Found problems: 85335
2004 Bulgaria Team Selection Test, 2
Find all primes $p \ge 3$ such that $p- \lfloor p/q \rfloor q$ is a square-free integer for any prime $q<p$.
1980 Tournament Of Towns, (003) 3
If permutations of the numbers $2, 3,4,..., 102$ are denoted by $a_i,a_2, a_3,...,a_{101}$, find all such permutations in which $a_k$ is divisible by $k$ for all $k$.
2008 Polish MO Finals, 2
A function $ f: R^3\rightarrow R$ for all reals $ a,b,c,d,e$ satisfies a condition:
\[ f(a,b,c)\plus{}f(b,c,d)\plus{}f(c,d,e)\plus{}f(d,e,a)\plus{}f(e,a,b)\equal{}a\plus{}b\plus{}c\plus{}d\plus{}e\]
Show that for all reals $ x_1,x_2,\ldots,x_n$ ($ n\geq 5$) equality holds:
\[ f(x_1,x_2,x_3)\plus{}f(x_2,x_3,x_4)\plus{}\ldots \plus{}f(x_{n\minus{}1},x_n,x_1)\plus{}f(x_n,x_1,x_2)\equal{}x_1\plus{}x_2\plus{}\ldots\plus{}x_n\]
2004 National High School Mathematics League, 3
For integer $n\geq4$, find the smallest integer $f(n)$, such that for any positive integer $m$, in any subset with $f(n)$ elements of the set $\{m, m+1, \cdots, m+n-1\}$ there are at least three elements that are relatively prime .
2019 Mathematical Talent Reward Programme, SAQ: P 6
Consider a finite set of points, $\Phi$, in the $\mathbb{R}^2$, such that they follow the following properties :
[list]
[*] $\Phi$ doesn't contain the origin $\{(0,0)\}$ and not all points are collinear.
[*] If $\alpha \in \Phi$, then $-\alpha \in \Phi$, $c\alpha \notin \Phi $ for $c\neq 1$ or $-1$
[*] If $\alpha, \ \beta$ are in $\Phi$, then the reflection of $\beta$ in the line passing through the origin and perpendicular to the line containing origin and $\alpha$ is in $\Phi$
[*] If $\alpha = (a,b) , \ \beta = (c,d)$, (both $\alpha, \ \beta \in \Phi$) then $\frac{2(ac+bd)}{c^2+d^2} \in \mathbb{Z}$
[/list]
Prove that there cannot be 5 collinear points in $\Phi$
2017 China National Olympiad, 4
Let $n \geq 2$ be a natural number. For any two permutations of $(1,2,\cdots,n)$, say $\alpha = (a_1,a_2,\cdots,a_n)$ and $\beta = (b_1,b_2,\cdots,b_n),$ if there exists a natural number $k \leq n$ such that
$$b_i = \begin{cases} a_{k+1-i}, & \text{ }1 \leq i \leq k; \\ a_i, & \text{} k < i \leq n, \end{cases}$$
we call $\alpha$ a friendly permutation of $\beta$.
Prove that it is possible to enumerate all possible permutations of $(1,2,\cdots,n)$ as $P_1,P_2,\cdots,P_m$ such that for all $i = 1,2,\cdots,m$, $P_{i+1}$ is a friendly permutation of $P_i$ where $m = n!$ and $P_{m+1} = P_1$.
1986 AMC 12/AHSME, 23
Let \[N = 69^{5} + 5\cdot 69^{4} + 10\cdot 69^{3} + 10\cdot 69^{2} + 5\cdot 69 + 1.\] How many positive integers are factors of $N$?
$ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 69\qquad\textbf{(D)}\ 125\qquad\textbf{(E)}\ 216 $
2015 Tournament of Towns, 7
Santa Clause had $n$ sorts of candies, $k$ candies of each sort. He distributed them at random between $k$ gift bags, $n$ candies per a bag and gave a bag to everyone of $k$ children at Christmas party. The children learned what they had in their bags and decided to trade. Two children trade one candy for one candy in case if each of them gets the candy of the sort which was absent in his/her bag. Prove that they can organize a sequence of trades so that finally every child would have candies of each sort.
2007 Stanford Mathematics Tournament, 3
Let $a, b, c$ be the roots of $x^3-7x^2-6x+5=0$. Compute $(a+b)(a+c)(b+c)$.
