10}}}\dots$$
evaluates to a real number. What is that number?
$\textbf{(A) }\sqrt{10}\qquad\textbf{(B) }\sqrt[3]{100}\qquad\textbf{(C) }\sqrt[4]{1000}\qquad\textbf{(D) }10\qquad\textbf{(E) }10\sqrt[3]{10}$
2018 Baltic Way, 17
Prove that for any positive integers $p,q$ such that $\sqrt{11}>\frac{p}{q}$, the following inequality holds:
\[\sqrt{11}-\frac{p}{q}>\frac{1}{2pq}.\]
1999 Poland - Second Round, 2
A cube of edge $2$ with one of the corner unit cubes removed is called a [i]piece[/i].
Prove that if a cube $T$ of edge $2^n$ is divided into $2^{3n}$ unit cubes and one of the unit cubes is removed, then the rest can be cut into [i]pieces[/i].
1993 Korea - Final Round, 3
Find the smallest $x \in\mathbb{N}$ for which $\frac{7x^{25}-10}{83}$ is an integer.
2024 Mexican Girls' Contest, 2
There are 50 slips of paper numbered from 1 to 50. It is desired to pick 3 slips such that for any of the three numbers, divided by the greatest common divisor of the other two, the square root of the result is a rational number.
How many unordered triples of slips satisfy this condition?
1971 Polish MO Finals, 5
Find the largest integer $A$ such that, for any permutation of the natural numbers not exceeding $100$, the sum of some ten successive numbers is at least $A$.
2011 Macedonia National Olympiad, 4
Find all functions $~$ $f: \mathbb{R} \to \mathbb{R}$ $~$ which satisfy the equation
\[ f(x+yf(x))\, =\, f(f(x)) + xf(y)\, . \]
2008 Sharygin Geometry Olympiad, 2
(A.Myakishev) Let triangle $ A_1B_1C_1$ be symmetric to $ ABC$ wrt the incenter of its medial triangle. Prove that the orthocenter of $ A_1B_1C_1$ coincides with the circumcenter of the triangle formed by the excenters of $ ABC$.
2013 IFYM, Sozopol, 5
Find all positive integers $n$ satisfying $2n+7 \mid n! -1$.
2006 India IMO Training Camp, 3
Let $A_1,A_2,\cdots , A_n$ be arithmetic progressions of integers, each of $k$ terms, such that any two of these arithmetic progressions have at least two common elements. Suppose $b$ of these arithmetic progressions have common difference $d_1$ and the remaining arithmetic progressions have common difference $d_2$ where $0<b<n$. Prove that
\[b \le 2\left(k-\frac{d_2}{gcd(d_1,d_2)}\right)-1.\]
2006 Miklós Schweitzer, 2
Let T be a finite tree graph that has more than one vertex. Let s be the largest number of vertices of a subtree $X \subset T$ for which every vertex of X has a neighbor other than X. Let t be the smallest positive integer for which each edge of T is contained in exactly t stars, and each vertex of T is contained in at most 2t - 1 stars. (That is, the stars can be represented by multiplicity.) Prove that s = t.
Note: a star of T is a vertex with degree $\geq$ 3 , including its neighouring edges and vertices.
2004 National Olympiad First Round, 36
If the function $f$ satisfies the equation $f(x) + f\left ( \dfrac{1}{\sqrt[3]{1-x^3}}\right ) = x^3$ for every real $x \neq 1$, what is $f(-1)$?
$
\textbf{(A)}\ -1
\qquad\textbf{(B)}\ \dfrac 14
\qquad\textbf{(C)}\ \dfrac 12
\qquad\textbf{(D)}\ \dfrac 74
\qquad\textbf{(E)}\ \text{None of above}
$