Found problems: 85335
2008 Hanoi Open Mathematics Competitions, 2
How many integers belong to ($a,2008a$), where $a$ ($a > 0$) is given.
2012 NIMO Problems, 1
Dan the dog spots Cate the cat 50m away. At that instant, Cate begins running away from Dan at 6 m/s, and Dan begins running toward Cate at 8 m/s. Both of them accelerate instantaneously and run in straight lines. Compute the number of seconds it takes for Dan to reach Cate.
[i]Proposed by Eugene Chen[/i]
2009 South East Mathematical Olympiad, 1
Find all pairs ($x,y$) of integers such that $x^2-2xy+126y^2=2009$.
1990 Kurschak Competition, 2
The incenter of $\triangle A_1A_2A_3$ is $I$, and the center of the $A_i$-excircle is $J_i$ ($i=1,2,3$). Let $B_i$ be the intersection point of side $A_{i+1}A_{i+2}$ and the bisector of $\angle A_{i+1}IA_{i+2}$ ($A_{i+3}:=A_i$ $\forall i$). Prove that the three lines $B_iJ_i$ are concurrent.
1983 IMO Longlists, 23
Let $p$ and $q$ be integers. Show that there exists an interval $I$ of length $1/q$ and a polynomial $P$ with integral coefficients such that
\[ \left|P(x)-\frac pq \right| < \frac{1}{q^2}\]for all $x \in I.$
1987 AMC 8, 20
"If a whole number $n$ is not prime, then the whole number $n-2$ is not prime." A value of $n$ which shows this statement to be false is
$\text{(A)}\ 9 \qquad \text{(B)}\ 12 \qquad \text{(C)}\ 13 \qquad \text{(D)}\ 16 \qquad \text{(E)}\ 23$
2002 National Olympiad First Round, 23
What is the arithmetic mean of the smallest elements of $r$-subsets of set $\{1, 2, \dots , n\}$ where $1 \leq r \leq n$?
$
\textbf{a)}\ \dfrac{n+1}{r+1}
\qquad\textbf{b)}\ \dfrac{r(n+1)}{r+1}
\qquad\textbf{c)}\ \dfrac{nr}{r+1}
\qquad\textbf{d)}\ \dfrac{r(n+1)}{(r+1)n}
\qquad\textbf{e)}\ \text{None of above}
$
2012 USA Team Selection Test, 2
Determine all functions $f:\mathbb{R}\to\mathbb{R}$ such that for every pair of real numbers $x$ and $y$,
\[f(x+y^2)=f(x)+|yf(y)|.\]
LMT Speed Rounds, 2010.16
Determine the number of three digit integers that are equal to $19$ times the sum of its digits.
1976 Euclid, 1
Source: 1976 Euclid Part B Problem 1
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Triangle $ABC$ has $\angle{B}=30^{\circ}$, $AB=150$, and $AC=50\sqrt{3}$. Determine the length of $BC$.
2008 Bulgarian Autumn Math Competition, Problem 11.4
a) Prove that $\lfloor x\rfloor$ is odd iff $\Big\lfloor 2\{\frac{x}{2}\}\Big\rfloor=1$ ($\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$ and $\{x\}=x-\lfloor x\rfloor$).
b) Let $n$ be a natural number. Find the number of [i]square free[/i] numbers $a$, such that $\Big\lfloor\frac{n}{\sqrt{a}}\Big\rfloor$ is odd. (A natural number is [i]square free[/i] if it's not divisible by any square of a prime number).
1962 Dutch Mathematical Olympiad, 4
Write using with the floor function: the last, the second last, and the first digit of the number $n$ written in the decimal system.
2023 AMC 10, 20
Four congruent semicircles are drawn on the surface of a sphere with radius $2$, as shown, creating a close curve that divides the surface into two congruent regions. The length of the curcve is $\pi \sqrt{n}$. What is $n$?
$\textbf{(A) } 32 \qquad \textbf{(B) } 12 \qquad \textbf{(C) } 48 \qquad \textbf{(D) } 36 \qquad \textbf{(E) } 27$
2019 VJIMC, 2
Find all twice differentiable functions $f : \mathbb{R} \to \mathbb{R}$ such that $$f''(x) \cos(f(x))\geq(f'(x))^2 \sin(f(x)) $$ for every $x\in \mathbb{R}$.
