Found problems: 85335
2022 Portugal MO, 3
The Proenc has a new $8\times 8$ chess board and requires composing it into rectangles that do not overlap, so that:
(i) each rectangle has as many white squares as black ones;
(ii) there are no two rectangles with the same number of squares.
Determines the maximum value of $n$ for which such a decomposition is possible. For this value of $n$, determine all possible sets ${A_1,... ,A_n}$, where $A_i$ is the number of rectangle $i$ in squares, for which a decomposition of the board under the conditions intended actions is possible.
2011 National Olympiad First Round, 34
Let $n$ be a positive integer number. The decimal representation of $2^n$ contains $m$ same numbers from the right. What is the largest value of $m$?
$\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ \text{None}$
2004 Iran MO (3rd Round), 3
Suppose $V= \mathbb{Z}_2^n$ and for a vector $x=(x_1,..x_n)$ in $V$ and permutation $\sigma$.We have $x_{\sigma}=(x_{\sigma(1)},...,x_{\sigma(n)})$
Suppose $ n=4k+2,4k+3$ and $f:V \to V$ is injective and if $x$ and $y$ differ in more than $n/2$ places then $f(x)$ and $f(y)$ differ in more than $n/2$ places.
Prove there exist permutaion $\sigma$ and vector $v$ that $f(x)=x_{\sigma}+v$
2006 Mexico National Olympiad, 6
Let n be the sum of the digits in a natural number A. The number A it's said to be "surtido" if every number 1,2,3,4....,n can be expressed as a sum of digits in A.
a)Prove that, if 1,2,3,4,5,6,7,8 are sums of digits in A, then A is "Surtido"
b)If 1,2,3,4,5,6,7 are sums of digits in A, does it follow that A is "Surtido"?
2009 All-Russian Olympiad, 4
On a circle there are 2009 nonnegative integers not greater than 100. If two numbers sit next to each other, we can increase both of them by 1. We can do this at most $ k$ times. What is the minimum $ k$ so that we can make all the numbers on the circle equal?
2000 National Olympiad First Round, 10
$N$ is a $50-$digit number (in the decimal scale). All digits except the $26^{\text{th}}$ digit (from the left) are $1$. If $N$ is divisible by $13$, what is the $26^{\text{th}}$ digit?
$ \textbf{(A)}\ 1
\qquad\textbf{(B)}\ 3
\qquad\textbf{(C)}\ 6
\qquad\textbf{(D)}\ 8
\qquad\textbf{(E)}\ \text{More information is needed}
$
2016 Mathematical Talent Reward Programme, MCQ: P 8
Let $p$ be a prime such that $16p+1$ is a perfect cube. A possible choice for $p$ is
[list=1]
[*] 283
[*] 307
[*] 593
[*] 691
[/list]
2009 AMC 8, 6
Steve's empty swimming pool will hold $ 24,000$ gallons of water when full. It will be filled by $ 4$ hoses, each of which supplies $ 2.5$ gallons of water per minute. How many hours will it take to fill Steve's pool?
$ \textbf{(A)}\ 40 \qquad
\textbf{(B)}\ 42 \qquad
\textbf{(C)}\ 44 \qquad
\textbf{(D)}\ 46 \qquad
\textbf{(E)}\ 48$
2012 Greece JBMO TST, 2
Find all pairs of coprime positive integers $(p,q)$ such that $p^2+2q^2+334=[p^2,q^2]$ where $[p^2,q^2]$ is the leact common multiple of $p^2,q^2$ .
1998 Junior Balkan Team Selection Tests - Romania, 2
Consider the rectangle $ ABCD $ and the points $ M,N,P,Q $ on the segments $ AB,BC,CD, $ respectively, $ DA, $ excluding its extremities. Denote with $ p_{\square} , A_{\square} $ the perimeter, respectively, the area of $ \square. $ Prove that:
[b]a)[/b] $ p_{MNPQ}\ge AC+BD. $
[b]b)[/b] $ p_{MNPQ} =AC+BD\implies A_{MNPQ}\le \frac{A_{ABCD}}{2} . $
[b]c)[/b] $ p_{MNPQ} =AC+BD\implies MP^2 +NQ^2\ge AC^2. $
[i]Dan Brânzei[/i] and [i]Gheorghe Iurea[/i]
1998 German National Olympiad, 6a
Find all real pairs $(x,y)$ that solve the system of equations \begin{align} x^5 &= 21x^3+y^3
\\ y^5 &= x^3+21y^3. \end{align}
1983 AIME Problems, 13
For $\{1, 2, 3, \dots, n\}$ and each of its nonempty subsets a unique [b]alternating sum[/b] is defined as follows: Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. (For example, the alternating sum for $\{1, 2, 4, 6, 9\}$ is $9 - 6 + 4 - 2 + 1 = 6$ and for $\{5\}$ it is simply 5.) Find the sum of all such alternating sums for $n = 7$.
