Found problems: 15460
DMM Team Rounds, 2011
[b]p1.[/b] How many primes $p < 100$ satisfy $p = a^2 + b^2$ for some positive integers $a$ and $b$?
[b]p2. [/b] For $a < b < c$, there exists exactly one Pythagorean triple such that $a + b + c = 2000$. Find $a + c - b$.
[b]p3.[/b] Five points lie on the surface of a sphere of radius $ 1$ such that the distance between any two points is at least $\sqrt2$. Find the maximum volume enclosed by these five points.
[b]p4.[/b] $ABCDEF$ is a convex hexagon with $AB = BC = CD = DE = EF = FA = 5$ and $AC = CE = EA = 6$. Find the area of $ABCDEF$.
[b]p5.[/b] Joe and Wanda are playing a game of chance. Each player rolls a fair $11$-sided die, whose sides are labeled with numbers $1, 2, ... , 11$. Let the result of the Joe’s roll be $X$, and the result of Wanda’s roll be $Y$ . Joe wins if $XY$ has remainder $ 1$ when divided by $11$, and Wanda wins otherwise. What is the probability that Joe wins?
[b]p6.[/b] Vivek picks a number and then plays a game. At each step of the game, he takes the current number and replaces it with a new number according to the following rule: if the current number $n$ is divisible by $3$, he replaces $n$ with $\frac{n}{3} + 2$, and otherwise he replaces $n$ with $\lfloor 3 \log_3 n \rfloor$. If he starts with the number $3^{2011}$, what number will he have after $2011$ steps?
Note that $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$.
[b]p7.[/b] Define a sequence an of positive real numbers with a$_1 = 1$, and $$a_{n+1} =\frac{4a^2_n - 1}{-2 + \frac{4a^2_n -1}{-2+ \frac{4a^2_n -1}{-2+...}}}.$$
What is $a_{2011}$?
[b]p8.[/b] A set $S$ of positive integers is called good if for any $x, y \in S$ either $x = y$ or $|x - y| \ge 3$. How many subsets of $\{1, 2, 3, ..., 13\}$ are good? Include the empty set in your count.
[b]p9.[/b] Find all pairs of positive integers $(a, b)$ with $a \le b$ such that $10 \cdot lcm \, (a, b) = a^2 + b^2$. Note that $lcm \,(m, n)$ denotes the least common multiple of $m$ and $n$.
[b]p10.[/b] For a natural number $n$, $g(n)$ denotes the largest odd divisor of $n$. Find $$g(1) + g(2) + g(3) + ... + g(2^{2011})$$
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2023 Thailand October Camp, 4
Find all pairs $(p, n)$ with $n>p$, consisting of a positive integer $n$ and a prime $p$, such that $n^{n-p}$ is an $n$-th power of a positive integer.
2004 Dutch Mathematical Olympiad, 1
Determine the number of pairs of positive integers $(a, b)$, with $a \le b$, for which lcm $(a, b) = 2004$.
lcm ($a, b$) means the least common multiple of $a$ and $b$. Example: lcm $(18, 24) = 72$.
1988 Tournament Of Towns, (191) 4
(a) Two identical cogwheels with $14$ teeth each are given . One is laid horizontally on top of the other in such a way that their teeth coincide (thus the projections of the teeth on the horizontal plane are identical ) . Four pairs of coinciding teeth are cut off. Is it always possible to rotate the two cogwheels with respect to each other so that their common projection looks like that of an entire cogwheel?
(The cogwheels may be rotated about their common axis, but not turned over.)
(b) Answer the same question , but with two $13$-tooth cogwheels and four pairs of cut-off teeth.
2019 Kosovo Team Selection Test, 3
Prove that there exist infinitely many positive integers $n$ such that $\frac{4^n+2^n+1}{n^2+n+1}$ is a positive integer.
2006 India Regional Mathematical Olympiad, 6
Prove that there are infinitely many positive integers $ n$ such that $ n(n\plus{}1)$ can be represented as a sum of two positive squares in at least two different ways. (Here $ a^{2}\plus{}b^{2}$ and $ b^{2}\plus{}a^{2}$ are considered as the same representation.)