[i]Proposed by Orif Ibrogimov (Czech Technical University of Prague), Karim Rakhimov (University of Pisa)[/i]
2011 Hanoi Open Mathematics Competitions, 12
Suppose that $|ax^2+bx+c| \geq |x^2-1|$ for all real numbers x. Prove that $|b^2-4ac|\geq 4$.
1996 AIME Problems, 9
A bored student walks down a hall that contains a row of closed lockers, numbered 1 to 1024. He opens the locker numbered 1, and then alternates between skipping and opening each closed locker thereafter. When he reaches the end of the hall, the student turns around and starts back. He opens the first closed locker he encounters, and then alternates between skipping and opening each closed locker thereafter. The student continues wandering back and forth in this manner until every locker is open. What is the number of the last locker he opens?
PEN L Problems, 1
An integer sequence $\{a_{n}\}_{n \ge 1}$ is defined by \[a_{0}=0, \; a_{1}=1, \; a_{n+2}=2a_{n+1}+a_{n}\] Show that $2^{k}$ divides $a_{n}$ if and only if $2^{k}$ divides $n$.
2010 Laurențiu Panaitopol, Tulcea, 2
Let be a real number $ c $ and a differentiable function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ such that
$$ f(c)\neq \frac{1}{b-a}\int_a^b f(x)dx, $$
for any real numbers $ a\neq b. $
Prove that $ f'(c)=0. $
[i]Florin Rotaru[/i]
2024 Sharygin Geometry Olympiad, 8.4
A square with side $1$ is cut from the paper. Construct a segment with length $1/2024$ using at most $20$ folds. No instruments are available. It is allowed only to fold the paper and to mark the common points of folding lines.
2013 Estonia Team Selection Test, 4
Let $D$ be the point different from $B$ on the hypotenuse $AB$ of a right triangle $ABC$ such that $|CB| = |CD|$. Let $O$ be the circumcenter of triangle $ACD$. Rays $OD$ and $CB$ intersect at point $P$, and the line through point $O$ perpendicular to side AB and ray $CD$ intersect at point $Q$. Points $A, C, P, Q$ are concyclic. Does this imply that $ACPQ$ is a square?
2002 JBMO ShortLists, 10
Let $ ABC$ be a triangle with area $ S$ and points $ D,E,F$ on the sides $ BC,CA,AB$. Perpendiculars at points $ D,E,F$ to the $ BC,CA,AB$ cut circumcircle of the triangle $ ABC$ at points $ (D_1,D_2), (E_1,E_2), (F_1,F_2)$. Prove that:
$ |D_1B\cdot D_1C \minus{} D_2B\cdot D_2C| \plus{} |E_1A\cdot E_1C \minus{} E_2A\cdot E_2C| \plus{} |F_1B\cdot F_1A \minus{} F_2B\cdot F_2A| > 4S$
2003 Singapore Team Selection Test, 1
Let $A = \{3 + 10k, 6 + 26k, 5 + 29k, k = 1, 2, 3, 4, ...\}$. Determine the smallest positive integer $r$ such that there exists an integer $b$ with the property that the set $B = \{b + rk, k = 1, 2, 3, 4, ...\}$ is disjoint from $A$.
1999 Harvard-MIT Mathematics Tournament, 4
Evaluate $\displaystyle\sum_{n=0}^\infty \dfrac{\cos n\theta}{2^n}$, where $\cos\theta = \dfrac{1}{5}$.
2014 Saudi Arabia Pre-TST, 4.3
Fatima and Asma are playing the following game. First, Fatima chooses $2013$ pairwise different numbers, called $a_1, a_2, ..., a_{2013}$. Then, Asma tries to know the value of each number $a_1, a_2, ..., a_{2013}$.. At each time, Asma chooses $1 \le i < j \le 2013$ and asks Fatima ''[i]What is the set $\{a_i,a_j\}$?[/i]'' (For example, if Asma asks what is the set $\{a_i,a_j\}$, and $a_1 = 17$ and $a_2 = 13$, Fatima will answer $\{13. 17\}$). Find the least number of questions Asma needs to ask, to know the value of all the numbers $a_1, a_2, ..., a_{2013}$.
2011 Oral Moscow Geometry Olympiad, 6
Let $AA_1 , BB_1$, and $CC_1$ be the altitudes of the non-isosceles acute-angled triangle $ABC$. The circles circumscibred around the triangles $ABC$ and $A_1 B_1 C$ intersect again at the point $P , Z$ is the intersection point of the tangents to the circumscribed circle of the triangle $ABC$ conducted at points $A$ and $B$ . Prove that lines $AP , BC$ and $ZC_1$ are concurrent.