Found problems: 85335
2013 AMC 8, 11
Ted's grandfather used his treadmill on 3 days this week. He went 2 miles each day. On Monday he jogged at a speed of 5 miles per hour. He walked at the rate of 3 miles per hour on Wednesday and at 4 miles per hour on Friday. If Grandfather had always walked at 4 miles per hour, he would have spent less time on the treadmill. How many minutes less?
$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$
1983 IMO Longlists, 37
The points $A_1,A_2, \ldots , A_{1983}$ are set on the circumference of a circle and each is given one of the values $\pm 1$. Show that if the number of points with the value $+1$ is greater than $1789$, then at least $1207$ of the points will have the property that the partial sums that can be formed by taking the numbers from them to any other point, in either direction, are strictly positive.
2019 Saint Petersburg Mathematical Olympiad, 7
In a square $10^{2019} \times 10^{2019}, 10^{4038}$ points are marked. Prove that there is such a rectangle with sides parallel to the sides of a square whose area differs from the number of points located in it by at least $6$.
2022 Stars of Mathematics, 1
A square grid $6 \times 6$ is tiled with $18$ dominoes. Prove that there is a line, intersecting no dominoes. Can this line be unique?
Ukraine Correspondence MO - geometry, 2006.7
Let $D$ and $E$ be the midpoints of the sides $BC$ and $AC$ of a right triangle $ABC$. Prove that if $\angle CAD=\angle ABE$, then $$\frac{5}{6} \le \frac{AD}{AB}\le \frac{\sqrt{73}}{10}.$$
2017 China Girls Math Olympiad, 1
(1) Find all positive integer $n$ such that for any odd integer $a$, we have $4\mid a^n-1$
(2) Find all positive integer $n$ such that for any odd integer $a$, we have $2^{2017}\mid a^n-1$
2017 CHKMO, Q1
A, B and C are three persons among a set P of n (n[u]>[/u]3) persons. It is known that A, B and C are friends of one another, and that every one of the three persons has already made friends with more than half the total number of people in P. Given that every three persons who are friends of one another form a [i]friendly group[/i], what is the minimum number of friendly groups that may exist in P?
2023-24 IOQM India, 2
Find the number of elements in the set
$$
\left\{(a, b) \in \mathbb{N}: 2 \leq a, b \leq 2023, \log _a(b)+6 \log _b(a)=5\right\}
$$
1999 All-Russian Olympiad Regional Round, 10.7
Each voter in an election puts $n$ names of candidates on the ballot. There are $n + 1$ at the polling station urn. After the elections it turned out that each ballot box contained at least at least one ballot, for every choice of the $(n + 1)$-th ballot, one from each ballot box, there is a candidate whose surname appears in each of the selected ballots. Prove that in at least one ballot box all ballots contain the name of the same candidate.
2004 Indonesia MO, 2
Quadratic equation $ x^2\plus{}ax\plus{}b\plus{}1\equal{}0$ have 2 positive integer roots, for integers $ a,b$. Show that $ a^2\plus{}b^2$ is not a prime.
2007 China Team Selection Test, 2
Find all positive integers $ n$ such that there exists sequence consisting of $ 1$ and $ - 1: a_{1},a_{2},\cdots,a_{n}$ satisfying $ a_{1}\cdot1^2 + a_{2}\cdot2^2 + \cdots + a_{n}\cdot n^2 = 0.$
2008 Postal Coaching, 2
Does there exist a triangle $ABC$ whose sides are rational numbers and $BC$ equals to the altitude from $A$?
2021 Harvard-MIT Mathematics Tournament., 1
A circle contains the points $(0, 11)$ and $(0, -11)$ on its circumference and contains all points $(x, y)$ with $x^2+y^2<1$ in its interior. Compute the largest possible radius of the circle.
2007 Stanford Mathematics Tournament, 6
What is the largest prime factor of $4^9+9^4$?
1985 Traian Lălescu, 2.2
We are given the line $ d, $ and a point $ A $ which is not on $ d. $ Two points $ B $ and $ C $ move on $ d $ such that the angle $ \angle BAC $ is constant. Prove that the circumcircle of $ ABC $ is tangent to a fixed circle.
