Found problems: 85335
2012 AMC 12/AHSME, 10
What is the area of the polygon whose vertices are the points of intersection of the curves $x^2+y^2=25$ and $(x-4)^2+9y^2=81$?
${{ \textbf{(A)}\ 24\qquad\textbf{(B)}\ 27\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 37.5}\qquad\textbf{(E)}\ 42} $
2013 India PRMO, 3
It is given that the equation $x^2 + ax + 20 = 0$ has integer roots. What is the sum of all possible values of $a$?
1988 Romania Team Selection Test, 6
Find all vectors of $n$ real numbers $(x_1,x_2,\ldots,x_n)$ such that
\[ \left\{ \begin{array}{ccc} x_1 & = & \dfrac 1{x_2} + \dfrac 1{x_3} + \cdots + \dfrac 1{x_n } \\ x_2 & = & \dfrac 1{x_1} + \dfrac 1{x_3} + \cdots + \dfrac 1{x_n} \\ \ & \cdots & \ \\ x_n & = & \dfrac 1{x_1} + \dfrac 1{x_2} + \cdots + \dfrac 1{x_{n-1}} \end{array} \right. \]
[i]Mircea Becheanu[/i]
2023 Caucasus Mathematical Olympiad, 6
Let $a, b, c$ be positive integers such that
$$\gcd(a, b) + \text{lcm}(a, b) = \gcd(a, c) + \text{lcm}(a, c).$$
Does it follow from this that $b = c$?
IV Soros Olympiad 1997 - 98 (Russia), 9.7
For any two points $A (x_1 , y_1)$ and $B (x_2, y_2)$, the distance $r (A, B)$ between them is determined by the equality $r(A, B) = | x_1- x_2 | + | y_1 - y_2 |$.
Prove that the triangle inequality $r(A, C) + r(C, B) \ge r(A, B)$. holds for the distance introduced in this way .
Let $A$ and $B$ be two points of the plane (you can take $A(1, 3)$, $B(3, 7)$). Find the locus of points $C$ for which
a) $r(A, C) + r(C, B) = r(A, B)$
b) $r(A, C) = r(C, B).$
2004 Germany Team Selection Test, 1
Let $D_1$, $D_2$, ..., $D_n$ be closed discs in the plane. (A closed disc is the region limited by a circle, taken jointly with this circle.) Suppose that every point in the plane is contained in at most $2003$ discs $D_i$. Prove that there exists a disc $D_k$ which intersects at most $7\cdot 2003 - 1 = 14020$ other discs $D_i$.
2019 Belarusian National Olympiad, 10.3
The polynomial of seven variables
$$
Q(x_1,x_2,\ldots,x_7)=(x_1+x_2+\ldots+x_7)^2+2(x_1^2+x_2^2+\ldots+x_7^2)
$$
is represented as the sum of seven squares of the polynomials with nonnegative integer coefficients:
$$
Q(x_1,\ldots,x_7)=P_1(x_1,\ldots,x_7)^2+P_2(x_1,\ldots,x_7)^2+\ldots+P_7(x_1,\ldots,x_7)^2.
$$
Find all possible values of $P_1(1,1,\ldots,1)$.
[i](A. Yuran)[/i]
2005 Swedish Mathematical Competition, 2
There are 12 people in a line in a bank. When the desk closes, the people form a new line at a newly opened desk. In how many ways can they do this in such a way that none of the 12 people changes his/her position in the line by more than one?
1971 IMO Longlists, 55
Prove that the polynomial $x^4+\lambda x^3+\mu x^2+\nu x+1$ has no real roots if $\lambda, \mu , \nu $ are real numbers satisfying
\[|\lambda |+|\mu |+|\nu |\le \sqrt{2} \]
2002 SNSB Admission, 2
Provided that the roots of the polynom $ X^n+a_1X^{n-1} +a_2X^{n-2} +\cdots +a_{n-1}X +a_n:\in\mathbb{R}[X] , $ of degree $ n\ge 2, $ are all real and pairwise distinct, prove that there exists is a neighbourhood $ \mathcal{V} $ of $ \left(
a_1,a_2,\ldots ,a_n \right) $ in $ \mathbb{R}^n $ and $ n $ functions $ x_1,x_2,\ldots ,x_n\in\mathcal{C}^{\infty } \left(
\mathcal{V} \right) $ whose values at $ \left( a_1,a_2,\ldots ,a_n \right) $ are roots of the mentioned polynom.
2009 Today's Calculation Of Integral, 446
Evaluate $ \int_0^1 \frac{(1\minus{}2x)e^{x}\plus{}(1\plus{}2x)e^{\minus{}x}}{(e^x\plus{}e^{\minus{}x})^3}\ dx.$
1981 Bulgaria National Olympiad, Problem 5
Find all positive values of $a$, for which there is a number $b$ such that the parabola $y=ax^2-b$ intersects the unit circle at four distinct points. Also prove that for every such a there exists $b$ such that the parabola $y=ax^2-b$ intersects the unit circle at four distinct points whose $x$-coordinates form an arithmetic progression.
