Found problems: 85335
PEN N Problems, 10
Let $a,b$ be integers greater than 2. Prove that there exists a positive integer $k$ and a finite sequence $n_1, n_2, \dots, n_k$ of positive integers such that $n_1 = a$, $n_k = b$, and $n_i n_{i+1}$ is divisible by $n_i + n_{i+1}$ for each $i$ ($1 \leq i < k$).
1993 Turkey Team Selection Test, 3
Let ($b_n$) be a sequence such that $b_n \geq 0 $ and $b_{n+1}^2 \geq \frac{b_1^2}{1^3}+\cdots+\frac{b_n^2}{n^3}$ for all $n \geq 1$. Prove that there exists a natural number $K$ such that
\[\sum_{n=1}^{K} \frac{b_{n+1}}{b_1+b_2+ \cdots + b_n} \geq \frac{1993}{1000}\]
1986 India National Olympiad, 5
If $ P(x)$ is a polynomial with integer coefficients and $ a$, $ b$, $ c$, three distinct integers, then show that it is impossible to have $ P(a)\equal{}b$, $ P(b)\equal{}c$, $ P(c)\equal{}a$.
2013 Online Math Open Problems, 30
Pairwise distinct points $P_1,P_2,\ldots, P_{16}$ lie on the perimeter of a square with side length $4$ centered at $O$ such that $\lvert P_iP_{i+1} \rvert = 1$ for $i=1,2,\ldots, 16$. (We take $P_{17}$ to be the point $P_1$.) We construct points $Q_1,Q_2,\ldots,Q_{16}$ as follows: for each $i$, a fair coin is flipped. If it lands heads, we define $Q_i$ to be $P_i$; otherwise, we define $Q_i$ to be the reflection of $P_i$ over $O$. (So, it is possible for some of the $Q_i$ to coincide.) Let $D$ be the length of the vector $\overrightarrow{OQ_1} + \overrightarrow{OQ_2} + \cdots + \overrightarrow{OQ_{16}}$. Compute the expected value of $D^2$.
[i]Ray Li[/i]
2015 Czech and Slovak Olympiad III A, 5
In given triangle $\triangle ABC$, difference between sizes of each pair of sides is at least $d>0$. Let $G$ and $I$ be the centroid and incenter of $\triangle ABC$ and $r$ be its inradius. Show that $$[AIG]+[BIG]+[CIG]\ge\frac{2}{3}dr,$$ where $[XYZ]$ is (nonnegative) area of triangle $\triangle XYZ$.
2010 ISI B.Math Entrance Exam, 9
Let $f(x)$ be a polynomial with integer co-efficients. Assume that $3$ divides the value $f(n)$ for each integer $n$. Prove that when $f(x)$ is divided by $x^3-x$ , the remainder is of the form $3r(x)$ where $r(x)$ is a polynomial with integer coefficients.
2013 Harvard-MIT Mathematics Tournament, 6
Find the number of integers $n$ such that \[1+\left\lfloor\dfrac{100n}{101}\right\rfloor=\left\lceil\dfrac{99n}{100}\right\rceil.\]
2021 Pan-African, 1
Let $n$ be an integer greater than $3$. A square of side length $n$ is divided by lines parallel to each side into $n^2$ squares of length $1$. Find the number of convex trapezoids which have vertices among the vertices of the $n^2$ squares of side length $1$, have side lengths less than or equal $3$ and have area equal to $2$
Note: Parallelograms are trapezoids.
2013 Regional Competition For Advanced Students, 1
For which integers between $2000$ and $2010$ (including) is the probability that a random divisor is smaller or equal $45$ the largest?
2013 CHMMC (Fall), 6
Let $a_1 < a_2 < a_3 < ... < a_n < ...$ be positive integers such that, for $n = 1, 2, 3, ...,$ $$a_{2n} = a_n + n.$$
Given that if $a_n$ is prime, then $n$ is also, find $a_{2014}$.
2020 Harvard-MIT Mathematics Tournament, 5
Let $S$ be a set of intervals defined recursively as follows:
[list][*] Initially, $[1,1000]$ is the only interval in $S$.
[*]If $l\neq r$ and $[l,r]\in S$, then both $\left[l,\left\lfloor \frac{l+r}{2}\right\rfloor\right], \left[\left\lfloor \frac{l+r}{2}\right\rfloor+1,r\right]\in S$. [/list]
(Note that $S$ can contain intervals such as $[1, 1]$, which contain a single integer.) An integer $i$ is chosen uniformly at random from the range $[1,1000]$. What is the expected number of intervals in $S$ which contain $i$?
[i]Proposed by Benjamin Qi.[/i]
2020 Brazil Team Selection Test, 4
A quadruple of integers $(a, b, c, d)$ is said good if $ad-bc=2020$. Two good quadruplets are said to be dissimilar if it is not possible to obtain one from the other using a finite number of applications of the following operations:
$$(a,b,c,d) \rightarrow (-c,-d,a,b)$$
$$(a,b,c,d) \rightarrow (a,b,c+a,d+b)$$
$$(a,b,c,d) \rightarrow (a,b,c-a,d-b)$$
Let $A$ be a set of $k$ good quadruples, two by two dissimilar. Show that $k \leq 4284$.
