Found problems: 85335
1998 Miklós Schweitzer, 8
X is a compact T2 space such that every subspace of cardinality $\aleph_1$ is first countable. Prove that X is first countable.
2018 Hanoi Open Mathematics Competitions, 8
Let $ABCD$ be rhombus, with $\angle ABC = 80^o$: Let $E$ be midpoint of $BC$ and $F$ be perpendicular projection of $A$ onto $DE$. Find the measure of $\angle DFC$ in degree.
2005 USAMTS Problems, 2
[i]Centered hexagonal numbers[/i] are the numbers of dots used to create hexagonal arrays of dots. The first four centered hexagonal numbers are 1, 7, 19, and 37 as shown below:
[asy]
size(250);defaultpen(linewidth(0.4));
dot(origin^^shift(-12,0)*origin^^shift(-24,0)*origin^^shift(-36,0)*origin);
int i;
for(i=0; i<360; i=i+60) {
dot(1*dir(i)^^2*dir(i)^^3*dir(i));
dot(shift(1/2, sqrt(3)/2)*1*dir(i)^^shift(1/2, sqrt(3)/2)*2*dir(i));
dot(shift(1, sqrt(3))*1*dir(i));
dot(shift(-12,0)*origin+1*dir(i)^^shift(-12,0)*origin+2*dir(i));
dot(shift(-12,0)*origin+sqrt(3)*dir(i+30));
dot(shift(-24,0)*origin+1*dir(i));
}
label("$1$", (-36, -5), S);
label("$7$", (-24, -5), S);
label("$19$", (-12, -5), S);
label("$37$", (0, -5), S);
label("Centered Hexagonal Numbers", (-18,-10), S);[/asy]
Consider an arithmetic sequence 1, $a$, $b$ and a geometric sequence 1,$c$,$d$, where $a$,$b$,$c$, and $d$ are all positive integers and $a+b=c+d$. Prove that each centered hexagonal number is a possible value of $a$, and prove that each possible value of $a$ is a centered hexagonal number.
2024 Brazil Team Selection Test, 3
Let $n$ be a positive integer and let $a_1, a_2, \ldots, a_n$ be positive reals. Show that $$\sum_{i=1}^{n} \frac{1}{2^i}(\frac{2}{1+a_i})^{2^i} \geq \frac{2}{1+a_1a_2\ldots a_n}-\frac{1}{2^n}.$$
1999 Turkey Team Selection Test, 2
Each of $A$, $B$, $C$, $D$, $E$, and $F$ knows a piece of gossip. They communicate by telephone via a central switchboard, which can connect only two of them at a time. During a conversation, each side tells the other everything he or she knows at that point. Determine the minimum number of calls for everyone to know all six pieces of gossip.
2012 ELMO Shortlist, 8
Find all functions $f : \mathbb{Q} \to \mathbb{R}$ such that $f(x)f(y)f(x+y) = f(xy)(f(x) + f(y))$ for all $x,y\in\mathbb{Q}$.
[i]Sammy Luo and Alex Zhu.[/i]
2023-IMOC, C4
A ghost leg is a game with some vertical lines and some horizontal lines. A player starts at the top of the vertical line and go downwards, and always walkthrough a horizontal line if he encounters one. We define a layer is some horizontal line with the same height and has no duplicated endpoints. Find the smallest number of layers needed to grant that you can walk from $(1, 2, \ldots , n)$ on the top to any permutation $(\sigma_1, \sigma_2, \ldots, \sigma_n)$ on the bottom.
1969 IMO Longlists, 27
$(GBR 4)$ The segment $AB$ perpendicularly bisects $CD$ at $X$. Show that, subject to restrictions, there is a right circular cone whose axis passes through $X$ and on whose surface lie the points $A,B,C,D.$ What are the restrictions?
2018 Ramnicean Hope, 3
Consider a complex number whose affix in the complex plane is situated on the first quadrant of the unit circle centered at origin. Then, the following inequality holds.
$$ \sqrt{2} +\sqrt{2+\sqrt{2}} \le |1+z|+|1+z^2|+|1+z^4|\le 6 $$
[i]Costică Ambrinoc[/i]
2017 Azerbaijan EGMO TST, 1
Given an equilateral triangle $ABC$ and a point $P$ so that the distances $P$ to $A$ and to $C$ are not farther than the distances $P$ to $B$. Prove that $PB = PA + PC$ if and only if $P$ lies on the circumcircle of $\vartriangle ABC$.
2022 Czech-Polish-Slovak Junior Match, 5
Given a regular nonagon $A_1A_2A_3A_4A_5A_6A_7A_8A_9$ with side length $1$. Diagonals $A_3A_7$ and $A_4A_8$ intersect at point $P$. Find the length of segment $P A_1$.
1999 China Team Selection Test, 2
For a fixed natural number $m \geq 2$, prove that
[b]a.)[/b] There exists integers $x_1, x_2, \ldots, x_{2m}$ such that \[x_i x_{m + i} = x_{i + 1} x_{m + i - 1} + 1, i = 1, 2, \ldots, m \hspace{2cm}(*)\]
[b]b.)[/b] For any set of integers $\lbrace x_1, x_2, \ldots, x_{2m}$ which fulfils (*), an integral sequence $\ldots, y_{-k}, \ldots, y_{-1}, y_0, y_1, \ldots, y_k, \ldots$ can be constructed such that $y_k y_{m + k} = y_{k + 1} y_{m + k - 1} + 1, k = 0, \pm 1, \pm 2, \ldots$ such that $y_i = x_i, i = 1, 2, \ldots, 2m$.
