Found problems: 15460
2022 Canadian Junior Mathematical Olympiad, 4
I think we are allowed to discuss since its after 24 hours
How do you do this
Prove that $d(1)+d(3)+..+d(2n-1)\leq d(2)+d(4)+...d(2n)$ which $d(x)$ is the divisor function
2007 Dutch Mathematical Olympiad, 4
Determine the number of integers $a$ satisfying $1 \le a \le 100$ such that $a^a$ is a perfect square.
(And prove that your answer is correct.)
1980 Poland - Second Round, 5
We print the terms of the sequence $ (n_1, n_2, \ldots, n_k) $, where $ n_1 = 1000 $, and $ n_j $ for $ j > 1 $ is an integer selected randomly from the range $ [0, n_{j-1 } - 1] $ (each number in this range is equally likely to be selected). We stop printing when the selected number is zero, i.e. $ n_{k-1} $, $ n_k = 0 $, The length $ k $ of the sequence $ (n_1, n_2, \ldots, n_k) $ is a random variable. Prove that the expected value of this random variable is greater than 7.
2018 Peru Iberoamerican Team Selection Test, P5
Find all positive integers $a, b$, and $c$ such that the numbers
$$\frac{a+1}{b}, \frac{b+1}{c} \quad \text{and} \quad \frac{c+1}{a}$$
are positive integers.
2004 Thailand Mathematical Olympiad, 11
Find the number of positive integer solutions to $(x_1 + x_2 + x_3)(y_1 + y_2 + y_3 + y_4) = 91$
2019 Saudi Arabia Pre-TST + Training Tests, 2.1
Let be given a positive integer $n \ge 3$. Consider integers $a_1,a_2,...,a_n >1$ with the product equals to $A$ such that: for each $k \in \{1, 2,..., n\}$ then the remainder when $\frac{A}{a_k}$ divided by $a_k$ are all equal to $r$. Prove that $r \le n- 2$
2024 UMD Math Competition Part I, #22
For how many angles $x$, in radians, satisfying $0\le x<2\pi$ do we have $\sin(14x)=\cos(68x)$?
\[\rm a. ~128\qquad \mathrm b. ~130\qquad \mathrm c. ~132 \qquad\mathrm d. ~134\qquad\mathrm e. ~136\]
2015 India National Olympiad, 6
Show that from a set of $11$ square integers one can select six numbers $a^2,b^2,c^2,d^2,e^2,f^2$ such that $a^2+b^2+c^2 \equiv d^2+e^2+f^2\pmod{12}$.
PEN R Problems, 10
Prove that if a lattice triangle has no lattice points on its boundary in addition to its vertices, and one point in its interior, then this interior point is its center of gravity.
2004 Vietnam National Olympiad, 3
Let $ S(n)$ be the sum of decimal digits of a natural number $ n$. Find the least value of $ S(m)$ if $ m$ is an integral multiple of $ 2003$.
1990 Tournament Of Towns, (251) 5
Find the number of pairs $(m, n)$ of positive integers, both of which are $\le 1000$, such that $\frac{m}{n+1}< \sqrt2 < \frac{m+1}{n}$
(recalling that $ \sqrt2 = 1.414213..$.).
(D. Fomin, Leningrad)
1997 Moldova Team Selection Test, 1
Let $a$ and $b$ be two odd positive integers. Define the sequence $(x_n)_{n\in\mathbb{N}}$ as such: $x_1=a, x_2=b,$ for every $n\geq3$ the term $x_n{}$ is the greatest odd integer of $x_{n-1}+x_{n-2}$. Show that starting with a term, all the following terms are constant.
2017 China Team Selection Test, 5
Show that there exists a positive real $C$ such that for any naturals $H,N$ satisfying $H \geq 3, N \geq e^{CH}$, for any subset of $\{1,2,\ldots,N\}$ with size $\lceil \frac{CHN}{\ln N} \rceil$, one can find $H$ naturals in it such that the greatest common divisor of any two elements is the greatest common divisor of all $H$ elements.
1988 Iran MO (2nd round), 3
Let $n$ be a positive integer. $1369^n$ positive rational numbers are given with this property: if we remove one of the numbers, then we can divide remain numbers into $1368$ sets with equal number of elements such that the product of the numbers of the sets be equal. Prove that all of the numbers are equal.
2022 Junior Balkan Team Selection Tests - Romania, P3
Let $p_i$ denote the $i^{\text{th}}$ prime number. For any positive integer $k$ let $a_k$ denote the number of positive integers $t$ such that $p_tp_{t+1}$ divides $k.$ Let $n$ be an arbitrary positive integer. Prove that \[a_1+a_2+\cdots+a_n<\frac{n}{3}.\]
2023 USAMTS Problems, 3
Lizzie and Alex are playing a game on the whiteboard. Initially, $n$ twos are written
on the board. On a player’s turn they must either
1. change any single positive number to 0, or
2. subtract one from any positive number of positive numbers on the board.
The game ends once all numbers are 0, and the last player who made a move wins. If Lizzie
always plays first, find all $n$ for which Lizzie has a winning strategy.
2019 Estonia Team Selection Test, 9
Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.
MMATHS Mathathon Rounds, 2018
[u]Round 1[/u]
[b]p1.[/b] Elaine creates a sequence of positive integers $\{s_n\}$. She starts with $s_1 = 2018$. For $n \ge 2$, she sets $s_n =\frac12 s_{n-1}$ if $s_{n-1}$ is even and $s_n = s_{n-1} + 1$ if $s_{n-1}$ is odd. Find the smallest positive integer $n$ such that $s_n = 1$, or submit “$0$” as your answer if no such $n$ exists.