2023 LMT Spring, 3
Phoenix is counting positive integers starting from $1$. When he counts a perfect square greater than $1$, he restarts at $1$, skipping that square the next time. For example, the first $10$ numbers Phoenix counts are $1$, $2$, $3$, $4$, $1$, $2$, $3$, $5$, $6$, $7$, $...$ How many numbers will Phoenix have counted after counting 1$00$ for the first time?
DMM Team Rounds, 2008
[b]p1.[/b] $ABCD$ is a convex quadrilateral such that $AB = 20$, $BC = 24$, $CD = 7$, $DA = 15$, and $\angle DAB$ is a right angle. What is the area of $ABCD$?
[b]p2.[/b] A triangular number is one that can be written in the form $1 + 2 +...·+n$ for some positive number $n$. $ 1$ is clearly both triangular and square. What is the next largest number that is both triangular and square?
[b]p3.[/b] Find the last (i.e. rightmost) three digits of $9^{2008}$.
[b]p4.[/b] When expressing numbers in a base $b \ge 11$, you use letters to represent digits greater than $9$. For example, $A$ represents $10$ and $B$ represents $11$, so that the number $110$ in base $10$ is $A0$ in base $11$. What is the smallest positive integer that has four digits when written in base $10$, has at least one letter in its base $12$ representation, and no letters in its base $16$ representation?
[b]p5.[/b] A fly starts from the point $(0, 16)$, then flies straight to the point $(8, 0)$, then straight to the point $(0, -4)$, then straight to the point $(-2, 0)$, and so on, spiraling to the origin, each time intersecting the coordinate axes at a point half as far from the origin as its previous intercept. If the fly flies at a constant speed of $2$ units per second, how many seconds will it take the fly to reach the origin?
[b]p6.[/b] A line segment is divided into two unequal lengths so that the ratio of the length of the short part to the length of the long part is the same as the ratio of the length of the long part to the length of the whole line segment. Let $D$ be this ratio. Compute $$D^{-1} + D^{[D^{-1}+D^{(D^{-1}+D^2)}]}.$$
[b]p7.[/b] Let $f(x) = 4x + 2$. Find the ordered pair of integers $(P, Q)$ such that their greatest common divisor is $1, P$ is positive, and for any two real numbers $a$ and $b$, the sentence:
“$P a + Qb \ge 0$”
is true if and only if the following sentence is true:
“For all real numbers x, if $|f(x) - 6| < b$, then $|x - 1| < a$.”
[b]p8.[/b] Call a rectangle “simple” if all four of its vertices have integers as both of their coordinates and has one vertex at the origin. How many simple rectangles are there whose area is less than or equal to $6$?
[b]p9.[/b] A square is divided into eight congruent triangles by the diagonals and the perpendicular bisectors of its sides. How many ways are there to color the triangles red and blue if two ways that are reflections or rotations of each other are considered the same?
[b]p10.[/b] In chess, a knight can move by jumping to any square whose center is $\sqrt5$ units away from the center of the square that it is currently on. For example, a knight on the square marked by the horse in the diagram below can move to any of the squares marked with an “X” and to no other squares. How many ways can a knight on the square marked by the horse in the diagram move to the square with a circle in exactly four moves?
[img]https://cdn.artofproblemsolving.com/attachments/d/9/2ef9939642362182af12089f95836d4e294725.png[/img]
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2002 Junior Balkan Team Selection Tests - Romania, 1
Let $n$ be an even positive integer and let $a, b$ be two relatively prime positive integers.
Find $a$ and $b$ such that $a + b$ is a divisor of $a^n + b^n$.
2020 APMO, 3
Determine all positive integers $k$ for which there exist a positive integer $m$ and a set $S$ of positive integers such that any integer $n > m$ can be written as a sum of distinct elements of $S$ in exactly $k$ ways.
2024 Belarus Team Selection Test, 4.1
Six integers $a,b,c,d,e,f$ satisfy:
$\begin{cases}
ace+3ebd-3bcf+3adf=5 \\
bce+acf-ade+3bdf=2
\end{cases}$
Find all possible values of $abcde$
[i]D. Bazyleu[/i]
2018 Hanoi Open Mathematics Competitions, 4
A pyramid of non-negative integers is constructed as follows
(a) The first row consists of only $0$,
(b) The second row consists of $1$ and $1$,
(c) The $n^{th}$ (for $n > 2$) is an array of $n$ integers among which the left most and right most elements are equal to $n - 1$ and the interior numbers are equal to the sum of two adjacent numbers from the $(n - 1)^{th}$ row (see Figure).
