Found problems: 85335
2023 Kyiv City MO Round 1, Problem 1
Find the integer which is closest to the value of the following expression:
$$((7 + \sqrt{48})^{2023} + (7 - \sqrt{48})^{2023})^2 - ((7 + \sqrt{48})^{2023} - (7 - \sqrt{48})^{2023})^2$$
2021 Science ON all problems, 4
The numbers $\frac 32$, $\frac 43$ and $\frac 65$ are intially written on the blackboard. A move consists of erasing one of the numbers from the blackboard, call it $a$, and replacing it with $bc-b-c+2$, where $b,c$ are the other two numbers currently written on the blackboard. Is it possible that $\frac{1000}{999}$ would eventually appear on the blackboard? What about $\frac{113}{108}$?
[i] (Andrei Bâra)[/i]
2001 All-Russian Olympiad Regional Round, 9.6
Is there such a natural number that the product of all its natural divisors (including $1$ and the number itself) ends exactly in $2001$ zeros?
2014 Nordic, 2
Given an equilateral triangle, find all points inside the triangle such that the distance from the point to one of the sides is equal to the geometric mean of the distances from the point to the other two sides of the triangle.
2018 China Team Selection Test, 2
A number $n$ is [i]interesting[/i] if 2018 divides $d(n)$ (the number of positive divisors of $n$). Determine all positive integers $k$ such that there exists an infinite arithmetic progression with common difference $k$ whose terms are all interesting.
2024 Regional Olympiad of Mexico West, 1
Initially, the numbers $1,3,4$ are written on a board. We do the following process repeatedly. Consider all of the numbers that can be obtained as the sum of $3$ distinct numbers written on the board and that aren't already written, and we write those numbers on the board. We repeat this process, until at a certain step, all of the numbers in that step are greater than $2024$. Determine all of the integers $1\leq k\leq 2024$ that were not written on the board.
2023 Saint Petersburg Mathematical Olympiad, 3
Find all positive integers $x, y$ and primes $p$, such that $x^5+y^4=pxy$.
2019 Thailand TSTST, 2
Let $\Omega$ be the inscribed circle of a triangle $\vartriangle ABC$. Let $D, E$ and $F$ be the tangency points of $\Omega$ and the sides $BC, CA$ and $AB$, respectively, and let $AD, BE$ and $CF$ intersect $\Omega$ at $K, L$ and $M$, respectively, such that $D, E, F, K, L$ and $M$ are all distinct. The tangent line of $\Omega$ at $K$ intersects $EF$ at $X$, the tangent line of $\Omega$ at $L$ intersects $DE$ at $Y$ , and the tangent line of $\Omega$ at M intersects $DF$ at $Z$. Prove that $X,Y$ and $Z$ are collinear.
1966 Bulgaria National Olympiad, Problem 2
Prove that for every four positive numbers $a,b,c,d$ the following inequality is true:
$$\sqrt{\frac{a^2+b^2+c^2+d^2}4}\ge\sqrt[3]{\frac{abc+abd+acd+bcd}4}.$$
1983 Kurschak Competition, 1
Let $x, y$ and $z$ be rational numbers satisfying $$x^3 + 3y^3 + 9z^3 - 9xyz = 0.$$
Prove that $x = y = z = 0$.
2009 AIME Problems, 11
For certain pairs $ (m,n)$ of positive integers with $ m\ge n$ there are exactly $ 50$ distinct positive integers $ k$ such that $ |\log m \minus{} \log k| < \log n$. Find the sum of all possible values of the product $ mn$.
1988 Tournament Of Towns, (183) 6
Consider a sequence of words , consisting of the letters $A$ and $B$ .
The first word in the sequence is "$A$" . The k-th word i s obtained from the $(k-1)$-th by means of the following transformation : each $A$ is substituted by $AAB$ , and each $B$ is substituted by $A$. It is easily seen that every word is an initial part of the next word. The initial parts of these words coincide to give a sequence of letters $AABAABAAA BAABAAB...$
(a) In which place of this sequence is the $1000$-th letter $A$?
(b ) Prove that this sequence is not periodic.
(V . Galperin , Moscows)
2010 Tuymaada Olympiad, 1
Misha and Sahsa play a game on a $100\times 100$ chessboard. First, Sasha places $50$ kings on the board, and Misha places a rook, and then they move in turns, as following (Sasha begins):
At his move, Sasha moves each of the kings one square in any direction, and Misha can move the rook on the horizontal or vertical any number of squares. The kings cannot be captured or stepped over. Sasha's purpose is to capture the rook, and Misha's is to avoid capture.
Is there a winning strategy available for Sasha?
