Found problems: 85335
2001 ITAMO, 5
Let $ABC$ be a triangle and $\gamma$ the circle inscribed in $ABC$. The circle $\gamma$ is tangent to side $AB$ at the point $T$. Let $D$ be the point of $\gamma$ diametrically opposite to $T$, and $S$ the intersection point of the line through $C$ and $D$ with side $AB$.
Prove that $AT=SB$.
2014 AMC 10, 14
Danica drove her new car on a trip for a whole number of hours, averaging $55$ miles per hour. At the beginning of the trip, $abc$ miles were displayed on the odometer, where $abc$ is a 3-digit number with $a \ge 1$ and $a+b+c \le 7$. At the end of the trip, where the odometer showed $cba$ miles. What is $a^2+b^2+c^2$?
$ \textbf{(A) } 26 \qquad\textbf{(B) }27\qquad\textbf{(C) }36\qquad\textbf{(D) }37\qquad\textbf{(E) }41\qquad $
2009 AMC 10, 13
As shown below, convex pentagon $ ABCDE$ has sides $ AB \equal{} 3$, $ BC \equal{} 4$, $ CD \equal{} 6$, $ DE \equal{} 3$, and $ EA \equal{} 7$. The pentagon is originally positioned in the plane with vertex $ A$ at the origin and vertex $ B$ on the positive $ x$-axis. The pentagon is then rolled clockwise to the right along the $ x$-axis. Which side will touch the point $ x \equal{} 2009$ on the $ x$-axis?
[asy]size(250);
defaultpen(linewidth(.8pt)+fontsize(8pt));
dotfactor=4;
pair A=(0,0), Ep=7*dir(105), B=3*dir(0);
pair D=Ep+B;
pair C=intersectionpoints(Circle(D,6),Circle(B,4))[1];
pair[] ds={A,B,C,D,Ep};
dot(ds);
draw(B--C--D--Ep--A);
draw((6,6)..(8,4)..(8,3),EndArrow(3));
xaxis("$x$",-8,14,EndArrow(3));
label("$E$",Ep,NW);
label("$D$",D,NE);
label("$C$",C,E);
label("$B$",B+(.2,.1),ENE);
label("$A$",A+(-.1,.1),WNW);
label("$(0,0)$",A,S);
label("$3$",midpoint(A--B),N);
label("$4$",midpoint(B--C),NW);
label("$6$",midpoint(C--D),NE);
label("$3$",midpoint(D--Ep),S);
label("$7$",midpoint(Ep--A),W);[/asy]$ \textbf{(A)}\ \overline{AB} \qquad \textbf{(B)}\ \overline{BC} \qquad \textbf{(C)}\ \overline{CD} \qquad \textbf{(D)}\ \overline{DE} \qquad \textbf{(E)}\ \overline{EA}$
2007 Puerto Rico Team Selection Test, 1
A rectangular field has dimensions $120$ meters and $192$ meters. You want to divide it into equal square plots. The measure of the sides of these squares must be an integer number . In addition, you want to place a post in each corner of plot. Determine the smallest number of plots in which you can divide the land and the number of posts needed.
[hide=Original wording]Un terreno de forma rectangular de 120 metros por 192 metros se quiere dividir en parcelas cuadradas iguales sin que sobre terreno. La medida de los lados de estos cuadrados debe ser un nu´mero entero. Adem´as se desea colocar un poste en cada esquina de parcela. Determinar el menor nu´mero de parcelas en que se puede dividir el terreno y el nu´mero de postes que se necesitan.[/hide]
2012 Today's Calculation Of Integral, 816
Find the volume of the solid of a circle $x^2+(y-1)^2=4$ generated by a rotation about the $x$-axis.
2004 AMC 12/AHSME, 16
The set of all real numbers $ x$ for which
\[ \log_{2004}(\log_{2003}(\log_{2002}(\log_{2001}{x})))
\]is defined is $ \{x|x > c\}$. What is the value of $ c$?
$ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 2001^{2002} \qquad \textbf{(C)}\ 2002^{2003} \qquad \textbf{(D)}\ 2003^{2004} \qquad \textbf{(E)}\ 2001^{2002^{2003}}$
2023 Simon Marais Mathematical Competition, B2
There are $256$ players in a tennis tournament who are ranked from $1$ to $256$, with $1$ corresponding to the highest rank and $256$ corresponding to the lowest rank. When two players play a match in the tournament, the player whose rank is higher wins the match with probability $\frac{3}{5}$.
In each round of the tournament, the player with the highest rank plays against the player with the second highest rank, the player with the third highest rank plays against the player with the fourth highest rank, and so on. At the end of the round, the players who win proceed to the next round and the players who lose exit the tournament. After eight rounds, there is one player remaining and they are declared the winner.
Determine the expected value of the rank of the winner.
2005 VTRMC, Problem 7
Let $A$ be a $5\times10$ matrix with real entries, and let $A^{\text T}$ denote its transpose. Suppose every $5\times1$ matrix with real entries can be written in the form $A\mathbf u$ where $\mathbf u$ is a $10\times1$ matrix with real entries. Prove that every $5\times1$ matrix with real entries can be written in the form $AA^{\text T}\mathbf v$ where $\mathbf v$ is a $5\times1$ matrix with real entries.
2015 HMNT, 5
Consider a $5 \times 5$ grid of squares. Vladimir colors some of these squares red, such that the centers of any four red squares do $\textbf{not}$ form an axis-parallel rectangle (i.e. a rectangle whose sides are parallel to those of the squares). What is the maximum number of squares he could have colored red?
1988 Putnam, B2
Prove or disprove: If $x$ and $y$ are real numbers with $y\geq0$ and $y(y+1) \leq (x+1)^2$, then $y(y-1)\leq x^2$.
1998 Greece National Olympiad, 2
For a regular $n$-gon, let $M$ be the set of the lengths of the segments joining its vertices. Show that the sum of the squares of the elements of $M$ is greater than twice the area of the polygon.
2024 ELMO Problems, 3
For some positive integer $n,$ Elmo writes down the equation
\[x_1+x_2+\dots+x_n=x_1+x_2+\dots+x_n.\]
Elmo inserts at least one $f$ to the left side of the equation and adds parentheses to create a valid functional equation. For example, if $n=3,$ Elmo could have created the equation
\[f(x_1+f(f(x_2)+x_3))=x_1+x_2+x_3.\]
Cookie Monster comes up with a function $f: \mathbb{Q}\to\mathbb{Q}$ which is a solution to Elmo's functional equation. (In other words, Elmo's equation is satisfied for all choices of $x_1,\dots,x_n\in\mathbb{Q})$. Is it possible that there is no integer $k$ (possibly depending on $f$) such that $f^k(x)=x$ for all $x$?
[i]Srinivas Arun[/i]
1988 Mexico National Olympiad, 5
If $a$ and $b$ are coprime positive integers and $n$ an integer, prove that the greatest common divisor of $a^2+b^2-nab$ and $a+b$ divides $n+2$.
2024 Bosnia and Herzegovina Junior BMO TST, 4.
Let $m$ and $n$ be natural numbers. Every one of the $m*n$ squares of the $m*n$ board is colored either black or white, so that no 2 neighbouring squares are the same color(the board is colored like in chess").In one step we can pick 2 neighbouring squares and change their colors like this:
[b]- [/b]a white square becomes black;
[b]-[/b]a black square becomes blue;
[b]-[/b]a blue square becomes white.
For which $m$ and $n$ can we ,in a finite sequence of these steps, switch the starting colors from white to black and vice versa.
2009 Peru Iberoamerican Team Selection Test, P6
Let $P$ be a set of $n \ge 2$ distinct points in the plane, which does not contain any triplet of aligned points. Let $S$ be the set of all segments whose endpoints are points of $P$. Given two segments $s_1, s_2 \in S$, we write $s_1 \otimes s_2$ if the intersection of $s_1$ with $s_2$ is a point other than the endpoints of $s_1$ and $s_2$. Prove that there exists a segment $s_0 \in S$ such that the set $\{s \in S | s_0 \otimes s\}$ has at least $\frac{1}{15}\dbinom{n-2}{2}$ elements
2020 BMT Fall, 5
Let $A_1 = (0, 0)$, $B_1 = (1, 0)$, $C_1 = (1, 1)$, $D_1 = (0, 1)$. For all $i > 1$, we recursively define
$$A_i =\frac{1}{2020} (A_{i-1} + 2019B_{i-1}),B_i =\frac{1}{2020} (B_{i-1} + 2019C_{i-1})$$
$$C_i =\frac{1}{2020} (C_{i-1} + 2019D_{i-1}), D_i =\frac{1}{2020} (D_{i-1} + 2019A_{i-1})$$
where all operations are done coordinate-wise.
