Found problems: 85335
1999 National Olympiad First Round, 15
2 squares are painted in blue and 2 squares are painted in red on a $ 3\times 3$ board in such a way that two square with same color is neither at same row nor at same column. In how many different ways can these four squares be painted?
$\textbf{(A)}\ 198 \qquad\textbf{(B)}\ 288 \qquad\textbf{(C)}\ 396 \qquad\textbf{(D)}\ 576 \qquad\textbf{(E)}\ 792$
1976 IMO Longlists, 41
Determine the greatest number, who is the product of some positive integers, and the sum of these numbers is $1976.$
2001 Portugal MO, 2
The trapezium $[ABCD]$ has bases $[AB]$ and $[CD]$ (with $[AB]$ being the largest base). Knowing that $BC = 2 DA$ and that $\angle DAB + \angle ABC =120^o$ , determines the measure of $\angle DAB$.
2017 China Western Mathematical Olympiad, 4
Let $n$ and $k$ be given integers such that $n\ge k\ge 2$. Alice and Bob play a game on an $n$ by $n$ table with white cells. They take turns to pick a white cell and color it black. Alice moves first. The game ends as soon as there is at least one black cell in every $k$ by $k$ square after a player moves, who is declared the winner of the game. Who has the winning strategy?
2023 Bulgarian Autumn Math Competition, 12.3
Solve in positive integers the equation $$m^{\frac{1}{n}}+n^{\frac{1}{m}}=2+\frac{2}{mn(m+n)^{\frac{1}{m}+\frac{1}{n}}}.$$
2016 Balkan MO Shortlist, A4
The positive real numbers $a, b, c$ satisfy the equality $a + b + c = 1$. For every natural number $n$ find the minimal possible value of the expression $$E=\frac{a^{-n}+b}{1-a}+\frac{b^{-n}+c}{1-b}+\frac{c^{-n}+a}{1-c}$$
2009 Oral Moscow Geometry Olympiad, 2
Trapezium $ABCD$ and parallelogram $MBDK$ are located so that the sides of the parallelogram are parallel to the diagonals of the trapezoid (see fig.). Prove that the area of the gray part is equal to the sum of the areas of the black part.
(Yu. Blinkov)
[img]https://cdn.artofproblemsolving.com/attachments/b/9/cfff83b1b85aea16b603995d4f3d327256b60b.png[/img]
1966 Spain Mathematical Olympiad, 6
They tell us that a married couple has $5$ children. Calculate the probability that among them there are at least two men and at least one woman. Probability of being born male is considered $1/2$.
LMT Team Rounds 2021+, 15
There are $28$ students who have to be separated into two groups such that the number of students in each group
is a multiple of $4$. The number of ways to split them into the groups can be written as
$$\sum_{k \ge 0} 2^k a_k = a_0 +2a_1 +4a_2 +...$$
where each $a_i$ is either $0$ or $1$. Find the value of
$$\sum_{k \ge 0} ka_k = 0+ a_1 +2a_2 +3a3_ +....$$
2022 Balkan MO Shortlist, N1
Let $n{}$ be a positive integer. What is the smallest sum of digits that $5^n + 6^n + 2022^n$ can take?
2015 Purple Comet Problems, 13
Given that x, y, and z are real numbers satisfying $x+y +z = 10$ and $x^2 +y^2 +z^2 = 50$, find the maximum possible value of $(x + 2y + 3z)^2 + (y + 2z + 3x)^2 + (z + 2x + 3y)^2$.
1998 Cono Sur Olympiad, 1
We have $98$ cards, in each one we will write one of the numbers: $1, 2, 3, 4,...., 97, 98$.
We can order the $98$ cards, in a sequence such that two consecutive numbers $X$ and $Y$ and the number $X - Y$ is greater than $48$, determine how and how many ways we can make this sequence!!
2019 Belarus Team Selection Test, 6.2
The numbers $1,2,\ldots,49,50$ are written on the blackboard. Ann performs the following operation: she chooses three arbitrary numbers $a,b,c$ from the board, replaces them by their sum $a+b+c$ and writes $(a+b)(b+c)(c+a)$ to her notebook. Ann performs such operations until only two numbers remain on the board (in total 24 operations). Then she calculates the sum of all $24$ numbers written in the notebook. Let $A$ and $B$ be the maximum and the minimum possible sums that Ann san obtain.
Find the value of $\frac{A}{B}$.
[i](I. Voronovich)[/i]
2014 Chile TST Ibero, 1
Consider a function $f: \mathbb{R} \to \mathbb{R}$ satisfying for all $x \in \mathbb{R}$:
\[
f(x+1) = \frac{1}{2} + \sqrt{f(x) - f(x)^2}.
\]
Prove that there exists a $b > 0$ such that $f(x + b) = f(x)$ for all $x \in \mathbb{R}$.
