Found problems: 85335
1991 AMC 12/AHSME, 19
Triangle $ABC$ has a right angle at $C$, $AC = 3$ and $BC = 4$. Triangle $ABD$ has a right angle at $A$ and $AD = 12$. Points $C$ and $D$ are on opposite sides of $\overline{AB}$. The line through $D$ parallel to $\overline{AC}$ meets $\overline{CB}$ extended at $E$. If $\frac{DE}{DB} = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, then $m + n = $
[asy]
size(170);
defaultpen(fontsize(10pt)+linewidth(.8pt));
pair C=origin, A=(0,3), B=(4,0), D=(7.2,12.6), E=(7.2,0);
draw(A--C--B--A--D--B--E--D);
label("$A$",A,W);
label("$B$",B,S);
label("$C$",C,SW);
label("$D$",D,NE);
label("$E$",E,SE);
[/asy]
$ \textbf{(A)}\ 25\qquad\textbf{(B)}\ 128\qquad\textbf{(C)}\ 153\qquad\textbf{(D)}\ 243\qquad\textbf{(E)}\ 256 $
1996 Nordic, 2
Determine all real numbers $x$, such that $x^n+x^{-n}$ is an integer for all integers $n$.
2011 Harvard-MIT Mathematics Tournament, 5
Let $f(x) = x^2 + 6x + c$ for all real number s$x$, where $c$ is some real number. For what values of $c$ does $f(f(x))$ have exactly $3$ distinct real roots?
2003 Tournament Of Towns, 5
Two players in turn play a game. First Player has cards with numbers $2, 4, \ldots, 2000$ while Second Player has cards with numbers $1, 3, \ldots, 2001$. In each his turn, a player chooses one of his cards and puts it on a table; the opponent sees it and puts his card next to the first one. Player, who put the card with a larger number, scores 1 point. Then both cards are discarded. First Player starts. After $1000$ turns the game is over; First Player has used all his cards and Second Player used all but one. What are the maximal scores, that players could guarantee for themselves, no matter how the opponent would play?
2006 Baltic Way, 4
Let $a,b,c,d,e,f$ be non-negative real numbers satisfying $a+b+c+d+e+f=6$. Find the maximal possible value of
$\color{white}\ .\quad \ \color{black}\ \quad abc+bcd+cde+def+efa+fab$
and determine all $6$-tuples $(a,b,c,d,e,f)$ for which this maximal value is achieved.
2021 Spain Mathematical Olympiad, 2
Given a positive integer $n$, we define $\lambda (n)$ as the number of positive integer solutions of $x^2-y^2=n$. We say that $n$ is [i]olympic[/i] if $\lambda (n) = 2021$. Which is the smallest olympic positive integer? Which is the smallest olympic positive odd integer?
2021 Indonesia TST, C
Anis, Banu, and Cholis are going to play a game. They are given an $n\times n$ board consisting of $n^2$ unit squares, where $n$ is an integer and $n > 5$. In the beginning of the game, the number $n$ is written on each unit square. Then Anis, Banu, and Cholis take turns playing the game, repeatedly in that order, according to the following procedure:
On every turn, an arrangement of $n$ squares on the same row or column is chosen, and every number from the chosen squares is subtracted by $1$. The turn cannot be done if it results in a negative number, that is, no arrangement of $n$ unit squares on the same column or row in which all of its unit squares contain a positive number can be found. The last person to get a turn wins.
Determine which player will win the game.
2010 Canadian Mathematical Olympiad Qualification Repechage, 2
Two tangents $AT$ and $BT$ touch a circle at $A$ and $B$, respectively, and meet perpendicularly at $T$. $Q$ is on $AT$, $S$ is on $BT$, and $R$ is on the circle, so that $QRST$ is a rectangle with $QT = 8$ and $ST = 9$. Determine the radius of the circle.
2022 Taiwan TST Round 3, 1
Let $ABCD$ be a quadrilateral inscribed in a circle $\Omega.$ Let the tangent to $\Omega$ at $D$ meet rays $BA$ and $BC$ at $E$ and $F,$ respectively. A point $T$ is chosen inside $\triangle ABC$ so that $\overline{TE}\parallel\overline{CD}$ and $\overline{TF}\parallel\overline{AD}.$ Let $K\ne D$ be a point on segment $DF$ satisfying $TD=TK.$ Prove that lines $AC,DT,$ and $BK$ are concurrent.
2018 Math Prize for Girls Problems, 20
A [i]smooth number[/i] is a positive integer of the form $2^m 3^n$, where $m$ and $n$ are nonnegative integers. Let $S$ be the set of all triples $(a, b, c)$ where $a$, $b$, and $c$ are smooth numbers such that $\gcd(a, b)$, $\gcd(b, c)$, and $\gcd(c, a)$ are all distinct. Evaluate the infinite sum $\sum_{(a,b,c) \in S} \frac{1}{abc}$. Recall that $\gcd(x, y)$ is the greatest common divisor of $x$ and $y$.
