Found problems: 85335
2020 Simon Marais Mathematics Competition, B4
[i]The following problem is open in the sense that no solution is currently known to part (b).[/i]
Let $n\geq 2$ be an integer, and $P_n$ be a regular polygon with $n^2-n+1$ vertices.
We say that $n$ is $\emph{taut}$ if it is possible to choose $n$ of the vertices of $P_n$ such that the pairwise distances between the chosen vertices are all distinct.
(a) show that if $n-1$ is prime then $n$ is taut.
(b) Which integers $n\geq 2$ are taut?
2002 HKIMO Preliminary Selection Contest, 2
A clock has an hour hand of length 3 and a minute hand of length 4. From 1:00 am to 1:00 pm of the same day, find the number of occurrences when the distance between the tips of the two hands is an integer.
PEN H Problems, 9
Determine all integers $a$ for which the equation \[x^{2}+axy+y^{2}=1\] has infinitely many distinct integer solutions $x, \;y$.
2019 Peru EGMO TST, 3
For a finite set $A$ of integers, define $s(A)$ as the number of values obtained by adding any two elements of $A$, not necessarily different. Analogously, define $r (A)$ as the number of values obtained by subtracting any two elements of $A$, not necessarily different.
For example, if $A = \{3,1,-1\}$
$\bullet$ The values obtained by adding any two elements of $A$ are $\{6,4,2,0,-2\}$ and so $s (A) = 5$.
$\bullet$ The values obtained by subtracting any two elements of $A$ are $\{4,2,0,-2,-4\}$ and as $r (A) = 5$.
Prove that for each positive integer $n$ there is a finite set $A$ of integers such that $r (A) \ge n s (A)$.
MOAA Team Rounds, 2019.10
Let $S$ be the set of all four digit palindromes (a palindrome is a number that reads the same forwards and backwards). The average value of $|m - n|$ over all ordered pairs $(m, n)$, where $m$ and $n$ are (not necessarily distinct) elements of $S$, is equal to $p/q$ , for relatively prime positive integers $p$ and $q$. Find $p + q$.
2014 Postal Coaching, 1
Let $d(n)$ denote the number of positive divisors of the positive integer $n$.Determine those numbers $n$ for which $d(n^3)=5d(n)$.
2022 Federal Competition For Advanced Students, P2, 6
(a) Prove that a square with sides $1000$ divided into $31$ squares tiles, at least one of which has a side length less than $1$.
(b) Show that a corresponding decomposition into $30$ squares is also possible.
[i](Walther Janous)[/i]
2015 Indonesia MO Shortlist, N6
Defined as $N_0$ as the set of all non-negative integers. Set $S \subset N_0$ with not so many elements is called beautiful if for every $a, b \in S$ with $a \ge b$ ($a$ and $b$ do not have to be different), exactly one of $a + b$ or $a - b$ is in $S$. Set $T \subset N_0$ with not so many elements is called charming if the largest number $k$ such that up to 3$^k | a$ is the same for each element $a \in T$. Prove that each beautiful set must be charming.
2014 Tuymaada Olympiad, 1
Given are three different primes. What maximum number of these primes can divide their sum?
[i](A. Golovanov)[/i]
2008 Regional Olympiad of Mexico Center Zone, 2
Let $ABC$ be a triangle with incenter $I $, the line $AI$ intersects $BC$ at $ L$ and the circumcircle of $ABC$ at $L'$. Show that the triangles $BLI$ and $L'IB$ are similar if and only if $AC = AB + BL$.
2016 Auckland Mathematical Olympiad, 5
In a city at every square exactly three roads meet, one is called street, one is an avenue, and one is a crescent. Most roads connect squares but three roads go outside of the city. Prove that among the roads going out of the city one is a street, one is an avenue and one is a crescent.
1989 Putnam, A6
Let $\alpha=1+a_1x+a_2x^2+\ldots$ be a formal power series with coefficients in the field of two elements. Let
$$a_n=\begin{cases}1&\text{if every block of zeroes in the binary expansion of }n\text{ has an even number of zeroes}\\0&\text{otherwise}\end{cases}$$(For example, $a_{36}=1$ since $36=100100_2$)
Prove that $\alpha^3+x\alpha+1=0$.
2006 Miklós Schweitzer, 9
Does the circle T = R / Z have a self-homeomorphism $\phi$ that is singular (that is, its derivative is almost everywhere 0), but the mapping $f:T \to T$ , $f(x) = \phi^{-1} (2\phi(x))$ is absolutely continuous?
