Found problems: 85335
2023 Miklós Schweitzer, 2
Let $G_0, G_1,\ldots$ be infinite open subsets of a Hausdorff space. Prove that there exist some infinite pairwise disjoint open sets $V_0,V_1,\ldots$ and some indices $n_0<n_1<\cdots$ such that $V_i\subseteq G_{n_i}$ for every $i\geqslant 0.$
2019 Mathematical Talent Reward Programme, SAQ: P 2
How many $n\times n$ matrices $A$, with all entries from the set $\{0, 1, 2\}$, are there, such that for all $i=1,2,\cdots,n$ $A_{ii} > \displaystyle{\sum \limits_{j=1 j\neq i}^n} A_{ij}$ [Where $A_{ij}$ is the $(i,j)$th element of the matrix $A$]
2016 Balkan MO Shortlist, C1
Let positive integers $K$ and $d$ be given. Prove that there exists a positive integer $n$ and a sequence of $K$ positive integers $b_1,b_2,..., b_K$ such that the number $n$ is a $d$-digit palindrome in all number bases $b_1,b_2,..., b_K$.
2022 HMIC, 3
For a nonnegative integer $n$, let $s(n)$ be the sum of the digits of the binary representation of $n$. Prove that
$$\sum_{n=1}^{2^{2022}-1} \frac{(-1)^{s(n)}}{n+2022}>0.$$
2012 VJIMC, Problem 4
Let $a,b,c,x,y,z,t$ be positive real numbers with $1\le x,y,z\le4$. Prove that
$$\frac x{(2a)^t}+\frac y{(2b)^t}+\frac z{(2c)^t}\ge\frac{y+z-x}{(b+c)^t}+\frac{z+x-y}{(c+a)^t}+\frac{x+y-z}{(a+b)^t}.$$
2007 Pre-Preparation Course Examination, 4
$a,b \in \mathbb Z$ and for every $n \in \mathbb{N}_0$, the number $2^na+b$ is a perfect square. Prove that $a=0$.
2012 Gheorghe Vranceanu, 1
Find the natural numbers $ n $ which have the property that $ \log_2 \left( 1+2^n \right) $ is rational.
[i]Cornel Berceanu[/i]
1983 Swedish Mathematical Competition, 3
The systems of equations
\[\left\{ \begin{array}{l}
2x_1 - x_2 = 1 \\
-x_1 + 2x_2 - x_3 = 1 \\
-x_2 + 2x_3 - x_4 = 1 \\
-x_3 + 3x_4 - x_5 =1 \\
\cdots\cdots\cdots\cdots\\
-x_{n-2} + 2x_{n-1} - x_n = 1 \\
-x_{n-1} + 2x_n = 1 \\
\end{array} \right.
\]
has a solution in positive integers $x_i$. Show that $n$ must be even.
2022 Argentina National Olympiad, 5
Find all pairs of positive integers $x,y$ such that $$x^3+y^3=4(x^2y+xy^2-5).$$
2017 Harvard-MIT Mathematics Tournament, 10
Compute the number of possible words $w=w_1w_2\dots w_{100}$ satisfying:
$\bullet$ $w$ has exactly $50$ $A$'s and $50$ $B$'s (and no other letter).
$\bullet$ For $i=1,2,\dots,100$, the number of $A$'s among $w_1, w_2, \dots, w_i$ is at most the number of $B$'s among $w_1, w_2, \dots, w_i$.
$\bullet$ For all $i=44,45,\dots,57$, if $w_i$ is a $B$, then $w_{i+1}$ must be a $B$.
2008 Peru IMO TST, 4
Let $\mathcal{S}_1$ and $\mathcal{S}_2$ be two non-concentric circumferences such that $\mathcal{S}_1$ is inside $\mathcal{S}_2$. Let $K$ be a variable point on $\mathcal{S}_1$. The line tangent to $\mathcal{S}_1$ at point $K$ intersects $\mathcal{S}_2$ at points $A$ and $B$. Let $M$ be the midpoint of arc $AB$ that is in the semiplane determined by $AB$ that does not contain $\mathcal{S}_1$. Determine the locus of the point symmetric to $M$ with respect to $K.$
2012 AMC 12/AHSME, 18
Let $(a_1,a_2, \dots ,a_{10})$ be a list of the first $10$ positive integers such that for each $2 \le i \le 10$ either $a_i+1$ or $a_i-1$ or both appear somewhere before $a_i$ in the list. How many such lists are there?
$ \textbf{(A)}\ 120\qquad\textbf{(B)}\ 512\qquad\textbf{(C)}\ 1024\qquad\textbf{(D)}\ 181,440\qquad\textbf{(E)}\ 362,880 $
2000 Harvard-MIT Mathematics Tournament, 8
Let $\vec{v_1},\vec{v_2},\vec{v_3},\vec{v_4}$ and $\vec{v_5}$ be vectors in three dimensions. Show that for some $i,j$ in $1,2,3,4,5$, $\vec{v_i}\cdot \vec{v_j}\ge 0$.