1999 IberoAmerican, 1
Find all the positive integers less than 1000 such that the cube of the sum of its digits is equal to the square of such integer.
2016 CHMMC (Fall), 3
For a positive integer $m$, let $f(m)$ be the number of positive integers $q \le m$ such that $\frac{q^2-4}{m}$ is an integer. How many positive square-free integers $m < 2016$ satisfy $f(m) \ge 16$?
2016 Kyiv Mathematical Festival, P1
Prove that for every positive integers $a$ and $b$ there exist positive integers $x$ and $y$ such that $\dfrac{x}{y+a}+\dfrac{y}{x+b}=\dfrac{3}{2}.$
2020 Iran MO (2nd Round), P6
Divide a circle into $2n$ equal sections. We call a circle [i]filled[/i] if it is filled with the numbers $0,1,2,\dots,n-1$. We call a filled circle [i] good[/i] if it has the following properties:
$i$. Each number $0 \leq a \leq n-1$ is used exactly twice
$ii$. For any $a$ we have that there are exactly $a$ sections between the two sections that have the number $a$ in them.
Here is an example of a good filling for $n=5$ (View attachment)
Prove that there doesn’t exist a good filling for $n=1399$
2007 National Olympiad First Round, 30
Let $(a_n)_{n=1}^{\infty}$ be an integer sequence such that $a_{n+48} \equiv a_n \pmod {35}$ for every $n \geq 1$. Let $i$ and $j$ be the least numbers satisfying the conditions $a_{n+i} \equiv a_n \pmod {5}$ and $a_{n+j} \equiv a_n \pmod {7}$ for every $n\geq 1$. Which one below cannot be an $(i,j)$ pair?
$
\textbf{(A)}\ (16,4)
\qquad\textbf{(B)}\ (3,16)
\qquad\textbf{(C)}\ (8,6)
\qquad\textbf{(D)}\ (1,48)
\qquad\textbf{(E)}\ (16,18)
$
2024 AMC 12/AHSME, 22
Let $\triangle{ABC}$ be a triangle with integer side lengths and the property that $\angle{B} = 2\angle{A}$. What is the least possible perimeter of such a triangle?
$
\textbf{(A) }13 \qquad
\textbf{(B) }14 \qquad
\textbf{(C) }15 \qquad
\textbf{(D) }16 \qquad
\textbf{(E) }17 \qquad
$
2010 Saudi Arabia BMO TST, 4
In quadrilateral $ABCD$, diagonals $AC$ and $BD$ intersect at $O$. Denote by $P, Q, R, S$ the orthogonal projections of $O$ onto $AB$ , $BC$ ,$CD$ , $DA$, respectively. Prove that $$PA \cdot AB + RC \cdot CD =\frac12 (AD^2 + BC^2)$$
if and only if $$QB \cdot BC + SD \cdot DA = \frac12(AB ^2 + CD^2)$$
2014 PUMaC Team, 1
The evilest number $666^{666}$ has $1881$ digits. Let $a$ be the sum of digits of $66^{666}$ and let $b$ be the sum of digits of $a$ and let $c$ be the sum of digits of $b$. Find $c$.
2018 lberoAmerican, 1
For each integer $n \ge 2$, find all integer solutions of the following system of equations:
\[x_1 = (x_2 + x_3 + x_4 + ... + x_n)^{2018}\]
\[x_2 = (x_1 + x_3 + x_4 + ... + x_n)^{2018}\]
\[\vdots\]
\[x_n = (x_1 + x_2 + x_3 + ... + x_{n - 1})^{2018}\]
2008 Junior Balkan Team Selection Tests - Romania, 3
Find all pairs $ (m,n)$ of integer numbers $ m,n > 1$ with property that $ mn \minus{} 1\mid n^3 \minus{} 1$.
2016 IMAR Test, 2
Given a positive integer $n$, does there exist a planar polygon and a point in its plane such that every line through that point meets the boundary of the polygon at exactly $2n$ points?
2007 Sharygin Geometry Olympiad, 2
By straightedge and compass, reconstruct a right triangle $ABC$ ($\angle C = 90^o$), given the vertices $A, C$ and a point on the bisector of angle $B$.
2002 Swedish Mathematical Competition, 1
$268$ numbers are written around a circle. The $17$th number is $3$, the $83$rd is $4$ and the $144$th is $9$. The sum of every $20$ consecutive numbers is $72$. Find the $210$th number.
PEN P Problems, 1
Show that any integer can be expressed as a sum of two squares and a cube.
2016 Argentina National Olympiad Level 2, 2
Point $D$ on the side $BC$ of the acute triangle $ABC$ is chosen so that $AD = AC$. Let $P$ and $Q$ be the feet of the perpendiculars from $C$ and $D$ on the side $AB$, respectively. Suppose that $AP^2 + 3BP^2 = AQ^2 + 3BQ^2$. Determine the measure of angle $\angle ABC$.