Found problems: 85335
2020 BMT Fall, 10
How many integers $100 \le x \le 999$ have the property that, among the six digits in $\lfloor 280 + \frac{x}{100} \rfloor$ and $x$, exactly two are identical?
2012 Online Math Open Problems, 21
A game is played with 16 cards laid out in a row. Each card has a black side and a red side, and initially the face-up sides of the cards alternate black and red with the leftmost card black-side-up. A move consists of taking a consecutive sequence of cards (possibly only containing 1 card) with leftmost card black-side-up and the rest of the cards red-side-up, and flipping all of these cards over. The game ends when a move can no longer be made. What is the maximum possible number of moves that can be made before the game ends?
[i]Ray Li.[/i]
[size=85][i]See a close variant [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=810&t=500913]here[/url].[/i][/size]
2015 Balkan MO Shortlist, N3
Let $a$ be a positive integer. For all positive integer n, we define $ a_n=1+a+a^2+\ldots+a^{n-1}. $
Let $s,t$ be two different positive integers with the following property:
If $p$ is prime divisor of $s-t$, then $p$ divides $a-1$.
Prove that number $\frac{a_{s}-a_{t}}{s-t}$ is an integer.
(FYROM)
2024 HMNT, 13
Let $f$ and $g$ be two quadratic polynomials with real coefficients such that the equation $f(g(x)) = 0$ has four distinct real solutions: $112, 131, 146,$ and $a.$ Compute the sum of all possible values of $a.$
2011 Tokio University Entry Examination, 5
Let $p,\ q$ be positive integers. Consider integrs $a,\ b,\ c$ satisfying conditions:
\[-q\leq b\leq 0\leq a\leq p,\ \ b\leq c\leq a\]
Call such $a,\ b,\ c$ in arranging in the form of $[a,\ b\ ;\ c]$ as $(p,\ q)$ pattern.
For each $(p,\ q)$ pattern $[a,\ b\ ;\ c]$, let $w([a,\ b\ ;\ c])=p-q-(a+b)$.
(1) Find the number of $(p,\ q)$ pattern such that $w([a,\ b\ ;\ c])=-q$, then find the number of $(p,\ q)$ pattern such that $w([a,\ b\ ;\ c])=p$.
From now on, we consider the case of $p=q$.
(2) Let $s$ be integer. Find the number of $(p,\ p)$ pattern such that $w([a,\ b\ ;\ c])=-p+s$.
(3) Find the total number of $(p,\ p)$ pattern.
[i]2011 Tokyo University entrance exam/Science, Problem 5[/i]
2012 BMT Spring, 3
Let $ABC$ be a triangle with side lengths $AB = 2011$, $BC = 2012$, $AC = 2013$. Create squares $S_1 =ABB'A''$, $S_2 = ACC''A'$ , and $S_3 = CBB''C'$ using the sides $AB$, $AC$, $BC$ respectively, so that the side $B'A''$ is on the opposite side of $AB$ from $C$, and so forth. Let square $S_4$ have side length $A''A' $, square $S_5$ have side length $C''C'$, and square $S_6$ have side length $B''B'$. Let $A(S_i)$ be the area of square $S_i$ . Compute $\frac{A(S_4)+A(S_5)+A(S_6)}{A(S_1)+A(S_2)+A(S_3)}$?
2001 Argentina National Olympiad, 4
Find all positive integers $k$ that can be expressed as the sum of $50$ fractions such that the numerators are the $50$ natural numbers from $1$ to $50$ and the denominators are positive integers, that is, $k = \dfrac{1}{a_1} + \dfrac{2}{a_2} + \ldots + \dfrac{50}{a_{50}}$ with a$_1 , a_2 , \ldots , a_n$ positive integers.
2002 Flanders Math Olympiad, 4
A lamp is situated at point $A$ and shines inside the cube. A (massive) square is hung on the midpoints of the 4 vertical faces. What's the area of its shadow?
[img]http://www.mathlinks.ro/Forum/album_pic.php?pic_id=285[/img]
2024 Sharygin Geometry Olympiad, 5
Points $A', B', C'$ are the reflections of vertices $A, B, C$ about the opposite sidelines of triangle $ABC$. Prove that the circles $AB'C', A'BC',$ and $A'B'C$ have a common point.
2025 AIME, 5
There are $8!= 40320$ eight-digit positive integers that use each of the digits 1, 2, 3, 4, 5, 6, 7, 8 exactly once. Let N be the number of these integers that are divisible by $22$. Find the difference between $N$ and 2025.
1999 Portugal MO, 1
A number is said to be [i]balanced [/i] if one of its digits is average of the others. How many [i]balanced [/i]$3$-digit numbers are there?
2011 Mongolia Team Selection Test, 3
We are given an acute triangle $ABC$. Let $(w,I)$ be the inscribed circle of $ABC$, $(\Omega,O)$ be the circumscribed circle of $ABC$, and $A_0$ be the midpoint of altitude $AH$. $w$ touches $BC$ at point $D$. $A_0 D$ and $w$ intersect at point $P$, and the perpendicular from $I$ to $A_0 D$ intersects $BC$ at the point $M$. $MR$ and $MS$ lines touch $\Omega$ at $R$ and $S$ respectively [note: I am not entirely sure of what is meant by this, but I am pretty sure it means draw the tangents to $\Omega$ from $M$]. Prove that the points $R,P,D,S$ are concyclic.
