Found problems: 85335
2007 F = Ma, 7
The chemical potential energy stored in a battery is converted into kinetic energy in a toy car that increases its speed first from $0 \text{ mph}$ to $2 \text{ mph}$ and then from $2 \text{ mph}$ up to $4 \text{ mph}$. Ignore the energy transferred to thermal energy due to friction and air resistance. Compared to the energy required to go from $0$ to $2 \text{ mph}$, the energy required to go from $2$ to $4 \text{ mph}$ is
$ \textbf{(A)}\ \text{half the amount.}$
$ \textbf{(B)}\ \text{the same amount.}$
$ \textbf{(C)}\ \text{twice the amount.}$
$ \textbf{(D)}\ \text{three times the amount.}$
$ \textbf{(E)}\ \text{four times the amount.} $
2011 AMC 12/AHSME, 25
For every $m$ and $k$ integers with $k$ odd, denote by $[\frac{m}{k}]$ the integer closest to $\frac{m}{k}$. For every odd integer $k$, let $P\left(k\right)$ be the probability that \[ [\frac{n}{k}] + [\frac{100-n}{k}] = [\frac{100}{k}] \] for an integer $n$ randomly chosen from the interval $1 \le n \le 99!$. What is the minimum possible value of $P\left(k\right)$ over the odd integers $k$ in the interval $1 \le k \le 99$?
$ \textbf{(A)}\ \frac{1}{2} \qquad
\textbf{(B)}\ \frac{50}{99} \qquad
\textbf{(C)}\ \frac{44}{87} \qquad
\textbf{(D)}\ \frac{34}{67} \qquad
\textbf{(E)}\ \frac{7}{13} $
1986 AIME Problems, 10
In a parlor game, the magician asks one of the participants to think of a three digit number (abc) where a, b, and c represent digits in base 10 in the order indicated. The magician then asks this person to form the numbers (acb), (bca), (bac), (cab), and (cba), to add these five numbers, and to reveal their sum, $N$. If told the value of $N$, the magician can identify the original number, (abc). Play the role of the magician and determine the (abc) if $N= 3194$.
2009 South East Mathematical Olympiad, 2
In the convex pentagon $ABCDE$ we know that $AB=DE, BC=EA$ but $AB \neq EA$.
$B,C,D,E$ are concyclic .
Prove that $A,B,C,D$ are concyclic if and only if $AC=AD.$
1981 Romania Team Selection Tests, 1.
Consider the polynomial $P(X)=X^{p-1}+X^{p-2}+\ldots+X+1$, where $p>2$ is a prime number. Show that if $n$ is an even number, then the polynomial \[-1+\prod_{k=0}^{n-1} P\left(X^{p^k}\right)\] is divisible by $X^2+1$.
[i]Mircea Becheanu[/i]
PEN H Problems, 67
Is there a positive integer $m$ such that the equation \[\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{abc}= \frac{m}{a+b+c}\] has infinitely many solutions in positive integers $a, b, c \;$?
2021 New Zealand MO, 4
Let $AB$ be a chord of circle $\Gamma$. Let $O$ be the centre of a circle which is tangent to $AB$ at $C$ and internally tangent to $\Gamma$ at $P$. Point $C$ lies between $A$ and $B$. Let the circumcircle of triangle $POC$ intersect $\Gamma$ at distinct points $P$ and $Q$. Prove that $\angle{AQP}=\angle{CQB}$.
2016 LMT, 1
Memories all must have at least one out of five different possible colors, two of which are red and green. Furthermore, they each can have at most two distinct colors. If all possible colorings are equally likely, what is the probability that a memory is at least partly green given that it has no red?
[i]Proposed by Matthew Weiss
2014 PUMaC Geometry B, 7
Consider quadrilateral $ABCD$. It is given that $\angle DAC=70^\circ$, $\angle BAC=40^\circ$, $\angle BDC=20^\circ$, $\angle CBD=35^\circ$. Let $P$ be the intersection of $AC$ and $BD$. Find $\angle BPC$.
2011 Saudi Arabia BMO TST, 3
Consider a triangle $ABC$. Let $A_1$ be the symmetric point of $A$ with respect to the line $BC$, $B_1$ the symmetric point of $B$ with respect to the line $CA$, and $C_1$ the symmetric point of $C$ with respect to the line $AB$. Determine the possible set of angles of triangle $ABC$ for which $A_1B_1C_1$ is equilateral.
2023 Azerbaijan National Mathematical Olympiad, 3
Find all the real roots of the system of equations:
$$ \begin{cases} x^3+y^3=19 \\ x^2+y^2+5x+5y+xy=12 \end{cases} $$
2016 Peru IMO TST, 6
Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ with the property that \[f(x-f(y))=f(f(x))-f(y)-1\] holds for all $x,y\in\mathbb{Z}$.
2015 Regional Olympiad of Mexico Center Zone, 5
In the triangle $ABC$, we have that $M$ and $N$ are points on $AB$ and $AC$, respectively, such that $BC$ is parallel to $MN$. A point $D$ is chosen inside the triangle $AMN$. Let $E$ and $F$ be the points of intersection of $MN$ with $BD$ and $CD$, respectively. Show that the line joining the centers of the circles circumscribed to the triangles $DEN$ and $DFM$ is perpendicular to $AD$.