[b]p2.[/b] Alice rolls a fair six-sided die with the numbers $1$ through $6$, and Bob rolls a fair eight-sided die with the numbers $1$ through $8$. Alice wins if her number divides Bob’s number, and Bob wins otherwise. What is the probability that Alice wins?
[b]p3.[/b] Four circles each of radius $\frac14$ are centered at the points $\left( \pm \frac14, \pm \frac14 \right)$, and ther exists a fifth circle is externally tangent to these four circles. What is the radius of this fifth circle?
[u]Round 2 [/u]
[b]p4.[/b] If Anna rows at a constant speed, it takes her two hours to row her boat up the river (which flows at a constant rate) to Bob’s house and thirty minutes to row back home. How many minutes would it take Anna to row to Bob’s house if the river were to stop flowing?
[b]p5.[/b] Let $a_1 = 2018$, and for $n \ge 2$ define $a_n = 2018^{a_{n-1}}$ . What is the ones digit of $a_{2018}$?
[b]p6.[/b] We can write $(x + 35)^n =\sum_{i=0}^n c_ix^i$ for some positive integer $n$ and real numbers $c_i$. If $c_0 = c_2$, what is $n$?
[u]Round 3[/u]
[b]p7.[/b] How many positive integers are factors of $12!$ but not of $(7!)^2$?
[b]p8.[/b] How many ordered pairs $(f(x), g(x))$ of polynomials of degree at least $1$ with integer coefficients satisfy $f(x)g(x) = 50x^6 - 3200$?
[b]p9.[/b] On a math test, Alice, Bob, and Carol are each equally likely to receive any integer score between $1$ and $10$ (inclusive). What is the probability that the average of their three scores is an integer?
[u]Round 4[/u]
[b]p10.[/b] Find the largest positive integer N such that $$(a-b)(a-c)(a-d)(a-e)(b-c)(b-d)(b-e)(c-d)(c-e)(d-e)$$ is divisible by $N$ for all choices of positive integers $a > b > c > d > e$.
[b]p11.[/b] Let $ABCDE$ be a square pyramid with $ABCD$ a square and E the apex of the pyramid. Each side length of $ABCDE$ is $6$. Let $ABCDD'C'B'A'$ be a cube, where $AA'$, $BB'$, $CC'$, $DD'$ are edges of the cube. Andy the ant is on the surface of $EABCDD'C'B'A'$ at the center of triangle $ABE$ (call this point $G$) and wants to crawl on the surface of the cube to $D'$. What is the length the shortest path from $G$ to $D'$? Write your answer in the form $\sqrt{a + b\sqrt3}$, where $a$ and $b$ are positive integers.
[b]p12.[/b] A six-digit palindrome is a positive integer between $100, 000$ and $999, 999$ (inclusive) which is the same read forwards and backwards in base ten. How many composite six-digit palindromes are there?
PS. You should use hide for answers. Rounds 5-7 have been posted [url=https://artofproblemsolving.com/community/c4h2784943p24473026]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2017 All-Russian Olympiad, 5
$n$ is composite. $1<a_1<a_2<...<a_k<n$ - all divisors of $n$. It is known, that $a_1+1,...,a_k+1$ are all divisors for some $m$ (except $1,m$). Find all such $n$.
2014 India PRMO, 11
For natural numbers $x$ and $y$, let $(x,y)$ denote the greatest common divisor of $x$ and $y$. How many pairs of natural numbers $x$ and $y$ with $x \le y$ satisfy the equation $xy = x + y + (x, y)$?
1957 Moscow Mathematical Olympiad, 364
(a) Prove that the number of all digits in the sequence $1, 2, 3,... , 10^8$ is equal to the number of all zeros in the sequence $1, 2, 3, ... , 10^9$.
(b) Prove that the number of all digits in the sequence $1, 2, 3, ... , 10^k$ is equal to the number of all zeros in the sequence $1, 2, 3, ... , 10^{k+1}$.
1976 IMO Longlists, 16
Prove that there is a positive integer $n$ such that the decimal representation of $7^n$ contains a block of at least $m$ consecutive zeros, where $m$ is any given positive integer.
1990 IMO Longlists, 93
Let $ {\mathbb Q}^ \plus{}$ be the set of positive rational numbers. Construct a function $ f : {\mathbb Q}^ \plus{} \rightarrow {\mathbb Q}^ \plus{}$ such that
\[ f(xf(y)) \equal{} \frac {f(x)}{y}
\]
for all $ x$, $ y$ in $ {\mathbb Q}^ \plus{}$.
2019 Taiwan APMO Preliminary Test, P2
Put $1,2,....,2018$ (2018 numbers) in a row randomly and call this number $A$. Find the remainder of $A$ divided by $3$.
1991 AIME Problems, 6
Suppose $r$ is a real number for which \[ \left\lfloor r + \frac{19}{100} \right\rfloor + \left\lfloor r + \frac{20}{100} \right\rfloor + \left\lfloor r + \frac{21}{100} \right\rfloor + \cdots + \left\lfloor r + \frac{91}{100} \right\rfloor = 546. \] Find $\lfloor 100r \rfloor$. (For real $x$, $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.)