1980 IMO Longlists, 7
The function $f$ is defined on the set $\mathbb{Q}$ of all rational numbers and has values in $\mathbb{Q}$. It satisfies the conditions $f(1) = 2$ and $f(xy) = f(x)f(y) - f(x+y) + 1$ for all $x,y \in \mathbb{Q}$. Determine $f$.
KoMaL A Problems 2018/2019, A.748
The circles $\Omega$ and $\omega$ in its interior are fixed. The distinct points $A,B,C,D,E$ move on $\Omega$ in such a way that the line segments $AB,BC,CD,DE$ are tangents to $\omega$ .The lines $AB$ and $CD$ meet at point
$P$, the lines $BC$ and $DE$ meet at $Q$ . Let $R$ be the second intersection of the circles $BCP$and $CDQ$, other than $C$. Show that $R$ moves either on a circle or on a line.
2023 IFYM, Sozopol, 8
Do there exist a natural number $n$ and real numbers $a_0, a_1, \dots, a_n$, each equal to $1$ or $-1$, such that the polynomial $a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0$ is divisible by the polynomial $x^{2023} - 2x^{2022} + c$, where: \\
(a) $c = 1$ \\
(b) $c = -1$?
[i]
(For polynomials $P(x)$ and $Q(x)$ with real coefficients, we say that $P(x)$ is divisible by $Q(x)$ if there exists a polynomial $R(x)$ with real coefficients such that $P(x) = Q(x)R(x)$.)[/i]
2000 National Olympiad First Round, 28
$$\begin{array}{ rlrlrl}
f_1(x)=&x^2+x & f_2(x)=&2x^2-x & f_3(x)=&x^2 +x \\
g_1(x)=&x-2 & g_2(x)=&2x \ \ & g_3(x)=&x+2 \\
\end{array}$$
If $h(x)=x$ can be get from $f_i$ and $g_i$ by using only addition, substraction, multiplication defined on those functions where $i\in\{1,2,3\}$, then $F_i=1$. Otherwise, $F_i=0$. What is $(F_1,F_2,F_3)$ ?
$ \textbf{(A)}\ (0,0,0)
\qquad\textbf{(B)}\ (0,0,1)
\qquad\textbf{(C)}\ (0,1,0)
\qquad\textbf{(D)}\ (0,1,1)
\qquad\textbf{(E)}\ \text{None}
$
2003 Moldova Team Selection Test, 4
A square-table of dimensions $ n\times n$, where $ n\in N^*$, is filled arbitrarly with the numbers $ 1,2,...,n^2$ such that every number appears on the table exactly one time. From each row of the table
is chosen the least number and then denote by $ x$ the biggest number from the numbers chosen. From each column of the table is chosen the least number and then denote by $ y$ the biggest number from the numbers chosen. The table is called [i]balanced [/i]iff $ x \equal{} y$. How many balanced tables we can obtain?
2023 SG Originals, Q2
Find all positive integers $k$ such that there exists positive integers $a, b$ such that
\[a^2+4=(k^2-4)b^2.\]
2015 Puerto Rico Team Selection Test, 6
Find all positive integers $n$ such that $7^n + 147$ is a perfect square.
1998 Irish Math Olympiad, 4
Show that a disk of radius $ 2$ can be covered by seven (possibly overlapping) disks of radius $ 1$.
1982 All Soviet Union Mathematical Olympiad, 337
All the natural numbers from $1$ to $1982$ are gathered in an array in an arbitrary order in computer's memory. The program looks through all the sequent pairs (first and second, second and third,...) and exchanges numbers in the pair, if the number on the lower place is greater than another. Then the program repeats the process, but moves from another end of the array. The number, that stand initially on the $100$-th place reserved its place. Find that number.
2012 Online Math Open Problems, 14
Al told Bob that he was thinking of $2011$ distinct positive integers. He also told Bob the sum of those $2011$ distinct positive integers. From this information, Bob was able to determine all $2011$ integers. How many possible sums could Al have told Bob?
[i]Author: Ray Li[/i]
2017 Korea - Final Round, 5
Let there be cyclic quadrilateral $ABCD$ with $L$ as the midpoint of $AB$ and $M$ as the midpoint of $CD$. Let $AC \cap BD = E$, and let rays $AB$ and $DC$ meet again at $F$. Let $LM \cap DE = P$. Let $Q$ be the foot of the perpendicular from $P$ to $EM$. If the orthocenter of $\triangle FLM$ is $E$, prove the following equality.
$$\frac{EP^2}{EQ} = \frac{1}{2} \left( \frac{BD^2}{DF} - \frac{BC^2}{CF} \right)$$
2023 Moldova EGMO TST, 9
Solve the equation $$\left[\frac{x^2+1}{x}\right]-\left[\frac{x}{x^2+1}\right]=3.$$
2011 Iran Team Selection Test, 4
Define a finite set $A$ to be 'good' if it satisfies the following conditions:
[list][*][b](a)[/b] For every three disjoint element of $A,$ like $a,b,c$ we have $\gcd(a,b,c)=1;$
[*][b](b)[/b] For every two distinct $b,c\in A,$ there exists an $a\in A,$ distinct from $b,c$ such that $bc$ is divisible by $a.$[/list]
Find all good sets.