[img]https://cdn.artofproblemsolving.com/attachments/8/7/9b6161656ed2bc70510286dc8cb75cc5bde6c8.png[/img]
If $[A_iB_iC_iD_i]$ denotes the area of $A_iB_iC_iD_i$, there are positive integers $a, b$, and $c$ such that $\sum_{i=1}^{\infty}[A_iB_iC_iD_i] = \frac{a^2b}{c}$, where $b$ is square-free and $c$ is as small as possible. Compute the value of $a + b + c$
1990 Rioplatense Mathematical Olympiad, Level 3, 1
How many positive integer solutions does the equation have $$\left\lfloor\frac{x}{10}\right\rfloor= \left\lfloor\frac{x}{11}\right\rfloor + 1?$$
($\lfloor x \rfloor$ denotes the integer part of $x$, for example $\lfloor 2\rfloor = 2$, $\lfloor \pi\rfloor = 3$, $\lfloor \sqrt2 \rfloor =1$)
1996 Hungary-Israel Binational, 2
$ n>2$ is an integer such that $ n^2$ can be represented as a difference of cubes of 2 consecutive positive integers. Prove that $ n$ is a sum of 2 squares of positive integers, and that such $ n$ does exist.
2015 CCA Math Bonanza, I12
Positive integers $x,y,z$ satisfy $x^3+xy+x^2+xz+y+z=301$. Compute $y+z-x$.
[i]2015 CCA Math Bonanza Individual Round #12[/i]
2015 Mexico National Olympiad, 6
Let $n$ be a positive integer and let $d_1, d_2, \dots, d_k$ be its positive divisors. Consider the number
$$f(n) = (-1)^{d_1}d_1 + (-1)^{d_2}d_2 + \dots + (-1)^{d_k}d_k$$
Assume $f(n)$ is a power of 2. Show if $m$ is an integer greater than 1, then $m^2$ does not divide $n$.
1994 All-Russian Olympiad Regional Round, 9.3
Does there exist a quadratic trinomial $p(x)$ with integer coefficients such that, for every natural number $n$ whose decimal representation consists of digits $1$, $p(n)$ also consists only of digits $1$?
2023 LMT Fall, 1
If $a \diamondsuit b = \vert a - b \vert \cdot \vert b - a \vert$ then find the value of $1 \diamondsuit (2 \diamondsuit (3 \diamondsuit (4 \diamondsuit 5)))$.
[i]Proposed by Muztaba Syed[/i]
[hide=Solution]
[i]Solution.[/i] $\boxed{9}$
$a\diamondsuit b = (a-b)^2$. This gives us an answer of $\boxed{9}$.
[/hide]
2005 Slovenia Team Selection Test, 3
Find all pairs $(m, n)$ of natural numbers for which the numbers $m^2 - 4n$ and $n^2 - 4m$ are both perfect squares.
2020-2021 OMMC, 6
Jason and Jared take turns placing blocks within a game board with dimensions $3 \times 300$, with Jason going first, such that no two blocks can overlap. The player who cannot place a block within the boundaries loses. Jason can only place $2 \times 100$ blocks, and Jared can only place $2 \times n$ blocks where $n$ is some positive integer greater than 3. Find the smallest integer value of $n$ that still guarantees Jason a win (given both players are playing optimally).
1974 IMO, 6
Let $P(x)$ be a polynomial with integer coefficients. We denote $\deg(P)$ its degree which is $\geq 1.$ Let $n(P)$ be the number of all the integers $k$ for which we have $(P(k))^{2}=1.$ Prove that $n(P)- \deg(P) \leq 2.$