Found problems: 85335
LMT Theme Rounds, 12
A round robin tournament is held with $2016$ participants. Each round, after seeing the results from the previous round, the tournament organizer chooses two players to play a game with each other that will result in a win for one of the players and a loss for the other. The tournament organizer wants each person to have a different total number of wins at the end of $k$ rounds. Find the minimum possible value of $k$ for which this can always be guaranteed.
[i]Proposed by Nathan Ramesh
2021 Balkan MO Shortlist, C5
Problem 4. Angel has a warehouse, which initially contains $100$ piles of $100$ pieces of rubbish each. Each morning, Angel performs exactly one of the following moves:
(a) He clears every piece of rubbish from a single pile.
(b) He clears one piece of rubbish from each pile.
However, every evening, a demon sneaks into the warehouse and performs exactly one of the
following moves:
(a) He adds one piece of rubbish to each non-empty pile.
(b) He creates a new pile with one piece of rubbish.
What is the first morning when Angel can guarantee to have cleared all the rubbish from the
warehouse?
JOM 2015 Shortlist, N4
Determine all triplet of non-negative integers $ (x,y,z) $ satisfy $$ 2^x3^y+1=7^z $$
2011 China Western Mathematical Olympiad, 4
In a circle $\Gamma_{1}$, centered at $O$, $AB$ and $CD$ are two unequal in length chords intersecting at $E$ inside $\Gamma_{1}$. A circle $\Gamma_{2}$, centered at $I$ is tangent to $\Gamma_{1}$ internally at $F$, and also tangent to $AB$ at $G$ and $CD$ at $H$. A line $l$ through $O$ intersects $AB$ and $CD$ at $P$ and $Q$ respectively such that $EP = EQ$. The line $EF$ intersects $l$ at $M$. Prove that the line through $M$ parallel to $AB$ is tangent to $\Gamma_{1}$
2014 Belarusian National Olympiad, 3
The angles at the vertices $A$ and $C$ in the convex quadrilateral $ABCD$ are not acute. Points $K, L, M$ and $N$ are marked on the sides $AB, BC, CD$ and $DA$ respectively. Prove that the perimeter of $KLMN$ is not less than the double length of the diagonal $AC$.
2019 Iran MO (3rd Round), 2
In acute-angled triangle $ABC$ altitudes $BE,CF$ meet at $H$. A perpendicular line is drawn from $H$ to $EF$ and intersects the arc $BC$ of circumcircle of $ABC$ (that doesn’t contain $A$) at $K$. If $AK,BC$ meet at $P$, prove that $PK=PH$.
1969 IMO Longlists, 32
$(GDR 4)$ Find the maximal number of regions into which a sphere can be partitioned by $n$ circles.
2015 Hanoi Open Mathematics Competitions, 13
Let $m$ be given odd number, and let $a, b$ denote the roots of equation $x^2 + mx - 1 = 0$ and $c = a^{2014} + b^{2014}$ , $d =a^{2015} + b^{2015}$ . Prove that $c$ and $d$ are relatively prime numbers.
2021 Princeton University Math Competition, A7
Consider the following expression
$$S = \log_2 \left( \sum^{2019}_{k=1}\sum^{2020}_{j=2}\log_{2^{1/k}} (j) \log_{j^2} \left(\sin \frac{\pi k}{2020}\right) \right).$$
Find the smallest integer $n$ which is bigger than $S$ (i.e. find $\lceil S \rceil$).
1958 AMC 12/AHSME, 12
If $ P \equal{} \frac{s}{(1 \plus{} k)^n}$ then $ n$ equals:
$ \textbf{(A)}\ \frac{\log{\left(\frac{s}{P}\right)}}{\log{(1 \plus{} k)}}\qquad
\textbf{(B)}\ \log{\left(\frac{s}{P(1 \plus{} k)}\right)}\qquad
\textbf{(C)}\ \log{\left(\frac{s \minus{} P}{1 \plus{} k}\right)}\qquad \\
\textbf{(D)}\ \log{\left(\frac{s}{P}\right)} \plus{} \log{(1 \plus{} k)}\qquad
\textbf{(E)}\ \frac{\log{(s)}}{\log{(P(1 \plus{} k))}}$
2005 Polish MO Finals, 1
Find all triplets $(x,y,n)$ of positive integers which satisfy:
\[ (x-y)^n=xy \]
2008 Bulgaria Team Selection Test, 1
For each positive integer $n$, denote by $a_{n}$ the first digit of $2^{n}$ (base ten). Is the number $0.a_{1}a_{2}a_{3}\cdots$ rational?
2005 ISI B.Math Entrance Exam, 5
Find the point in the closed unit disc $D=\{ (x,y) | x^2+y^2\le 1 \}$ at which the function $f(x,y)=x+y$ attains its maximum .
2013 CHMMC (Fall), 4
Let
$$A =\frac12 +\frac13 +\frac15 +\frac19,$$
$$B =\frac{1}{2 \cdot 3}+\frac{1}{2 \cdot 5}+\frac{1}{2 \cdot 9}+\frac{1}{3 \cdot 5}+\frac{1}{3 \cdot 9}
+\frac{1}{5 \cdot 9},$$
$$C =\frac{1}{2 \cdot 3 \cdot 5} + \frac{1}{2 \cdot 3 \cdot 9} + \frac{1}{2 \cdot 5 \cdot 9}
+\frac{1}{3 \cdot 5 \cdot 9}.$$
Compute the value of $A + B + C$.
