Found problems: 85335
2019 Junior Balkan Team Selection Tests - Romania, 4
Consider two disjoint finite sets of positive integers, $A$ and $B$, have $n$ and $m$ elements, respectively. It is knows that all $k$ belonging to $A \cup B$ satisfies at least one of the conditions $k + 17 \in A$ and $k - 31 \in B$.
Prove that $17n = 31m$.
2014 Singapore Senior Math Olympiad, 25
Find the number of ordered pairs of integers (p,q) satisfying the equation $p^2-q^2+p+q=2014$.
2014 Contests, 4
Let $x_1,x_2,\dots,x_{2014}$ be integers among which no two are congurent modulo $2014$. Let $y_1,y_2,\dots,y_{2014}$ be integers among which no two are congurent modulo $2014$. Prove that one can rearrange $y_1,y_2,\dots,y_{2014}$ to $z_1,z_2,\dots,z_{2014}$, so that among \[x_1+z_1,x_2+z_2,\dots,x_{2014}+z_{2014}\] no two are congurent modulo $4028$.
2005 Belarusian National Olympiad, 3
Solve in positive integers $a>b$:
$$(a-b)^{ab}=a^bb^a$$
2007 District Olympiad, 3
Let $b>a\geq 2$ be positive integers. Prove that if number $a+k$ is coprime to number $b+k$, for all $k=1,2,...,b-a$, then $a,b$ are consecutive numbers
1990 India Regional Mathematical Olympiad, 2
For all positive real numbers $ a,b,c$, prove that
\[ \frac {a}{b \plus{} c} \plus{} \frac {b}{c \plus{} a} \plus{} \frac {c}{a \plus{} b} \geq \frac {3}{2}.\]
2021 Bangladeshi National Mathematical Olympiad, 10
A positive integer $n$ is called [i]nice[/i] if it has at least $3$ proper divisors and it is equal to the sum of its three largest proper divisors. For example, $6$ is [i]nice[/i] because its largest three proper divisors are $3,2,1$ and $6=3+2+1$. Find the number of [i]nice[/i] integers not greater than $3000$.
2007 Stanford Mathematics Tournament, 2
Given that $x_1>0$ and $x_2=4x_1$ are solutions to $ax^2+bx+c$ and that $3a=2(c-b)$, what is $x_1$?
2001 District Olympiad, 3
Consider four points $A,B,C,D$ not in the same plane such that
\[AB=BD=CD=AC=\sqrt{2} AD=\frac{\sqrt{2}}{2}BC=a\]
Prove that:
a)There is a point $M\in [BC]$ such that $MA=MB=MC=MD$.
b)$2m(\sphericalangle(AD,BC))=3m(\sphericalangle((ABC),(BCD)))$
c)$6(d(A,CD))^2=7(d(A,(BCD)))^2$
[i]Ion Trandafir[/i]
1991 Denmark MO - Mohr Contest, 4
Let $a, b, c$ and $d$ be arbitrary real numbers. Prove that if $$a^2+b^2+c^2+d^2=ab+bc+cd+da,$$ then $a=b=c=d$.
1993 Chile National Olympiad, 5
Let $a,b,c$ three positive numbers less than $ 1$. Prove that cannot occur simultaneously these three inequalities:
$$a (1- b)>\frac14$$
$$b (1-c)>\frac14 $$
$$c (1-a)>\frac14$$
2006 CentroAmerican, 1
For $0 \leq d \leq 9$, we define the numbers \[S_{d}=1+d+d^{2}+\cdots+d^{2006}\]Find the last digit of the number \[S_{0}+S_{1}+\cdots+S_{9}.\]
2004 AMC 10, 6
Bertha has $ 6$ daughters and no sons. Some of her daughters have $ 6$ daughters and the rest have none. Bertha has a total of $ 30$ daughters and granddaughters, and no great-grand daughters. How many of Bertha's daughters and granddaughters have no daughters?