Let $S_n$ be the sum of numbers in row $n^{th}$. Determine the remainder when dividing $S_{2018}$ by $2018$:
A. $2$ B. $4$ C. $6$ D. $11$ E. $17$
2013 Bangladesh Mathematical Olympiad, 5
Higher Secondary P5
Let $x>1$ be an integer such that for any two positive integers $a$ and $b$, if $x$ divides $ab$ then $x$ either divides $a$ or divides $b$. Find with proof the number of positive integers that divide $x$.
2001 Moldova Team Selection Test, 5
Find $ a,b,c \in N$ such that $ ab$ divides $ a^2\plus{}b^2\plus{}1$.
2021 Tuymaada Olympiad, 7
A pile contains $2021^{2021}$ stones. In a move any pile can be divided into two piles so that the numbers of stones in them differ by a power
of $2$ with non-negative integer exponent. After some move it turned out that the number of stones in each pile is a power of $2$ with non-negative integer exponent. Prove that the number of moves performed was even.
1999 Brazil National Olympiad, 2
Show that, if $\sqrt{2}$ is written in decimal notation, there is at least one nonzero digit at the interval of 1,000,000-th and 3,000,000-th digits.
2013 Saint Petersburg Mathematical Olympiad, 7
Let $a_1,a_2$ - two naturals, and $1<b_1<a_1,1<b_2<a_2$ and $b_1|a_1,b_2|a_2$. Prove that $a_1b_1+a_2b_2-1$ is not divided by $a_1a_2$
2014 Spain Mathematical Olympiad, 2
Let $M$ be the set of all integers in the form of $a^2+13b^2$, where $a$ and $b$ are distinct itnegers.
i) Prove that the product of any two elements of $M$ is also an element of $M$.
ii) Determine, reasonably, if there exist infinite pairs of integers $(x,y)$ so that $x+y\not\in M$ but $x^{13}+y^{13}\in M$.
2024 UMD Math Competition Part II, #2
Consider a set $S = \{a_1, \ldots, a_{2024}\}$ consisting of $2024$ distinct positive integers that satisfies the following property:
[center] "For every positive integer $m < 2024,$ the sum of no $m$ distinct elements of $S$ is a multiple of $2024.$" [/center]
Prove $a_1, \ldots, a_{2024}$ all leave the same remainder when divided by $2024.$ Justify your answer.
2010 Turkey Team Selection Test, 1
Let $0 \leq k < n$ be integers and $A=\{a \: : \: a \equiv k \pmod n \}.$ Find the smallest value of $n$ for which the expression
\[ \frac{a^m+3^m}{a^2-3a+1} \]
does not take any integer values for $(a,m) \in A \times \mathbb{Z^+}.$
2025 Romania EGMO TST, P2
Let $m$ and $n$ be positive integers with $m > n \ge 2$. Set $S =\{1,2,...,m\}$, and set $T = \{a_1,a_2,...,a_n\}$ is a subset of $S$ such that every element of $S$ is not divisible by any pair of distinct elements of $T$. Prove that
$$\frac{1}{a_1}+\frac{1}{a_2}+ ...+ \frac{1}{a_n} < \frac{m+n}{m}$$
2010 Belarus Team Selection Test, 6.1
Let $f$ be a non-constant function from the set of positive integers into the set of positive integer, such that $a-b$ divides $f(a)-f(b)$ for all distinct positive integers $a$, $b$. Prove that there exist infinitely many primes $p$ such that $p$ divides $f(c)$ for some positive integer $c$.
[i]Proposed by Juhan Aru, Estonia[/i]
2015 Moldova Team Selection Test, 3
Let $p$ be a fixed odd prime. Find the minimum positive value of $E_{p}(x,y) = \sqrt{2p}-\sqrt{x}-\sqrt{y}$ where $x,y \in \mathbb{Z}_{+}$.
2005 All-Russian Olympiad, 3
Positive integers $x>1$ and $y$ satisfy an equation $2x^2-1=y^{15}$. Prove that 5 divides $x$.
2021 Tuymaada Olympiad, 6
Given are real $y>1$ and positive integer $n \leq y^{50}$ such that all prime divisors of $n$ do not exceed $y$. Prove that $n$ is a product of $99$ positive integer factors (not necessarily primes) not exceeding $y$.