LMT Guts Rounds, 34
A [i]prime power[/i] is an integer of the form $p^k,$ where $p$ is a prime and $k$ is a nonnegative integer. How many prime powers are there less than or equal to $10^6?$ Your score will be $16-80|\frac{\textbf{Your Answer}}{\textbf{Actual Answer}}-1|$ rounded to the nearest integer or $0,$ whichever is higher.
2008 SDMO (Middle School), 5
For a positive integer $n$, let $f\left(n\right)$ be the sum of the first $n$ terms of the sequence $$0,1,1,2,2,3,3,4,4,\ldots,r,r,r+1,r+1,\ldots.$$ For example, $f\left(5\right)=0+1+1+2+2=6$.
(a) Find a formula for $f\left(n\right)$.
(b) Prove that $f\left(s+t\right)-f\left(s-t\right)=st$ for all positive integers $s$ and $t$, where $s>t$.
2016 IMO Shortlist, A2
Find the smallest constant $C > 0$ for which the following statement holds: among any five positive real numbers $a_1,a_2,a_3,a_4,a_5$ (not necessarily distinct), one can always choose distinct subscripts $i,j,k,l$ such that
\[ \left| \frac{a_i}{a_j} - \frac {a_k}{a_l} \right| \le C. \]
2023 ISI Entrance UGB, 2
Let $a_0 = \frac{1}{2}$ and $a_n$ be defined inductively by
\[a_n = \sqrt{\frac{1+a_{n-1}}{2}} \text{, $n \ge 1$.} \]
[list=a]
[*] Show that for $n = 0,1,2, \ldots,$
\[a_n = \cos(\theta_n) \text{ for some $0 < \theta_n < \frac{\pi}{2}$, }\]
and determine $\theta_n$.
[*] Using (a) or otherwise, calculate
\[ \lim_{n \to \infty} 4^n (1 - a_n).\]
[/list]
1988 IMO Longlists, 26
The circle $x^2+ y^2 = r^2$ meets the coordinate axis at $A = (r,0), B = (-r,0), C = (0,r)$ and $D = (0,-r).$ Let $P = (u,v)$ and $Q = (-u,v)$ be two points on the circumference of the circle. Let $N$ be the point of intersection of $PQ$ and the $y$-axis, and $M$ be the foot of the perpendicular drawn from $P$ to the $x$-axis. If $r^2$ is odd, $u = p^m > q^n = v,$ where $p$ and $q$ are prime numbers and $m$ and $n$ are natural numbers, show that
\[ |AM| = 1, |BM| = 9, |DN| = 8, |PQ| = 8. \]
2012 Moldova Team Selection Test, 8
Let $p\geq5$ be a prime and $S_k=1^k+2^k+...+(p-1)^k,\forall k\in\mathbb{N}.$ Prove that there is an infinity of numbers $n\in\mathbb{N}$ such that $p^3$ divides $S_n$ and $ p $ divides $S_{n-1}$ and $S_{n-2}.$
2016 Uzbekistan National Olympiad, 5
Solve following system equations:
\[\left\{ \begin{array}{c}
3x+4y=26\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\
\sqrt{x^2+y^2-4x+2y+5}+\sqrt{x^2+y^2-20x-10y+125}=10\ \end{array}
\right.\ \ \]
2011 AIME Problems, 15
For some integer $m$, the polynomial $x^3-2011x+m$ has the three integer roots $a$, $b$, and $c$. Find $|a|+|b|+|c|$.
2002 Iran MO (3rd Round), 14
A subset $S$ of $\mathbb N$ is [i]eventually linear[/i] iff there are $k,N\in\mathbb N$ that for $n>N,n\in S\Longleftrightarrow k|n$. Let $S$ be a subset of $\mathbb N$ that is closed under addition. Prove that $S$ is eventually linear.
1997 Romania National Olympiad, 1
Let $m \ge 2$ and $n \ge 1$ be integers and $A=(a_{ij})$ a square matrix of order $n$ with integer entries. Prove that for any permutation $\sigma \in S_n$ there is a function $\varepsilon : \{1,2,\ldots,n\} \to \{0,1\}$ such that replacing the entries $a_{\sigma(1)1},$ $a_{\sigma(2)2}, $ $\ldots,$ $a_{\sigma(n)n}$ of $A$ respectively by $$a_{\sigma(1)1}+\varepsilon(1), ~a_{\sigma(2)2}+\varepsilon(2), ~\ldots, ~a_{\sigma(n)n}+\varepsilon(n),$$ the determinant of the matrix $A_{\varepsilon}$ thus obtained is not divisible by $m.$
2021 Polish Junior MO Finals, 5
Natural numbers $a$, $b$ are written in decimal using the same digits (i.e. every digit from 0 to 9 appears the same number of times in $a$ and in $b$). Prove that if $a+b=10^{1000}$ then both numbers $a$ and $b$ are divisible by $10$.