Found problems: 85335
2014 Saudi Arabia Pre-TST, 2.1
Prove that $2014$ divides $53n^{55}- 57n^{53} + 4n$ for all integer $n$.
1984 Kurschak Competition, 1
Writing down the first $4$ rows of the Pascal triangle in the usual way and then adding up the numbers in vertical columns, we obtain $7$ numbers as shown above. If we repeat this procedure with the first $1024$ rows of the Pascal triangle, how many of the $2047$ numbers thus obtained will be odd?
[img]https://cdn.artofproblemsolving.com/attachments/8/a/4dc4a815d8b002c9f36a6da7ad6e1c11a848e9.png[/img]
2015 BMT Spring, P1
Find two disjoint sets $N_1$ and $N_2$ with $N_1\cup N_2=\mathbb N$, so that neither set contains an infinite arithmetic progression.
1988 Tournament Of Towns, (178) 4
Pawns are placed on an infinite chess board so that they form an infinite square net (along any row or column containing pawns ther is a pawn , three free squares , pawn , three squares, and so on , with only every fourth row and every fourth column containing pawns). Prove that it is not possible for a knight to tour every free square once and only once.
(An old problem of A . K . Tolpugo)
2013 Indonesia MO, 3
Determine all positive real $M$ such that for any positive reals $a,b,c$, at least one of $a + \dfrac{M}{ab}, b + \dfrac{M}{bc}, c + \dfrac{M}{ca}$ is greater than or equal to $1+M$.
2023 CMIMC Team, 9
A positive integer $N$ is a [i]triple-double[/i] if there exists non-negative integers $a$, $b$, $c$ such that $2^a + 2^b + 2^c = N$. How many three-digit numbers are triple-doubles?
[i]Proposed by Giacomo Rizzo[/i]
2021 AMC 10 Fall, 2
What is the area of the shaded figure shown below?
[asy]
size(200);
defaultpen(linewidth(0.4)+fontsize(12));
pen s = linewidth(0.8)+fontsize(8);
pair O,X,Y;
O = origin;
X = (6,0);
Y = (0,5);
fill((1,0)--(3,5)--(5,0)--(3,2)--cycle, palegray+opacity(0.2));
for (int i=1; i<7; ++i)
{
draw((i,0)--(i,5), gray+dashed);
label("${"+string(i)+"}$", (i,0), 2*S);
if (i<6)
{
draw((0,i)--(6,i), gray+dashed);
label("${"+string(i)+"}$", (0,i), 2*W);
}
}
label("$0$", O, 2*SW);
draw(O--X+(0.15,0), EndArrow);
draw(O--Y+(0,0.15), EndArrow);
draw((1,0)--(3,5)--(5,0)--(3,2)--(1,0), black+1.5);
[/asy]
2023 CMIMC TCS, 3
You are given a deck of $n \cdot m$ different cards where $n$ and $m$ are fixed numbers both between $10^{99}$ and $10^{100}$. You perform an [i]$m$-perfect shuffle[/i] for some times. In a single $m$-perfect shuffle, you divide the deck into $m$ piles with $n$ consecutive cards in each pile. You take one card from each pile, in order of the piles, for $n$ times to form the new deck. (The $m$-perfect shuffle is deterministic)
For example, if the cards are labeled 12345678 where $n=4$ and $m=2$, you divide the deck into 1234 and 5678, and after one $2$-perfect shuffle you get 15263748. In another example, if the cards are labeled 123456789 where $n=3$ and $m=3$, you divide the deck into 123, 456, and 789, and after one $3$-perfect shuffle you get 147258369.
Find an algorithm that, in at most $k$ steps, outputs the smallest positive number of $m$-perfect shuffle after which the deck is exactly the same as the original deck. In each step, you can do one arithmetic operation in $\{+, -, *, /, \bmod\}$, do one comparison, break out of a loop, or store one number to a specific location of an array. You can use the following precomputed numbers of steps in your solution:
[list]
[*] Checking if $a$ divides $b$ for any two integers $a$ and $b$ takes 2 steps because you need to compute $b \bmod a$ then compare with $0$.
[*]A loop over $k$ iterations takes $2k$ steps because you need to increment the loop index by $1$ $k$ times and check the loop guard $k$ times.
[*]Simulating one "$m$-perfect shuffle" takes $7nm$ steps because there is one loop index increment, four arithmetic operations, and one store in each iteration of the loop.
[/list]
[b]Scoring:[/b]
An algorithm that completes in at most $k$ steps will be awarded:
[list]
[*] 1 pt for $k > 10^{10^{10^{10}}}$
[*] 10 pts for $k = 10^{10^{10^{10}}}$
[*] 20 pts for $k = 10^{420}$
[*] 30 pts for $k = 10^{360}$
[*] 50 pts for $k = 10^{240}$
[*] 70 pts for $k = 10^{202}$
[*] 80 pts for $k = 10^{201}$
[*] 95 pts for $k = 10^{120}$
[*] 98 pts for $k = 10^{102}$
[*] 100 pts for $k = 10^{101}$
[/list]
[i]Proposed by Mingkuan Xu[/i]
2022 IFYM, Sozopol, 5
Find the number of subsets of $\{1, 2,... , 2100\}$ such that each has sum of the elements giving a remainder of $3$ when divided by $7$.