$ \textbf{(A)}\ 22\qquad
\textbf{(B)}\ 23\qquad
\textbf{(C)}\ 24\qquad
\textbf{(D)}\ 25\qquad
\textbf{(E)}\ 26$
2022 Israel TST, 1
A triangle $ABC$ with orthocenter $H$ is given. $P$ is a variable point on line $BC$. The perpendicular to $BC$ through $P$ meets $BH$, $CH$ at $X$, $Y$ respectively. The line through $H$ parallel to $BC$ meets $AP$ at $Q$. Lines $QX$ and $QY$ meet $BC$ at $U$, $V$ respectively. Find the shape of the locus of the incenters of the triangles $QUV$.
2022 Azerbaijan Junior National Olympiad, A3
Let $x,y,z \in \mathbb{R}^{+}$ and $x^2+y^2+z^2=x+y+z$. Prove that
$$x+y+z+3 \ge 6 \sqrt[3]{\frac{xy+yz+zx}{3}}$$
2021 Bangladesh Mathematical Olympiad, Problem 5
How many ways can you roll three 20-sided dice such that the sum of the three rolls is exactly $42$? Here the order of the rolls matter. [i](Note that a 20-sided die is is very much like a regular 6-sided die other than the fact that it has $20$ faces instead of $6$)[/i]
2011 Abels Math Contest (Norwegian MO), 2b
The diagonals $AD, BE$, and $CF$ of a convex hexagon $ABCDEF$ intersect in a common point.
Show that $a(ABE) a(CDA) a(EFC) = a(BCE) a(DEA) a(FAC)$,
where $a(KLM)$ is the area of the triangle $KLM$.
[img]https://cdn.artofproblemsolving.com/attachments/0/a/bcbbddedde159150fe3c26b1f0a2bfc322aa1a.png[/img]
2014-2015 SDML (High School), 2
Sally is thinking of a positive four-digit integer. When she divides it by any one-digit integer greater than $1$, the remainder is $1$. How many possible values are there for Sally's four-digit number?
2014 Contests, 2
Let $A$ be the $n\times n$ matrix whose entry in the $i$-th row and $j$-th column is \[\frac1{\min(i,j)}\] for $1\le i,j\le n.$ Compute $\det(A).$
2010 Harvard-MIT Mathematics Tournament, 6
Let $f(x)=x^3-x^2$. For a given value of $x$, the graph of $f(x)$, together with the graph of the line $c+x$, split the plane up into regions. Suppose that $c$ is such that exactly two of these regions have finite area. Find the value of $c$ that minimizes the sum of the areas of these two regions.
2022 Czech-Austrian-Polish-Slovak Match, 1
Let $k \leq 2022$ be a positive integer. Alice and Bob play a game on a $2022 \times 2022$ board. Initially, all cells are white. Alice starts and the players alternate. In her turn, Alice can either color one white cell in red or pass her turn. In his turn, Bob can either color a $k \times k$ square of white cells in blue or pass his turn. Once both players pass, the game ends and the person who colored more cells wins (a draw can occur). For each $1 \leq k \leq 2022$, determine which player (if any) has a winning strategy.
2005 Lithuania Team Selection Test, 1
Find the smallest integer $n$ such that an $n\times n$ square can be partitioned into $40\times 40$ and $49\times 49$ squares, with both types of squares present in the partition, if
a) $40|n$; b) $49|n$; c) $n\in \mathbb N$.
1984 AMC 12/AHSME, 11
A calculator has a key which replaces the displayed entry with its square, and another key which replaces the displayed entry with its reciprocal. Let $y$ be the final result if one starts with an entry $x \neq 0$ and alternately squares and reciprocates $n$ times each. Assuming the calculator is completely accurate (e.g., no roundoff or overflow), then $y$ equals
A. $x^{((-2)^n)}$
B. $x^{2n}$
C. $x^{-2n}$
D. $x^{-(2^n)}$
E. $x^{((-1)^n 2n)}$
2017 Regional Olympiad of Mexico Northeast, 3
Prove that there is no pair of relatively prime positive integers $(a, b)$ that satisfy the equation
$$a^3 + 2017a = b^3 -2017b.$$
1952 Putnam, A7
Directed lines are drawn from the center of a circle, making angles of $0, \pm 1, \pm 2, \pm 3, \ldots$ (measured in radians from a prime direction). If these lines meet the circle in points $P_0, P_1, P_{-1}, P_2, P_{-2}, \ldots,$ show that there is no interval on the circumference of the circle which does not contain some $P_{\pm i}.$ (You may assume that $\pi$ is irrational.)