2011 Iran Team Selection Test, 8
Let $p$ be a prime and $k$ a positive integer such that $k \le p$. We know that $f(x)$ is a polynomial in $\mathbb Z[x]$ such that for all $x \in \mathbb{Z}$ we have $p^k | f(x)$.
[b](a)[/b] Prove that there exist polynomials $A_0(x),\ldots,A_k(x)$ all in $\mathbb Z[x]$ such that
\[ f(x)=\sum_{i=0}^{k} (x^p-x)^ip^{k-i}A_i(x),\]
[b](b)[/b] Find a counter example for each prime $p$ and each $k > p$.
2009 Bosnia And Herzegovina - Regional Olympiad, 1
Find all triplets of integers $(x,y,z)$ such that $$xy(x^2-y^2)+yz(y^2-z^2)+zx(z^2-x^2)=1$$
2022 VIASM Summer Challenge, Problem 2
Give $P(x) = {x^{2022}} + {a_{2021}}{x^{2021}} + ... + {a_1}x + 1$ is a polynomial with real coefficents.
a) Assume that $2021a_{2021}^2 - 4044{a_{2020}} < 0.$ Prove that: $P(x)$ can't have $2022$ real roots.
b) Assume that $a_1^2 + a_2^2 + ... + a_{2021}^2 \le \frac{4}{{2021}}.$ Prove that: $P(x)\ge 0$, for all $x\in \mathbb{R}.$
2008 AIME Problems, 7
Let $ r$, $ s$, and $ t$ be the three roots of the equation
\[ 8x^3\plus{}1001x\plus{}2008\equal{}0.\]Find $ (r\plus{}s)^3\plus{}(s\plus{}t)^3\plus{}(t\plus{}r)^3$.
2021 Brazil EGMO TST, 4
The [i][b]duchess[/b][/i] is a chess piece such that the duchess attacks all the cells in two of the four diagonals which she is contained(the directions of the attack can vary to two different duchesses). Determine the greatest integer $n$, such that we can put $n$ duchesses in a table $8\times 8$ and none duchess attacks other duchess.
Note: The attack diagonals can be "outside" the table; for instance, a duchess on the top-leftmost cell we can choose attack or not the main diagonal of the table $8\times 8$.
2023 Harvard-MIT Mathematics Tournament, 4
The cells of a $5\times5$ grid are each colored red, white, or blue. Sam starts at the bottom-left cell of the grid and walks to the top-right cell by taking steps one cell either up or to the right. Thus, he passes through $9$ cells on his path, including the start and end cells. Compute the number of colorings for which Sam is guaranteed to pass through a total of exactly $3$ red cells, exactly $3$ white cells, and exactly $3$ blue cells no matter which route he takes.
2006 Iran MO (3rd Round), 2
Find all real polynomials that \[p(x+p(x))=p(x)+p(p(x))\]
2009 May Olympiad, 2
Let $ABCD$ be a convex quadrilateral such that the triangle $ABD$ is equilateral and the triangle $BCD$ is isosceles, with $\angle C = 90^o$. If $E$ is the midpoint of the side $AD$, determine the measure of the angle $\angle CED$.
2005 Estonia National Olympiad, 2
Let $a, b$, and $n$ be integers such that $a + b$ is divisible by $n$ and $a^2 + b^2$ is divisible by $n^2$. Prove that $a^m + b^m$ is divisible by $n^m$ for all positive integers $m$.
2012 Princeton University Math Competition, A4 / B6
How many (possibly empty) sets of lattice points $\{P_1, P_2, ... , P_M\}$, where each point $P_i =(x_i, y_i)$ for $x_i
, y_i \in \{0, 1, 2, 3, 4, 5, 6\}$, satisfy that the slope of the line $P_iP_j$ is positive for each $1 \le i < j \le M$ ? An infinite slope, e.g. $P_i$ is vertically above $P_j$ , does not count as positive.
2014 ELMO Shortlist, 11
Let $ABC$ be a triangle with circumcenter $O$. Let $P$ be a point inside $ABC$, so let the points $D, E, F$ be on $BC, AC, AB$ respectively so that the Miquel point of $DEF$ with respect to $ABC$ is $P$. Let the reflections of $D, E, F$ over the midpoints of the sides that they lie on be $R, S, T$. Let the Miquel point of $RST$ with respect to the triangle $ABC$ be $Q$. Show that $OP = OQ$.
[i]Proposed by Yang Liu[/i]
1977 IMO Longlists, 28
Let $n$ be an integer greater than $1$. Define
\[x_1 = n, y_1 = 1, x_{i+1} =\left[ \frac{x_i+y_i}{2}\right] , y_{i+1} = \left[ \frac{n}{x_{i+1}}\right], \qquad \text{for }i = 1, 2, \ldots\ ,\]
where $[z]$ denotes the largest integer less than or equal to $z$. Prove that
\[ \min \{x_1, x_2, \ldots, x_n \} =[ \sqrt n ]\]