2021 Girls in Math at Yale, R2
4. Suppose that $\overline{A2021B}$ is a six-digit integer divisible by $9$. Find the maximum possible value of $A \cdot B$.
5. In an arbitrary triangle, two distinct segments are drawn from each vertex to the opposite side. What is the minimum possible number of intersection points between these segments?
6. Suppose that $a$ and $b$ are positive integers such that $\frac{a}{b-20}$ and $\frac{b+21}{a}$ are positive integers. Find the maximum possible value of $a + b$.
2007 Tournament Of Towns, 5
A triangular pie has the same shape as its box, except that they are mirror images of each other. We wish to cut the pie in two pieces which can t together in the box without turning either piece over. How can this be done if
[list][b](a)[/b] one angle of the triangle is three times as big as another;
[b](b)[/b] one angle of the triangle is obtuse and is twice as big as one of the acute angles?[/list]
2008 ISI B.Stat Entrance Exam, 4
Suppose $P$ and $Q$ are the centres of two disjoint circles $C_1$ and $C_2$ respectively, such that $P$ lies outside $C_2$ and $Q$ lies outside $C_1$. Two tangents are drawn from the point $P$ to the circle $C_2$, which intersect the circle $C_1$ at point $A$ and $B$. Similarly, two tangents are drawn from the point $Q$ to the circle $C_1$, which intersect the circle $C_2$ at points $M$ and $N$. Show that $AB=MN$
Kvant 2020, M2606
Three circles $\omega_1,\omega_2$ and $\omega_3$ pass through one point $D{}$. Let $A{}$ be the intersection of $\omega_1$ and $\omega_3$, and $E{}$ be the intersections of $\omega_3$ and $\omega_2$ and $F{}$ be the intersection of $\omega_2$ and $\omega_1$. It is known that $\omega_3$ passes through the center $B{}$ of the circle $\omega_2$. The line $EF$ intersects $\omega_1$ a second time at the point $G{}$. Prove that $\angle GAB=90^\circ$.
[i]Proposed by K. Knop[/i]
2015 JBMO Shortlist, C1
A board $ n \times n$ ($n \ge 3$) is divided into $n^2$ unit squares. Integers from $O$ to $n$ included, are written down: one integer in each unit square, in such a way that the sums of integers in each $2\times 2$ square of the board are different. Find all $n$ for which such boards exist.
1977 IMO Longlists, 32
In a room there are nine men. Among every three of them there are two mutually acquainted. Prove that some four of them are mutually acquainted.
1967 Poland - Second Round, 3
Two circles touch internally at point $A$. A chord $ BC $ of the larger circle is drawn tangent to the smaller one at point $ D $. Prove that $ AD $ is the bisector of angle $ BAC $.
2001 Croatia National Olympiad, Problem 3
Let $p_1,p_2,p_3,p_4$ be four distinct primes, and let $1=d_1<d_2<\ldots<d_{16}=n$ be the divisors of $n=p_1p_2p_3p_4$. Determine all $n<2001$ with the property that
$d_9-d_8=22$.
1998 IberoAmerican, 1
There are representants from $n$ different countries sit around a circular table ($n\geq2$), in such way that if two representants are from the same country, then, their neighbors to the right are not from the same country. Find, for every $n$, the maximal number of people that can be sit around the table.
the 12th XMO, Problem 4
求最小的 $n,$ 使得对任意有 ${1000}$ 个顶点且每个顶点度均为 ${4}$ 的简单图 $G,$ 都一定可以从中取掉 ${n}$ 条边$,$ 使 ${G}$ 变为二部图$.$
1988 AMC 8, 19
What is the $100th$ number in the arithmetic sequence: $ 1,5,9,13,17,21,25,... $
$ \text{(A)}\ 397\qquad\text{(B)}\ 399\qquad\text{(C)}\ 401\qquad\text{(D)}\ 403\qquad\text{(E)}\ 405 $
1981 Romania Team Selection Tests, 3.
Determine the lengths of the edges of a right tetrahedron of volume $a^3$ so that the sum of its edges' lengths is minumum.
2010 ISI B.Math Entrance Exam, 8
Let $f$ be a real-valued differentiable function on the real line $\mathbb{R}$ such that
$\lim_{x\to 0} \frac{f(x)}{x^2}$ exists, and is finite . Prove that $f'(0)=0$.