1952 AMC 12/AHSME, 43
The diameter of a circle is divided into $ n$ equal parts. On each part a semicircle is construced. As $ n$ becomes very large, the sum of the lengths of the arcs of the semicircles approaches a length:
$ \textbf{(A)}$ equal to the semi-circumference of the original circle
$ \textbf{(B)}$ equal to the diameter of the original circle
$ \textbf{(C)}$ greater than the diameter but less than the semi-circumeference of the original circle
$ \textbf{(D)}$ that is infinite
$ \textbf{(E)}$ greater than the semi-circumference but finite
2023 MOAA, 7
Andy flips a strange coin for which the probability of flipping heads is $\frac{1}{2^k+1}$, where $k$ is the number of heads that appeared previously. If Andy flips the coin repeatedly until he gets heads 10 times, what is the expected number of total flips he performs?
[i]Proposed by Harry Kim[/i]
2022 AMC 12/AHSME, 15
One of the following numbers is not divisible by any prime number less than 10. Which is it?
(A) $2^{606} - 1 \ \ $ (B) $2^{606} + 1 \ \ $ (C) $2^{607} - 1 \ \ $ (D) $2^{607} + 1 \ \ $ (E) $2^{607} + 3^{607} \ \ $
2022 Yasinsky Geometry Olympiad, 3
Reconstruct the triangle$ ABC$, in which $\angle B - \angle C = 90^o$ , by the orthocenter $H$ and points $M_1$ and $L_1$ the feet of the median and angle bisector drawn from vertex $A$, respectively.
(Gryhoriy Filippovskyi)
2002 AMC 10, 9
The function $f$ is given by the table
\[\begin{array}{|c||c|c|c|c|c|}\hline x & 1 & 2 & 3 & 4 & 5 \\ \hline f(x) & 4 & 1 & 3 & 5 & 2 \\ \hline \end{array}\]
If $u_0=4$ and $u_{n+1}=f(u_n)$ for $n\geq 0$, find $u_{2002}$.
$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }5$
2023 MIG, 13
Five cards numbered $1,2,3,4,$ and $5$ are given to Paige, Quincy, Ronald, Selena, and Terrence. Paige, Quincy, and Ronald have the following conversation:
[list=disc]
[*]Paige: My number is between is between Selena's number and Quincy's number.
[*]Quincy: My number is between Ronald's number and Terrence's number.
[*]Ronald: My number is between Paige's number and Quincy's number.
[/list]
Who received the card numbered $3$?
$\textbf{(A) } \text{Paige}\qquad\textbf{(B) } \text{Quincy}\qquad\textbf{(C) } \text{Ronald}\qquad\textbf{(D) } \text{Selena}\qquad\textbf{(E) } \text{Terrence}$
2008 Thailand Mathematical Olympiad, 6
Let $f : R \to R$ be a function satisfying the inequality $|f(x + y) -f(x) - f(y)| < 1$ for all reals $x, y$.
Show that $\left| f\left( \frac{x}{2008 }\right) - \frac{f(x)}{2008} \right| < 1$ for all real numbers $x$.
2024 Malaysian IMO Training Camp, 3
Find all functions $f:\mathbb{Z}\rightarrow \mathbb{Z}$ such that for all integers $x$, $y$, $$f(x-f(y))=f(f(y))+f(x-2y)$$
[i]Proposed by Ivan Chan Kai Chin[/i]
2014 District Olympiad, 2
Let $f:[0,1]\rightarrow{\mathbb{R}}$ be a differentiable function, with continuous derivative, and let
\[ s_{n}=\sum_{k=1}^{n}f\left( \frac{k}{n}\right) \]
Prove that the sequence $(s_{n+1}-s_{n})_{n\in{\mathbb{N}}^{\ast}}$ converges to $\int_{0}^{1}f(x)\mathrm{d}x$.
2015 BMT Spring, 19
It is known that $4$ people $A, B, C$, and $D$ each have a $1/3$ probability of telling the truth. Suppose that
$\bullet$ $A$ makes a statement.
$\bullet$ $B$ makes a statement about the truthfulness of $A$’s statement.
$\bullet$ $C$ makes a statement about the truthfulness of $B$’s statement.
$\bullet$ $D$ says that $C$ says that $B$ says that $A$ was telling the truth.
What is the probability that $A$ was actually telling the truth?
2015 Singapore Senior Math Olympiad, 4
Is it possible to color each square on a $9\times 9$ board so that each $2\times 3$ or $3\times 2$ block contains exactly $2$ black squares? If so, what is/are the possible total number(s) of black squares?
2023 Princeton University Math Competition, A2 / B4
If $\theta$ is the unique solution in $(0,\pi)$ to the equation $2\sin(x)+3\sin(\tfrac{3x}{2})+\sin(2x)+3\sin(\tfrac{5x}{2})=0,$ then $\cos(\theta)=\tfrac{a-\sqrt{b}}{c}$ for positive integers $a,b,c$ such that $a$ and $c$ are relatively prime. Find $a+b+c.$
1999 Romania Team Selection Test, 12
Two circles intersect at two points $A$ and $B$. A line $\ell$ which passes through the point $A$ meets the two circles again at the points $C$ and $D$, respectively. Let $M$ and $N$ be the midpoints of the arcs $BC$ and $BD$ (which do not contain the point $A$) on the respective circles. Let $K$ be the midpoint of the segment $CD$. Prove that $\measuredangle MKN = 90^{\circ}$.