2015 Online Math Open Problems, 12
At the Intergalactic Math Olympiad held in the year 9001, there are 6 problems, and on each problem you can earn an integer score from 0 to 7. The contestant's score is the [i]product[/i] of the scores on the 6 problems, and ties are broken by the sum of the 6 problems. If 2 contestants are still tied after this, their ranks are equal. In this olympiad, there are $8^6=262144$ participants, and no two get the same score on every problem. Find the score of the participant whose rank was $7^6 = 117649$.
[i]Proposed by Yang Liu[/i]
2005 Iran Team Selection Test, 1
Find all $f : N \longmapsto N$ that there exist $k \in N$ and a prime $p$ that:
$\forall n \geq k \ f(n+p)=f(n)$ and also if $m \mid n$ then $f(m+1) \mid f(n)+1$
1961 AMC 12/AHSME, 30
If $\log_{10}2=a$ and $\log_{10}3=b$, then $\log_{5}12=?$
${{ \textbf{(A)}\ \frac{a+b}{a+1} \qquad\textbf{(B)}\ \frac{2a+b}{a+1} \qquad\textbf{(C)}\ \frac{a+2b}{1+a} \qquad\textbf{(D)}\ \frac{2a+b}{1-a} }\qquad\textbf{(E)}\ \frac{a+2b}{1-a}} $
2005 India IMO Training Camp, 2
Determine all positive integers $n > 2$ , such that \[ \frac{1}{2} \varphi(n) \equiv 1 ( \bmod 6) \]
1987 Dutch Mathematical Olympiad, 3
There are two kinds of creatures living in the flatland of Pentagonia: the Spires ($S$) and the Bones ($B$). They all have the shape of an isosceles triangle: the Spiers have an apical angle of $36^o$ and the bones an apical angle of $108^o$.
Every year on [i]Great Day of Division[/i] (September 11 - the day this Olympiad was held) they divide into pieces: each $S$ into two smaller $S$'s and a $B$; each $B$ in an $S$ and a $B$. Over the course of the year they then grow back to adult proportions. In the distant past, the population originated from one $B$-being. Deaths do not occur.
Investigate whether the ratio between the number of Spires and the number of Bones will eventually approach a limit value and if so, calculate that limit value.
2002 Tournament Of Towns, 7
[list]
[*] A power grid with the shape of a $3\times 3$ lattice with $16$ nodes (vertices of the lattice) joined by wires (along the sides of squares. It may have happened that some of the wires have burned out. In one test technician can choose any two nodes and check if electrical current circulates between them (i.e there is a chain of intact wires joining the chosen nodes) . Technicial knows that current will circulate from any node to another node. What is the least number of tests required to demonstrate this?
[*] Previous problem for the grid of $7\times 7$ lattice.[/list]
1980 AMC 12/AHSME, 9
A man walks $x$ miles due west, turns $150^\circ$ to his left and walks 3 miles in the new direction. If he finishes a a point $\sqrt{3}$ from his starting point, then $x$ is
$\text{(A)} \ \sqrt 3 \qquad \text{(B)} \ 2\sqrt{5} \qquad \text{(C)} \ \frac 32 \qquad \text{(D)} \ 3 \qquad \text{(E)} \ \text{not uniquely determined}$
2002 AIME Problems, 4
Consider the sequence defined by $a_k=\frac 1{k^2+k}$ for $k\ge 1.$ Given that $a_m+a_{m+1}+\cdots+a_{n-1}=1/29,$ for positive integers $m$ and $n$ with $m<n$, find $m+n.$
2015 Caucasus Mathematical Olympiad, 2
Let $a$ and $b$ be arbitrary distinct numbers.
Prove that the equation $(x +a) (x+b)=2x+a+b$ has two different roots.
2018 Czech and Slovak Olympiad III A, 6
Determine the least positive integer $n$ with the following property – for every 3-coloring of numbers $1,2,\ldots,n$ there are two (different) numbers $a,b$ of the same color such that $|a-b|$ is a perfect square.
2016 German National Olympiad, 3
Let $I_a$ be the $A$-excenter of a scalene triangle $ABC$. And let $M$ be the point symmetric to $I_a$ about line $BC$.
Prove that line $AM$ is parallel to the line through the circumcenter and the orthocenter of triangle $I_aCB$.
2007 Harvard-MIT Mathematics Tournament, 6
There are three video game systems: the Paystation, the WHAT, and the ZBoz2$\pi$, and none of these systems will play games for the other systems. Uncle Riemann has three nephews: Bernoulli, Galois, and Dirac. Bernoulli owns a Paystation and a WHAT, Galois owns a WHAT and a ZBoz2$\pi$, and Dirac owns a ZBoz2$\pi$ and a Paystation. A store sells $4$ different games for the Paystation, $6$ different games for the WHAT, and $10$ different games for the ZBoz2$\pi$. Uncle Riemann does not understand the difference between the systems, so he walks into the store and buys $3$ random games (not necessarily distinct) and randomly hands them to his nephews. What is the probability that each nephew receives a game he can play?