Found problems: 85335
2022-IMOC, N3
Find all positive integer $n$ satifying $$2n+3|n!-1$$
[i]Proposed by ltf0501[/i]
1954 Polish MO Finals, 4
Find the values of $ x $ that satisfy the inequality
$$ \sqrt{x} - \sqrt{x- a} > 2,$$
where $ a $ is a gicen poistive number.
1958 AMC 12/AHSME, 45
A check is written for $ x$ dollars and $ y$ cents, $ x$ and $ y$ both two-digit numbers. In error it is cashed for $ y$ dollars and $ x$ cents, the incorrect amount exceeding the correct amount by $ \$17.82$. Then:
$ \textbf{(A)}\ {x}\text{ cannot exceed }{70}\qquad \\
\textbf{(B)}\ {y}\text{ can equal }{2x}\qquad\\
\textbf{(C)}\ \text{the amount of the check cannot be a multiple of }{5}\qquad \\
\textbf{(D)}\ \text{the incorrect amount can equal twice the correct amount}\qquad \\
\textbf{(E)}\ \text{the sum of the digits of the correct amount is divisible by }{9}$
2024 Argentina Iberoamerican TST, 1
Find all positive prime numbers $p$, $q$ that satisfy the equation
$$p(p^4+p^2+10q)=q(q^2+3).$$
2011 Purple Comet Problems, 5
Let $a_1 = 2,$ and for $n\ge 1,$ let $a_{n+1} = 2a_n + 1.$ Find the smallest value of an $a_n$ that is not a prime number.
PEN M Problems, 5
Show that there is a unique sequence of integers $\{a_{n}\}_{n \ge 1}$ with \[a_{1}=1, \; a_{2}=2, \; a_{4}=12, \; a_{n+1}a_{n-1}=a_{n}^{2}\pm1 \;\; (n \ge 2).\]
2007 AIME Problems, 4
The workers in a factory produce widgets and whoosits. For each product, production time is constant and identical for all workers, but not necessarily equal for the two products. In one hour, 100 workers can produce 300 widgets and 200 whoosits. In two hours, 60 workers can produce 240 widgets and 300 whoosits. In three hours, 50 workers can produce 150 widgets and m whoosits. Find m.
2018 PUMaC Number Theory A, 7
Find the smallest positive integer $G$ such that there exist distinct positive integers $a, b, c$ with the following properties:
$\: \bullet \: \gcd(a, b, c) = G$.
$\: \bullet \: \text{lcm}(a, b) = \text{lcm}(a, c) = \text{lcm}(b, c)$.
$\: \bullet \: \frac{1}{a} + \frac{1}{b}, \frac{1}{a} + \frac{1}{c},$ and $\frac{1}{b} + \frac{1}{c}$ are reciprocals of integers.
$\: \bullet \: \gcd(a, b) + \gcd(a, c) + \gcd(b, c) = 16G$.
2019 Tournament Of Towns, 5
Let us say that the pair $(m, n)$ of two positive different integers m and n is [i]nice [/i] if $mn$ and $(m + 1)(n + 1)$ are perfect squares. Prove that for each positive integer m there exists at least one $n > m$ such that the pair $(m, n)$ is nice.
(Yury Markelov)
2020 CCA Math Bonanza, I8
Compute the remainder when the largest integer below $\frac{3^{123}}{5}$ is divided by $16$.
[i]2020 CCA Math Bonanza Individual Round #8[/i]
1994 Italy TST, 3
Find all functions $f : R \to R$ satisfying the condition $f(x- f(y)) = 1+x-y$ for all $x,y \in R$.
2010 Stanford Mathematics Tournament, 3
Bob sends a secret message to Alice using her RSA public key $n = 400000001.$ Eve wants to listen in on their conversation. But to do this, she needs Alice's private key, which is the factorization of $n.$ Eve knows that $n = pq,$ a product of two prime factors. Find $p$ and $q.$
2020 March Advanced Contest, 3
A [i]simple polygon[/i] is a polygon whose perimeter does not self-intersect. Suppose a simple polygon $\mathcal P$ can be tiled with a finite number of parallelograms. Prove that regardless of the tiling, the sum of the areas of all rectangles in the tiling is fixed.\\
[i]Note:[/i] Points will be awarded depending on the generality of the polygons for which the result is proven.
2014 Contests, 2
Let $n \ge 2$ be an integer. Consider an $n \times n$ chessboard consisting of $n^2$ unit squares. A configuration of $n$ rooks on this board is [i]peaceful[/i] if every row and every column contains exactly one rook. Find the greatest positive integer $k$ such that, for each peaceful configuration of $n$ rooks, there is a $k \times k$ square which does not contain a rook on any of its $k^2$ unit squares.
1949-56 Chisinau City MO, 3
Prove that the number $N = 10 ...050...01$ (1, 49 zeros, 5 , 99 zeros, 1) is a not cube of an integer.
