Found problems: 85335
2014-2015 SDML (High School), 9
The quadrilateral $ABCD$ can be inscribed in a circle and $\angle{ABD}$ is a right angle. $M$ is the midpoint of $BD$, where $CM$ is an altitude of $\triangle{BCD}$. If $AB=14$ and $CD=6\sqrt{11}$, what [is] the length of $AD$?
$\text{(A) }36\qquad\text{(B) }38\qquad\text{(C) }41\qquad\text{(D) }42\qquad\text{(E) }44$
1994 Putnam, 5
Let $(r_n)_{n\ge 0}$ be a sequence of positive real numbers such that $\lim_{n\to \infty} r_n = 0$. Let $S$ be the set of numbers representable as a sum
\[ r_{i_1} + r_{i_2} +\cdots + r_{i_{1994}} ,\]
with $i_1 < i_2 < \cdots< i_{1994}.$ Show that every nonempty interval $(a, b)$ contains a nonempty subinterval $(c, d)$ that does not intersect $S$.
2006 Thailand Mathematical Olympiad, 3
Let $P(x), Q(x)$ and $R(x)$ be polynomials satisfying the equation $2xP(x^3) + Q(-x -x^3) = (1 + x + x^2)R(x)$.
Show that $x - 1$ divides $P(x) - Q(x)$.
1993 Mexico National Olympiad, 5
$OA, OB, OC$ are three chords of a circle. The circles with diameters $OA, OB$ meet again at $Z$, the circles with diameters $OB, OC$ meet again at $X$, and the circles with diameters $OC, OA$ meet again at $Y$. Show that $X, Y, Z$ are collinear.
2015 Taiwan TST Round 3, 1
For any positive integer $n$, let $a_n=\sum_{k=1}^{\infty}[\frac{n+2^{k-1}}{2^k}]$, where $[x]$ is the largest integer that is equal or less than $x$. Determine the value of $a_{2015}$.
2005 Italy TST, 1
A stage course is attended by $n \ge 4$ students. The day before the final exam, each group of three students conspire against another student to throw him/her out of the exam. Prove that there is a student against whom there are at least $\sqrt[3]{(n-1)(n- 2)} $conspirators.
2017 Korea Winter Program Practice Test, 3
For a number consists of $0$ and $1$, one can perform the following operation: change all $1$ into $100$, all $0$ into $1$. For all nonnegative integer $n$, let $A_n$ be the number obtained by performing the operation $n$ times on $1$(starts with $100,10011,10011100100,\dots$), and $a_n$ be the $n$-th digit(from the left side) of $A_n$. Prove or disprove that there exists a positive integer $m$ satisfies the following:
For every positive integer $l$, there exists a positive integer $k\le m$ satisfying$$a_{l+k+1}=a_1,\ a_{l+k+2}=a_2,\ \dots,\ a_{l+k+2017}=a_{2017}$$
2022-23 IOQM India, 7
Find the number of ordered pairs $(a,b)$ such that $a,b \in \{10,11,\cdots,29,30\}$ and \\
$\hspace{1cm}$ $GCD(a,b)+LCM(a,b)=a+b$.
Kvant 2022, M2720
Let $\Omega$ be the circumcircle of the triangle $ABC$. The points $M_a,M_b$ and $M_c$ are the midpoints of the sides $BC, CA$ and $AB{}$, respectively. Let $A_l, B_l$ and $C_l$ be the intersection points of $\Omega$ with the rays $M_cM_b, M_aM_c$ and $M_bM_a$ respectively. Similarly, let $A_r, B_r$ and $C_r$ be the intersection points of $\Omega$ with the rays $M_bM_c, M_cM_a$ and $M_aM_b$ respectively. Prove that the mean of the areas of the triangles $A_lB_lC_l$ and $A_rB_rC_r$ is not less than the area of the triangle $ABC$.
[i]Proposed by L. Shatunov and T. Kazantseva[/i]
2012 India National Olympiad, 1
Let $ABCD$ be a quadrilateral inscribed in a circle. Suppose $AB=\sqrt{2+\sqrt{2}}$ and $AB$ subtends $135$ degrees at center of circle . Find the maximum possible area of $ABCD$.
2005 AMC 10, 12
Twelve fair dice are rolled. What is the probability that the product of the numbers on the top faces is prime?
$ \textbf{(A)}\ \left(\frac{1}{12}\right)^{12}\qquad
\textbf{(B)}\ \left(\frac{1}{6}\right)^{12}\qquad
\textbf{(C)}\ 2\left(\frac{1}{6}\right)^{11}\qquad
\textbf{(D)}\ \frac{5}{2}\left(\frac{1}{6}\right)^{11}\qquad
\textbf{(E)}\ \left(\frac{1}{6}\right)^{10}$
2019 Regional Olympiad of Mexico Southeast, 4
Let $\Gamma$ a circumference. $T$ a point in $\Gamma$, $P$ and $A$ two points outside $\Gamma$ such that $PT$ is tangent to $\Gamma$ and $PA=PT$. Let $C$ a point in $\Gamma (C\neq T)$, $AC$ and $PC$ intersect again $\Gamma$ in $D$ and $B$, respectively. $AB$ intersect $\Gamma$ in $E$. Prove that $DE$ it´s parallel to $AP$
2021 MIG, 15
Which of the following answer choices is the closest approximation to
\[\dfrac34+\dfrac78+\dfrac{15}{16}+\cdots+\dfrac{1023}{1024} = \dfrac{2^2-1}{2^2}+\dfrac{2^3-1}{2^3}+\cdots+\dfrac{2^{10}-1}{2^{10}}?\]
$\textbf{(A) }\dfrac{15}{2}\qquad\textbf{(B) }8\qquad\textbf{(C) }\dfrac{17}{2}\qquad\textbf{(D) }9\qquad\textbf{(E) }\dfrac{19}{2}$
2006 AMC 8, 9
What is the product of $ \dfrac{3}{2}\times \dfrac{4}{3}\times \dfrac{5}{4}\times \cdots \times \dfrac{2006}{2005}$?