2014 IFYM, Sozopol, 8
Let $c>1$ be a real constant. For the sequence $a_1,a_2,...$ we have: $a_1=1$, $a_2=2$,
$a_{mn}=a_m a_n$, and $a_{m+n}\leq c(a_m+a_n)$. Prove that $a_n=n$.
1976 Bulgaria National Olympiad, Problem 4
Let $0<x_1\le x_2\le\ldots\le x_n$. Prove that
$$\frac{x_1}{x_2}+\frac{x_2}{x_3}+\ldots+\frac{x_{n-1}}{x_n}+\frac{x_n}{x_1}\ge\frac{x_2}{x_1}+\frac{x_3}{x_2}+\ldots+\frac{x_n}{x_{n-1}}+\frac{x_1}{x_n}$$
[i]I. Tonov[/i]
1972 Bundeswettbewerb Mathematik, 4
Which natural numbers cannot be presented in that way: $[n+\sqrt{n}+\frac{1}{2}]$, $n\in\mathbb{N}$
$[y]$ is the greatest integer function.
2017 Hanoi Open Mathematics Competitions, 15
Show that an arbitrary quadrilateral can be divided into nine isosceles triangles.
2013 Bosnia Herzegovina Team Selection Test, 6
In triangle $ABC$, $I$ is the incenter. We have chosen points $P,Q,R$ on segments $IA,IB,IC$ respectively such that $IP\cdot IA=IQ \cdot IB=IR\cdot IC$.
Prove that the points $I$ and $O$ belong to Euler line of triangle $PQR$ where $O$ is circumcenter of $ABC$.
1985 AMC 12/AHSME, 1
If $ 2x \plus{} 1 \equal{} 8$, then $ 4x \plus{} 1 \equal{}$
$ \textbf{(A)}\ 15 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 17 \qquad \textbf{(D)}\ 18 \qquad \textbf{(E)}\ 19$
1995 India Regional Mathematical Olympiad, 2
Call a positive integer $n$ [i]good[/i] if there are $n$ integers, positive or negative, and not necessarily distinct, such that their sum and products are both equal to $n$. Show that the integers of the form $4k+1$ and $4l$ are good.
1993 Bundeswettbewerb Mathematik, 2
For the real number $a$ it holds that there is exactly one square whose vertices are all on the graph with the equation $y = x^3 + ax$. Find the side length of this square.
1954 Moscow Mathematical Olympiad, 272
Find all real solutions of the equation $x^2 + 2x \sin (xy) + 1 = 0$.
2023 Greece National Olympiad, 1
Find all quadruplets (x, y, z, w) of positive real numbers that satisfy the following system:
$\begin{cases}
\frac{xyz+1}{x+1}= \frac{yzw+1}{y+1}= \frac{zwx+1}{z+1}= \frac{wxy+1}{w+1}\\
x+y+z+w= 48
\end{cases}$
1975 IMO, 2
Let $a_{1}, \ldots, a_{n}$ be an infinite sequence of strictly positive integers, so that $a_{k} < a_{k+1}$ for any $k.$ Prove that there exists an infinity of terms $ a_{m},$ which can be written like $a_m = x \cdot a_p + y \cdot a_q$ with $x,y$ strictly positive integers and $p \neq q.$
1973 Bulgaria National Olympiad, Problem 3
Let $a_1,a_2,\ldots,a_n$ are different integer numbers in the range $[100,200]$ for which: $a_1+a_2+\ldots+a_n\ge11100$. Prove that it can be found at least number from the given in the representation of decimal system on which there are at least two equal digits.
[i]L. Davidov[/i]
2003 Spain Mathematical Olympiad, Problem 3
The altitudes of the triangle ${ABC}$ meet in the point ${H}$. You know that ${AB = CH}$. Determine the value of the angle $\widehat{BCA}$.
1969 IMO Longlists, 25
$(GBR 2)$ Let $a, b, x, y$ be positive integers such that $a$ and $b$ have no common divisor greater than $1$. Prove that the largest number not expressible in the form $ax + by$ is $ab - a - b$. If $N(k)$ is the largest number not expressible in the form $ax + by$ in only $k$ ways, find $N(k).$
MBMT Guts Rounds, 2015.30
Estimate the number of positive integers less than or equal to $1,000,000$ that can be expressed as the sum of two nonnegative integer squares. Your estimate must be an integer, or you will receive a zero.
2021 Ecuador NMO (OMEC), 6
Find all positive integers $a, b, c$ such that $ab+1$ and $c$ are coprimes and:
$$a(ba+1)(ca^2+ba+1)=2021^{2021}$$