2021 CCA Math Bonanza, I9
Points $A$, $B$, $C$, $D$, and $E$ are on the same plane such that $A,E,C$ lie on a line in that order, $B,E,D$ lie on a line in that order, $AE = 1$, $BE = 4$, $CE = 3$, $DE = 2$, and $\angle AEB = 60^\circ$. Let $AB$ and $CD$ intersect at $P$. The square of the area of quadrilateral $PAED$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
[i]2021 CCA Math Bonanza Individual Round #9[/i]
1996 Israel National Olympiad, 4
Eight guests arrive to a hotel with four rooms. Each guest dislikes at most three other guests and doesn’t want to share a room with any of them (this feeling is mutual). Show that the guests can reside in the four rooms, with two persons in each room
2010 China Team Selection Test, 1
Let $ABCD$ be a convex quadrilateral with $A,B,C,D$ concyclic. Assume $\angle ADC$ is acute and $\frac{AB}{BC}=\frac{DA}{CD}$. Let $\Gamma$ be a circle through $A$ and $D$, tangent to $AB$, and let $E$ be a point on $\Gamma$ and inside $ABCD$.
Prove that $AE\perp EC$ if and only if $\frac{AE}{AB}-\frac{ED}{AD}=1$.
2014 India IMO Training Camp, 2
For $j=1,2,3$ let $x_{j},y_{j}$ be non-zero real numbers, and let $v_{j}=x_{j}+y_{j}$.Suppose that the following statements hold:
$x_{1}x_{2}x_{3}=-y_{1}y_{2}y_{3}$
$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=y_{1}^{2}+y_{2}^{2}+y_{3}^2$
$v_{1},v_{2},v_{3}$ satisfy triangle inequality
$v_{1}^{2},v_{2}^{2},v_{3}^{2}$ also satisfy triangle inequality.
Prove that exactly one of $x_{1},x_{2},x_{3},y_{1},y_{2},y_{3}$ is negative.
2013 AMC 12/AHSME, 1
Square $ ABCD $ has side length $ 10 $. Point $ E $ is on $ \overline{BC} $, and the area of $ \bigtriangleup ABE $ is $ 40 $. What is $ BE $?
$\textbf{(A)} \ 4 \qquad \textbf{(B)} \ 5 \qquad \textbf{(C)} \ 6 \qquad \textbf{(D)} \ 7 \qquad \textbf{(E)} \ 8 \qquad $
[asy]
pair A,B,C,D,E;
A=(0,0);
B=(0,50);
C=(50,50);
D=(50,0);
E = (30,50);
draw(A--B);
draw(B--E);
draw(E--C);
draw(C--D);
draw(D--A);
draw(A--E);
dot(A);
dot(B);
dot(C);
dot(D);
dot(E);
label("A",A,SW);
label("B",B,NW);
label("C",C,NE);
label("D",D,SE);
label("E",E,N);
[/asy]
2021-2022 OMMC, 9
$12$ people stand in a row. Each person is given a red shirt or a blue shirt. Every minute, exactly one pair of two people with the same color currently standing next to each other in the row leave. After $6$ minutes, everyone has left. How many ways could the shirts have been assigned initially?
[i]Proposed by Evan Chang[/i]
2019 China Team Selection Test, 4
Does there exist a finite set $A$ of positive integers of at least two elements and an infinite set $B$ of positive integers, such that any two distinct elements in $A+B$ are coprime, and for any coprime positive integers $m,n$, there exists an element $x$ in $A+B$ satisfying $x\equiv n \pmod m$ ?
Here $A+B=\{a+b|a\in A, b\in B\}$.
2002 Romania Team Selection Test, 3
Let $M$ and $N$ be the midpoints of the respective sides $AB$ and $AC$ of an acute-angled triangle $ABC$. Let $P$ be the foot of the perpendicular from $N$ onto $BC$ and let $A_1$ be the midpoint of $MP$. Points $B_1$ and $C_1$ are obtained similarly. If $AA_1$, $BB_1$ and $CC_1$ are concurrent, show that the triangle $ABC$ is isosceles.
[i]Mircea Becheanu[/i]
2008 AMC 12/AHSME, 3
A semipro baseball league has teams with $ 21$ players each. League rules state that a player must be paid at least $ \$15,000$, and that the total of all players' salaries for each team cannot exceed $ \$700,000$. What is the maximum possiblle salary, in dollars, for a single player?
$ \textbf{(A)}\ 270,000 \qquad
\textbf{(B)}\ 385,000 \qquad
\textbf{(C)}\ 400,000 \qquad
\textbf{(D)}\ 430,000 \qquad
\textbf{(E)}\ 700,000$
2013 NIMO Problems, 1
Find the sum of all primes that can be written both as a sum of two primes and as a difference of two primes.
[i]Anonymous Proposal[/i]