(proposed by E. Enkzaya, inspired by Vietnamese olympiad problem)
2002 Austrian-Polish Competition, 7
Find all real functions $f$ definited on positive integers and satisying:
(a) $f(x+22)=f(x)$,
(b) $f\left(x^{2}y\right)=\left(f(x)\right)^{2}f(y)$
for all positive integers $x$ and $y$.
2006 IberoAmerican Olympiad For University Students, 4
Prove that for any interval $[a,b]$ of real numbers and any positive integer $n$ there exists a positive integer $k$ and a partition of the given interval
\[a = x (0) < x (1) < x (2) < \cdots < x (k-1) < x (k) = b\]
such that
\[\int_{x(0)}^{x(1)}f(x)dx+\int_{x(2)}^{x(3)}f(x)dx+\cdots=\int_{x(1)}^{x(2)}f(x)dx+\int_{x(3)}^{x(4)}f(x)dx+\cdots\]
for all polynomials $f$ with real coefficients and degree less than $n$.
1971 AMC 12/AHSME, 5
Points $A,B,Q,D,$ and $C$ lie on the circle shown and the measures of arcs $\widehat{BQ}$ and $\widehat{QD}$ are $42^\circ$ and $38^\circ$ respectively. The sum of the measures of angles $P$ and $Q$ is
$\textbf{(A) }80^\circ\qquad\textbf{(B) }62^\circ\qquad\textbf{(C) }40^\circ\qquad\textbf{(D) }46^\circ\qquad \textbf{(E) }\text{None of these}$
[asy]
size(3inch);
draw(Circle((1,0),1));
pair A, B, C, D, P, Q;
P = (-2,0);
B=(sqrt(2)/2+1,sqrt(2)/2);
D=(sqrt(2)/2+1,-sqrt(2)/2);
Q = (2,0);
A = intersectionpoints(Circle((1,0),1),B--P)[1];
C = intersectionpoints(Circle((1,0),1),D--P)[0];
draw(B--P--D);
draw(A--Q--C);
label("$A$",A,NW);
label("$B$",B,NE);
label("$C$",C,SW);
label("$D$",D,SE);
label("$P$",P,W);
label("$Q$",Q,E);
//Credit to chezbgone2 for the diagram[/asy]
PEN M Problems, 14
Let $x_{1}$ and $x_{2}$ be relatively prime positive integers. For $n \ge 2$, define $x_{n+1}=x_{n}x_{n-1}+1$.[list=a][*] Prove that for every $i>1$, there exists $j>i$ such that ${x_{i}}^{i}$ divides ${x_{j}}^{j}$. [*] Is it true that $x_{1}$ must divide ${x_{j}}^{j}$ for some $j>1$? [/list]
2013 CHMMC (Fall), 4
The numbers $25$ and $76$ have the property that when squared in base 10, their squares also end in the same two digits. A positive integer that has at most $3$ digits when expressed in base 21 and also has the property that its base $21$ square ends in the same $3$ digits is called amazing. Find the sum of all amazing numbers. Express your answer in base $21$.
2021 Girls in Math at Yale, R3
7. Peggy picks three positive integers between $1$ and $25$, inclusive, and tells us the following information about those numbers:
[list]
[*] Exactly one of them is a multiple of $2$;
[*] Exactly one of them is a multiple of $3$;
[*] Exactly one of them is a multiple of $5$;
[*] Exactly one of them is a multiple of $7$;
[*] Exactly one of them is a multiple of $11$.
[/list]
What is the maximum possible sum of the integers that Peggy picked?
8. What is the largest positive integer $k$ such that $2^k$ divides $2^{4^8}+8^{2^4}+4^{8^2}$?
9. Find the smallest integer $n$ such that $n$ is the sum of $7$ consecutive positive integers and the sum of $12$ consecutive positive integers.
1992 APMO, 5
Find a sequence of maximal length consisting of non-zero integers in which the sum of any seven consecutive terms is positive and that of any eleven consecutive terms is negative.
2022 CMIMC, 2.7 1.3
For a family gathering, $8$ people order one dish each. The family sits around a circular table. Find the number of ways to place the dishes so that each person’s dish is either to the left, right, or directly in front of them.
[i]Proposed by Nicole Sim[/i]
2001 Estonia National Olympiad, 1
A convex $n$-gon has exactly three obtuse interior angles. Find all possible values of $n$.
2005 Taiwan TST Round 3, 2
Find all primes $p$ such that the number of distinct positive factors of $p^2+2543$ is less than 16.
2017 Online Math Open Problems, 2
A positive integer $n$ is called[i] bad [/i]if it cannot be expressed as the product of two distinct positive integers greater than $1$. Find the number of bad positive integers less than $100. $
[i]Proposed by Michael Ren[/i]
2016 Azerbaijan JBMO TST, 4
There are three stacks of tokens on the table: the first contains $a,$ the second contains $b,$ and the third contains $c$ where $a \ge b \ge c.$ Players $A$ and $B$ take turns playing a game of swapping tokens. $A$ starts first. On each turn, the player who gets his turn chooses two stacks, then takes at least one token from the stack with the lowest number of tokens and places them on the stack with the highest number of tokens. If the number of tokens in the two piles he/she chooses is equal, then he/she takes at least one token from any of them and puts it in the other. If only one pile remains after a player's move, that player is considered a winner. At what values of $a, b, c$ who has the winning strategy ($A$ or $B$)?
2012 Thailand Mathematical Olympiad, 12
Let $a, b, c$ be positive integers. Show that if $\frac{a}{b} +\frac{b}{c} +\frac{c}{a}$ is an integer then $abc$ is a perfect cube.