$ \textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 1002 \qquad
\textbf{(C)}\ 1003 \qquad
\textbf{(D)}\ 2005 \qquad
\textbf{(E)}\ 2006$
2006 Iran MO (3rd Round), 6
Assume that $C$ is a convex subset of $\mathbb R^{d}$. Suppose that $C_{1},C_{2},\dots,C_{n}$ are translations of $C$ that $C_{i}\cap C\neq\emptyset$ but $C_{i}\cap C_{j}=\emptyset$. Prove that \[n\leq 3^{d}-1\] Prove that $3^{d}-1$ is the best bound.
P.S. In the exam problem was given for $n=3$.
2020 LMT Spring, 2
In tetrahedron $ABCD,$ as shown below, compute the number of ways to start at $A,$ walk along some path of edges, and arrive back at $A$ without walking over the same edge twice.
[Insert Diagram]
[i]Proposed by Richard Chen[/i]
1997 Swedish Mathematical Competition, 5
Let $s(m)$ denote the sum of (decimal) digits of a positive integer $m$. Prove that for every integer $n > 1$ not equal to $10$ there is a unique integer $f(n) \ge 2$ such that $s(k)+s(f(n)-k) = n$ for all integers $k$ with $0 < k < f(n)$.
2009 AMC 10, 4
Eric plans to compete in a triathlon. He can average $ 2$ miles per hour in the $ \tfrac{1}{4}$-mile swim and $ 6$ miles per hour in the $ 3$-mile run. His goal is to finish the triathlon in $ 2$ hours. To accomplish his goal what must his average speed, in miles per hour, be for the $ 15$-mile bicycle ride?
$ \textbf{(A)}\ \frac{120}{11} \qquad
\textbf{(B)}\ 11 \qquad
\textbf{(C)}\ \frac{56}{5} \qquad
\textbf{(D)}\ \frac{45}{4} \qquad
\textbf{(E)}\ 12$
2007 Today's Calculation Of Integral, 176
Let $f_{n}(x)=\sum_{k=1}^{n}\frac{\sin kx}{\sqrt{k(k+1)}}.$
Find $\lim_{n\to\infty}\int_{0}^{2\pi}\{f_{n}(x)\}^{2}dx.$
2008 Moldova National Olympiad, 9.5
Determine the polynomial P(X) satisfying simoultaneously the conditions:
a) The remainder obtained when dividing P(X) to the polynomial X^3 −2 is equal
to the fourth power of quotient.
b) P(−2) + P(2) = −34.
2000 Harvard-MIT Mathematics Tournament, 9
Find all positive primes of the form $4x^4 + 1$, for $x$ an integer.
2014 ISI Entrance Examination, 2
Let us consider a triangle $\Delta{PQR}$ in the co-ordinate plane. Show for every function $f: \mathbb{R}^2\to \mathbb{R}\;,f(X)=ax+by+c$ where $X\equiv (x,y) \text{ and } a,b,c\in\mathbb{R}$ and every point $A$ on $\Delta PQR$ or inside the triangle we have the inequality:
\begin{align*} & f(A)\le \text{max}\{f(P),f(Q),f(R)\} \end{align*}
2017 China Northern MO, 4
Let \(Q\) be a set of permutations of \(1,2,...,100\) such that for all \(1\leq a,b \leq 100\), \(a\) can be found to the left of \(b\) and adjacent to \(b\) in at most one permutation in \(Q\). Find the largest possible number of elements in \(Q\).
2004 Italy TST, 1
At the vertices $A, B, C, D, E, F, G, H$ of a cube, $2001, 2002, 2003, 2004, 2005, 2008, 2007$ and $2006$ stones respectively are placed. It is allowed to move a stone from a vertex to each of its three neighbours, or to move a stone to a vertex from each of its three neighbours. Which of the following arrangements of stones at $A, B, \ldots , H$ can be obtained?
$(\text{a})\quad 2001, 2002, 2003, 2004, 2006, 2007, 2008, 2005;$
$(\text{b})\quad 2002, 2003, 2004, 2001, 2006, 2005, 2008, 2007;$
$(\text{c})\quad 2004, 2002, 2003, 2001, 2005, 2008, 2007, 2006.$
2007 iTest Tournament of Champions, 3
Find the largest natural number $n$ such that \[2^n + 2^{11} + 2^8\] is a perfect square.