2023 BMT, 20
Call a positive integer, $n$, [i]ready [/i] if all positive integer divisors of $n$ have a ones digit of either $1$ or $3$. Let S be the sum of all positive integer divisors of $32!$ that are ready. Compute the remainder when S is divided by $131$.
Novosibirsk Oral Geo Oly VII, 2019.4
Two squares and an isosceles triangle are positioned as shown in the figure (the up left vertex of the large square lies on the side of the triangle). Prove that points $A, B$ and $C$ are collinear.
[img]https://cdn.artofproblemsolving.com/attachments/d/c/03515e40f74ced1f8243c11b3e610ef92137ac.png[/img]
2024 Belarus Team Selection Test, 3.4
Points $A_1, \ldots A_n$ with rational coordinates lie on a plane. It turned out that the distance between every pair of points is an integer. Prove that there exist points $B_1, \ldots ,B_n$ with integer coordinates such that $A_iA_j=B_iB_j$ for every pair $1 \leq i \leq j \leq n$
[i]N. Sheshko, D. Zmiaikou[/i]
2020 Romanian Master of Mathematics Shortlist, N1
Determine all pairs of positive integers $(m, n)$ for which there exists a bijective function \[f : \mathbb{Z}_m \times \mathbb{Z}_n \to \mathbb{Z}_m \times \mathbb{Z}_n\]such that the vectors $f(\mathbf{v}) + \mathbf{v}$, as $\mathbf{v}$ runs through all of $\mathbb{Z}_m \times \mathbb{Z}_n$, are pairwise distinct.
(For any integers $a$ and $b$, the vectors $[a, b], [a + m, b]$ and $[a, b + n]$ are treated as equal.)
[i]Poland, Wojciech Nadara[/i]
2010 Tournament Of Towns, 2
Pete has an instrument which can locate the midpoint of a line segment, and also the point which divides the line segment into two segments whose lengths are in a ratio of $n : (n + 1)$, where $n$ is any positive integer. Pete claims that with this instrument, he can locate the point which divides a line segment into two segments whose lengths are at any given rational ratio. Is Pete right?
2022 CMIMC, 2.6 1.3
Find the smallest positive integer $N$ such that each of the $101$ intervals $$[N^2, (N+1)^2), [(N+1)^2, (N+2)^2), \cdots, [(N+100)^2, (N+101)^2)$$ contains at least one multiple of $1001.$
[i]Proposed by Kyle Lee[/i]
1999 Gauss, 7
If the numbers $\dfrac{4}{5},81\%$ and $0.801$ are arranged from smallest to largest, the correct order is
$\textbf{(A)}\ \dfrac{4}{5},81\%,0.801 \qquad \textbf{(B)}\ 81\%,0.801,\dfrac{4}{5} \qquad \textbf{(C)}\ 0.801,\dfrac{4}{5},81\% \qquad \textbf{(D)}\ 81\%,\dfrac{4}{5},0.801 \qquad \textbf{(E)}\ \dfrac{4}{5},0.801,81\%$
1993 National High School Mathematics League, 13
In triangular pyramid $S-ABC$, any two of $SA,SB,SC$ are perpendicular. $M$ is the centre of gravity of $\triangle ABC$. $D$ is the midpoint of $AB$, line $DP//SC$. Prove:
[b](a)[/b] $DP$ and $SM$ intersect.
[b](b)[/b] $DP\cap SM=D'$, then $D'$ is the center of circumsphere of $S-ABC$.
2020 Indonesia MO, 5
A set $A$ contains exactly $n$ integers, each of which is greater than $1$ and every of their prime factors is less than $10$. Determine the smallest $n$ such that $A$ must contain at least two distinct elements $a$ and $b$ such that $ab$ is the square of an integer.
2009 IberoAmerican Olympiad For University Students, 5
Let $\mathbb{N}$ and $\mathbb{N}^*$ be the sets containing the natural numbers/positive integers respectively.
We define a binary relation on $\mathbb{N}$ by $a\acute{\in}b$ iff the $a$-th bit in the binary representation of $b$ is $1$.
We define a binary relation on $\mathbb{N}^*$ by $a\tilde{\in}b$ iff $b$ is a multiple of the $a$-th prime number $p_a$.
i) Prove that there is no bijection $f:\mathbb{N}\to \mathbb{N}^*$ such that $a\acute{\in}b\Leftrightarrow f(a)\tilde{\in}f(b)$.
ii) Prove that there is a bijection $g:\mathbb{N}\to \mathbb{N}^*$ such that $(a\acute{\in}b \vee b\acute{\in}a)\Leftrightarrow (g(a)\tilde{\in}g(b) \vee g(b)\tilde{\in}g(a))$.
Indonesia Regional MO OSP SMA - geometry, 2012.4
Given an acute triangle $ABC$. Point $H$ denotes the foot of the altitude drawn from $A$. Prove that $$AB + AC \ge BC cos \angle BAC + 2AH sin \angle BAC$$