2016 AMC 12/AHSME, 19
Jerry starts at 0 on the real number line. He tosses a fair coin 8 times. When he gets heads, he moves 1 unit in the positive direction; when he gets tails, he moves 1 unit in the negative direction. The probability that he reaches 4 at some time during this process is $a/b$, where $a$ and $b$ are relatively prime positive integers. What is $a+b$? (For example, he succeeds if his sequence of tosses is $HTHHHHHH$.)
$\textbf{(A)}\ 69\qquad\textbf{(B)}\ 151\qquad\textbf{(C)}\ 257\qquad\textbf{(D)}\ 293\qquad\textbf{(E)}\ 313$
2009 Today's Calculation Of Integral, 442
Evaluate $ \int_0^{\frac{\pi}{2}} \frac{\cos \theta \minus{}\sin \theta}{(1\plus{}\cos \theta)(1\plus{}\sin \theta)}\ d\theta$
2022 IMO Shortlist, A1
Let $(a_n)_{n\geq 1}$ be a sequence of positive real numbers with the property that
$$(a_{n+1})^2 + a_na_{n+2} \leq a_n + a_{n+2}$$
for all positive integers $n$. Show that $a_{2022}\leq 1$.
2000 USAMO, 5
Let $A_1A_2A_3$ be a triangle and let $\omega_1$ be a circle in its plane passing through $A_1$ and $A_2.$ Suppose there exist circles $\omega_2, \omega_3, \dots, \omega_7$ such that for $k = 2, 3, \dots, 7,$ $\omega_k$ is externally tangent to $\omega_{k-1}$ and passes through $A_k$ and $A_{k+1},$ where $A_{n+3} = A_{n}$ for all $n \ge 1$. Prove that $\omega_7 = \omega_1.$
2017 Princeton University Math Competition, A1/B3
Shaq sees the numbers $1$ through $2017$ written on a chalkboard. He repeatedly chooses three numbers, erases them, and writes one plus their median. (For instance, if he erased $-2, -1, 0$ he would replace them with $0$.) If $M$ is the maximum possible final value remaining on the board, and if m is the minimum, compute $M - m$.
2003 JBMO Shortlist, 2
Is there a triangle with $12 \, cm^2$ area and $12$ cm perimeter?
2005 Today's Calculation Of Integral, 54
evaluate
\[\int_{-1}^0 \sqrt{\frac{1+x}{1-x}}dx\]
May Olympiad L1 - geometry, 2023.3
On a straight line $\ell$ there are four points, $A$, $B$, $C$ and $D$ in that order, such that $AB=BC=CD$. A point $E$ is chosen outside the straight line so that when drawing the segments $EB$ and $EC$, an equilateral triangle $EBC$ is formed . Segments $EA$ and $ED$ are drawn, and a point $F$ is chosen so that when drawing the segments $FA$ and $FE$, an equilateral triangle $FAE$ is formed outside the triangle $EAD$. Finally, the lines $EB$ and $FA$ are drawn , which intersect at the point $G$. If the area of triangle $EBD$ is $10$, calculate the area of triangle $EFG$.
2017 Peru IMO TST, 8
The leader of an IMO team chooses positive integers $n$ and $k$ with $n > k$, and announces them to the deputy leader and a contestant. The leader then secretly tells the deputy leader an $n$-digit binary string, and the deputy leader writes down all $n$-digit binary strings which differ from the leader’s in exactly $k$ positions. (For example, if $n = 3$ and $k = 1$, and if the leader chooses $101$, the deputy leader would write down $001, 111$ and $100$.) The contestant is allowed to look at the strings written by the deputy leader and guess the leader’s string. What is the minimum number of guesses (in terms of $n$ and $k$) needed to guarantee the correct answer?
2016 Bulgaria JBMO TST, 3
Let $ M (x,y)=x^2+xy-2y $ , x,y are positive integers
a) Solve in positive integers $ x^2+xy-2y=64 $
b) Prove that if M (x,y) is a perfect square, then x+y+2 is composite if x>2.
2019 AMC 8, 25
Alice has 24 apples. In how many ways can she share them with Becky and Chris so that each of the people has at least 2 apples?
$\textbf{(A) }105\qquad\textbf{(B) }114\qquad\textbf{(C) }190\qquad\textbf{(D) }210\qquad\textbf{(E) }380$
2019 IMO Shortlist, N4
Find all functions $f:\mathbb Z_{>0}\to \mathbb Z_{>0}$ such that $a+f(b)$ divides $a^2+bf(a)$ for all positive integers $a$ and $b$ with $a+b>2019$.