2009 Ukraine National Mathematical Olympiad, 2
Let $M = \{1, 2, 3, 4, 6, 8,12,16, 24, 48\} .$ Find out which of four-element subsets of $M$ are more: those with product of all elements greater than $2009$ or those with product of all elements less than $2009.$
2018 Canadian Open Math Challenge, A3
Source: 2018 Canadian Open Math Challenge Part A Problem 3
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Points $(0,0)$ and $(3\sqrt7,7\sqrt3)$ are the endpoints of a diameter of circle $\Gamma.$ Determine the other $x$ intercept of $\Gamma.$
2018 CHKMO, 2
Suppose $ABCD$ is a cyclic quadrilateral. Extend $DA$ and $DC$ to $P$ and $Q$ respectively such that $AP=BC$ and $CQ=AB$. Let $M$ be the midpoint of $PQ$. Show that $MA\perp MC$.
2017 BMT Spring, 4
How many lattice points $(v, w, x, y, z)$ does a $5$-sphere centered on the origin, with radius $3$, contain on its surface or in its interior?
2022 Assam Mathematical Olympiad, 17
Consider a rectangular grid of points consisting of $4$ rows and $84$ columns. Each point is coloured with one of the colours red, blue or green. Show that no matter whatever way the colouring is done, there always exist four points
of the same colour that form the vertices of a rectangle. An illustration is shown in the figure below.
2014 Dutch BxMO/EGMO TST, 1
Find all non-negative integer numbers $n$ for which there exists integers $a$ and $b$ such that $n^2=a+b$ and $n^3=a^2+b^2.$
2016 ASDAN Math Tournament, 8
Consider all fractions $\tfrac{a}{b}$ where $1\leq b\leq100$ and $0\leq a\leq b$. Of these fractions, let $\tfrac{m}{n}$ be the smallest fraction such that $\tfrac{m}{n}>\tfrac{2}{7}$. What is $\tfrac{m}{n}$?
2000 Moldova National Olympiad, Problem 1
Let $a,b,c$ be real numbers with $a,c\ne0$. Prove that if $r$ is a real root of $ax^2+bx+c=0$ and $s$ a real root of $-ax^2+bx+c=0$, then there is a root of a
$\frac a2x^2+bx+c=0$ between $r$ and $s$.
1986 Iran MO (2nd round), 4
Find all positive integers $n$ for which the number $1!+2!+3!+\cdots+n!$ is a perfect power of an integer.
2014 Contests, 2
Let $k\ge 2$, $n\ge 1$, $a_1, a_2,\dots, a_k$ and $b_1, b_2, \dots, b_n$ be integers such that $1<a_1<a_2<\dots <a_k<b_1<b_2<\dots <b_n$. Prove that if $a_1+a_2+\dots +a_k>b_1+b_2+\dots + b_n$, then $a_1\cdot a_2\cdot \ldots \cdot a_k>b_1\cdot b_2 \cdot \ldots \cdot b_n$.
Estonia Open Junior - geometry, 2012.2.5
Is it possible that the perimeter of a triangle whose side lengths are integers, is divisible by the double of the longest side length?
2014 Peru IMO TST, 1
a) Find at least two functions $f: \mathbb{R}^+ \rightarrow \mathbb{R}^+$ such that $$\displaystyle{2f(x^2)\geq xf(x) + x,}$$ for all $x \in \mathbb{R}^+.$
b) Let $f: \mathbb{R}^+ \rightarrow \mathbb{R}^+$ be a function such that $$\displaystyle{2f(x^2)\geq xf(x) + x,}$$ for all $x \in \mathbb{R}^+.$ Show that $ f(x^3)\geq x^2,$ for all $x \in \mathbb{R}^+.$
Can we find the best constant $a\in \Bbb{R}$ such that $f(x)\geq x^a,$ for all $x \in \mathbb{R}^+?$
2015 ASDAN Math Tournament, 4
In trapezoid $ABCD$ with $AD\parallel BC$, $AB=6$, $AD=9$, and $BD=12$. If $\angle ABD=\angle DCB$, find the perimeter of the trapezoid.
MathLinks Contest 5th, 5.2
Prove or disprove the existence of a function $f : S \to R$ such that for all $x \ne y \in S$ we have $|f(x) - f(y)| \ge \frac{1}{x^2 + y^2}$, in each of the cases:
a) $S = R$
b) $S = Q$.
1956 Putnam, B6
Given $T_1 =2, T_{n+1}= T_{n}^{2} -T_n +1$ for $n>0.$ Prove:
(i) If $m \ne n,$ $T_m$ and $T_n$ have no common factor greater than $1.$
(ii) $\sum_{i=1}^{\infty} \frac{1}{T_i }=1.$
2012 AMC 8, 16
Each of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 is used only once to make two five-digit numbers so that they have the largest possible sum. Which of the following could be one of the numbers?
$\textbf{(A)}\hspace{.05in}76531 \qquad \textbf{(B)}\hspace{.05in}86724 \qquad \textbf{(C)}\hspace{.05in}87431 \qquad \textbf{(D)}\hspace{.05in}96240 \qquad \textbf{(E)}\hspace{.05in}97403 $