PEN M Problems, 4
The sequence $ \{a_{n}\}_{n \ge 1}$ is defined by \[ a_{1}=1, \; a_{2}=2, \; a_{3}=24, \; a_{n}=\frac{ 6a_{n-1}^{2}a_{n-3}-8a_{n-1}a_{n-2}^{2}}{a_{n-2}a_{n-3}}\ \ \ \ (n\ge4).\] Show that $ a_{n}$ is an integer for all $ n$, and show that $ n|a_{n}$ for every $ n\in\mathbb{N}$.
2012 Princeton University Math Competition, A8
Cyclic quadrilateral $ABCD$ has side lengths $AB = 2, BC = 3, CD = 5, AD = 4$.
Find $\sin A \sin B(\cot A/2 + \cot B/2 + \cot C/2 + \cot D/2)^2$.
Your answer can be written in simplest form as $a/b$. Find $a + b$.
2014 IMC, 1
For a positive integer $x$, denote its $n^{\mathrm{th}}$ decimal digit by $d_n(x)$, i.e. $d_n(x)\in \{ 0,1, \dots, 9\}$ and $x=\sum_{n=1}^{\infty} d_n(x)10^{n-1}$. Suppose that for some sequence $(a_n)_{n=1}^{\infty}$, there are only finitely many zeros in the sequence $(d_n(a_n))_{n=1}^{\infty}$. Prove that there are infinitely many positive integers that do not occur in the sequence $(a_n)_{n=1}^{\infty}$.
(Proposed by Alexander Bolbot, State University, Novosibirsk)
2022 Bosnia and Herzegovina IMO TST, 2
Let $p$ be an odd prime number. Around a circular table, $p$ students sit. We give $p$ pieces of candy to those students in the following manner. The first candy we give to an arbitrary student. Then, going around clockwise, we skip two students and give the next student a piece of candy, then we skip 4 students and give another piece of candy to the next student... In general in the $k-$th turn we skip $2k$ students and give the next student a piece of candy. We do this until we don't give out all $p$ pieces of candy.
$a)$ How many students won't get any pieces of candy?
$b)$ How many pairs of neighboring students (those students who sit next to each other on the table) both got at least a piece of candy?
2020 Ecuador NMO (OMEC), 4
Find all polynomials $P(x)$ such that, for all real numbers $x, y, z$ that satisfy $x+ y +z =0$,
$$P(x) +P(y) +P(z)=0$$
2014 Purple Comet Problems, 6
Nora drove $82$ miles in $90$ minutes. She averaged $50$ miles per hour for the fi rst half-hour and averaged $55$ miles per hour for the last half-hour. What was her average speed in miles per hour over the middle half-hour (during the $30$ minutes beginning after the fi rst half-hour)?
1998 Putnam, 3
Let $f$ be a real function on the real line with continuous third derivative. Prove that there exists a point $a$ such that \[f(a)\cdot f^\prime(a)\cdot f^{\prime\prime}(a)\cdot f^{\prime\prime\prime}(a)\geq 0.\]
1969 German National Olympiad, 3
A set $M$ of elements $u, v, w$ is called a semigroup if an operation is defined in it is which uniquely assigns an element $w$ from $M$ to every ordered pair $(u, v)$ of elements from $M$ (you write $u \otimes v = w$) and if this algebraic operation is associative, i.e. if for all elements $u, v,w$ from $M$:
$$(u \otimes v) \otimes w = u \otimes (v \otimes w).$$
Now let $c$ be a positive real number and let $M$ be the set of all non-negative real numbers that are smaller than $c$. For each two numbers $u, v$ from $M$ we define:
$$u \otimes v = \dfrac{u + v}{1 + \dfrac{uv}{c^2}}$$
Investigate
a) whether $M$ is a semigroup;
b) whether this semigroup is regular, i.e. whether from $u \otimes v_1 = u\otimes v_2$ always $v_1 = v_2$ and from $v_1 \otimes u = v_2 \otimes u$ also $v_1 = v_2$ follows.
2007 Today's Calculation Of Integral, 251
Evaluate $ \int_0^{n\pi} e^x\sin ^ 4 x\ dx\ (n\equal{}1,\ 2,\ \cdots).$
1964 Polish MO Finals, 1
Prove that the inequality $$ \frac{1}{3} \leq \frac{\tan 3\alpha}{\tan \alpha} \leq 3 $$ is not true for any value of $ \alpha $.
2010 Brazil Team Selection Test, 4
Let $f$ be any function that maps the set of real numbers into the set of real numbers. Prove that there exist real numbers $x$ and $y$ such that \[f\left(x-f(y)\right)>yf(x)+x\]
[i]Proposed by Igor Voronovich, Belarus[/i]
2000 Slovenia National Olympiad, Problem 3
The diagonals of a cyclic quadrilateral $ABCD$ intersect at $E$. Let $F$ and $G$ be the midpoints of $AB$ and $CD$ respectively. Prove that the lines through $E,F$ and $G$ perpendicular to $AD,BD$ and $AC$, respectively, intersect in a single point.
2003 AMC 12-AHSME, 23
The number of $ x$-intercepts on the graph of $ y \equal{} \sin(1/x)$ in the interval $ (0.0001,0.001)$ is closest to
$ \textbf{(A)}\ 2900 \qquad \textbf{(B)}\ 3000 \qquad \textbf{(C)}\ 3100 \qquad \textbf{(D)}\ 3200 \qquad \textbf{(E)}\ 3300$