2017 Hanoi Open Mathematics Competitions, 12
Let $(O)$ denote a circle with a chord $AB$, and let $W$ be the midpoint of the minor arc $AB$. Let $C$ stand for an arbitrary point on the major arc $AB$. The tangent to the circle $(O)$ at $C$ meets the tangents at $A$ and $B$ at points $X$ and $Y$, respectively. The lines $W X$ and $W Y$ meet $AB$ at points $N$ and $M$ , respectively. Does the length of segment $NM$ depend on position of $C$ ?
2024 Azerbaijan National Mathematical Olympiad, 5
In a scalene triangle $ABC$, the points $E$ and $F$ are the foot of altitudes drawn from $B$ and $C$, respectively. The points $X$ and $Y$ are the reflections of the vertices $B$ and $C$ to the line $EF$, respectively. Let the circumcircles of the $\triangle ABC$ and $\triangle AEF$ intersect at $T$ for the second time. Show that the four points $A, X, Y, T$ lie on a single circle.
2012 Purple Comet Problems, 12
Ted flips seven fair coins. there are relatively prime positive integers $m$ and $n$ so that $\frac{m}{n}$ is the probability that Ted flips at least two heads given that he flips at least three tails. Find $m+n$.
2024 Nordic, 1
Let $T(a)$ be the sum of digits of $a$. For which positive integers $R$ does there exist a positive
integer $n$ such that $\frac{T(n^2)}{T(n)}=R$?
2024 Indonesia Regional, 4
Find the number of positive integer pairs $1\leqslant a,b \leqslant 2027$ that satisfy
\[ 2027 \mid a^6+b^5+b^2.\]
(Note: For integers $a$ and $b$, the notation $a \mid b$ means that there is an integer $c$ such that $ac=b$.)
[i]Proposed by Valentio Iverson, Indonesia[/i]
2003 Croatia National Olympiad, Problem 3
For positive numbers $a_1,a_2,\ldots,a_n$ ($n\ge2$) denote $s=a_1+\ldots+a_n$. Prove that
$$\frac{a_1}{s-a_1}+\ldots+\frac{a_n}{s-a_n}\ge\frac n{n-1}.$$
2006 China Team Selection Test, 3
Given positive integers $m$ and $n$ so there is a chessboard with $mn$ $1 \times 1$ grids. Colour the grids into red and blue (Grids that have a common side are not the same colour and the grid in the left corner at the bottom is red). Now the diagnol that goes from the left corner at the bottom to the top right corner is coloured into red and blue segments (Every segment has the same colour with the grid that contains it). Find the sum of the length of all the red segments.
2018 AIME Problems, 4
In \(\triangle ABC, AB = AC = 10\) and \(BC = 12\). Point \(D\) lies strictly between \(A\) and \(B\) on \(\overline{AB}\) and point \(E\) lies strictly between \(A\) and \(C\) on \(\overline{AC}\) so that \(AD = DE = EC\). Then \(AD\) can be expressed in the form \(\tfrac{p}{q}\), where \(p\) and \(q\) are relatively prime positive integers. Find \(p + q\).
2004 Uzbekistan National Olympiad, 4
In triangle $ABC$ $CL$ is a bisector($L$ lies $AB$) $I$ is center incircle of $ABC$. $G$ is intersection medians. If $a=BC, b=AC, c=AB$ and $CL\perp GI$ then prove that $\frac{a+b+c}{3}=\frac{2ab}{a+b}$
2009 Croatia Team Selection Test, 3
A triangle $ ABC$ is given with $ \left|AB\right| > \left|AC\right|$. Line $ l$ tangents in a point $ A$ the circumcirle of $ ABC$. A circle centered in $ A$ with radius $ \left|AC\right|$ cuts $ AB$ in the point $ D$ and the line $ l$ in points $ E, F$ (such that $ C$ and $ E$ are in the same halfplane with respect to $ AB$). Prove that the line $ DE$ passes through the incenter of $ ABC$.
1994 Tournament Of Towns, (427) 4
From the sequence $1,\frac12, \frac13, ...$ can one choose
(a) a subsequence of $100$ different numbers,
(b) an infinite subsequence
such that each number (beginning from the third) is equal to the difference between the two preceding numbers ($a_k=a_{k-2}-a_{k-1}$)?
(SI Tokarev)
1975 AMC 12/AHSME, 30
Let $x=\cos 36^{\circ} - \cos 72^{\circ}$. Then $x$ equals
$ \textbf{(A)}\ \frac{1}{3} \qquad\textbf{(B)}\ \frac{1}{2} \qquad\textbf{(C)}\ 3-\sqrt{6} \qquad\textbf{(D)}\ 2\sqrt{3}-3 \qquad\textbf{(E)}\ \text{none of these} $