2019 Auckland Mathematical Olympiad, 5
$2019$ coins are on the table. Two students play the following game making alternating moves. The first player can in one move take the odd number of coins from $ 1$ to $99$, the second player in one move can take an even number of coins from $2$ to $100$. The player who can not make a move is lost. Who has the winning strategy in this game?
2024 Princeton University Math Competition, A5 / B7
Call a positive integer [I]nice[/I] if the sum of its even proper divisors is larger than the sum of its odd proper divisors. What is the smallest nice number that is congruent to $2 \text{ mod } 4$?
2009 Germany Team Selection Test, 1
Let $ ABCD$ be a chordal/cyclic quadrilateral. Consider points $ P,Q$ on $ AB$ and $ R,S$ on $ CD$ with
\[ \overline{AP}: \overline{PB} \equal{} \overline{CS}: \overline{SD}, \quad \overline{AQ}: \overline{QB} \equal{} \overline{CR}: \overline{RD}.\]
How to choose $ P,Q,R,S$ such that $ \overline{PR} \cdot \overline{AB} \plus{} \overline{QS} \cdot \overline{CD}$ is minimal?
1981 Bundeswettbewerb Mathematik, 1
A sequence $a_1, a_2, a_3, \ldots $ is defined as follows: $a_1$ is a positive integer and
$$a_{n+1} = \left\lfloor \frac{3}{2} a_n \right\rfloor +1$$
for all $n \in \mathbb{N}$. Can $a_1$ be chosen in such a way that the first $100000$ terms of the sequence are even, but the $100001$-th term is odd?
2020 Switzerland - Final Round, 1
Let $\mathbb N$ be the set of positive integers. Find all functions $f\colon\mathbb N\to \mathbb N$ such that for every $m,n\in \mathbb N$, \[
f(m)+f(n)\mid m+n.
\]
1969 Swedish Mathematical Competition, 3
$a_1 \le a_2 \le ... \le a_n$ is a sequence of reals $b_1, _b2, b_3,..., b_n$ is any rearrangement of the sequence $B_1 \le B_2 \le ...\le B_n$. Show that $\sum a_ib_i \le \sum a_i B_i$.
2019 Caucasus Mathematical Olympiad, 3
Points $A'$ and $B'$ lie inside the parallelogram $ABCD$ and points $C'$ and $D'$ lie outside of it, so that all sides of 8-gon $AA'BB'CC'DD'$ are equal. Prove that $A'$, $B'$, $C'$, $D'$ are concyclic.
2011-2012 SDML (High School), 5
In triangle $ABC$, $\angle{BAC}=15^{\circ}$. The circumcenter $O$ of triangle $ABC$ lies in its interior. Find $\angle{OBC}$.
[asy]
size(3cm,0);
dot((0,0));
draw(Circle((0,0),1));
draw(dir(70)--dir(220));
draw(dir(220)--dir(310));
draw(dir(310)--dir(70));
draw((0,0)--dir(220));
label("$A$",dir(70),NE);
label("$B$",dir(220),SW);
label("$C$",dir(310),SE);
label("$O$",(0,0),NE);
[/asy]
$\text{(A) }30^{\circ}\qquad\text{(B) }75^{\circ}\qquad\text{(C) }45^{\circ}\qquad\text{(D) }60^{\circ}\qquad\text{(E) }15^{\circ}$
2016 Taiwan TST Round 1, 5
Let $ABC$ be an acute triangle and let $M$ be the midpoint of $AC$. A circle $\omega$ passing through $B$ and $M$ meets the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $T$ be the point such that $BPTQ$ is a parallelogram. Suppose that $T$ lies on the circumcircle of $ABC$. Determine all possible values of $\frac{BT}{BM}$.
2010 Indonesia TST, 2
Let $\Gamma_1$, $\Gamma_2$, $\Gamma_3$, $\Gamma_4$ be distinct circles such that $\Gamma_1$, $\Gamma_3$ are externally tangent at $P$, and $\Gamma_2$, $\Gamma_4$ are externally tangent at the same point $P$. Suppose that $\Gamma_1$ and $\Gamma_2$; $\Gamma_2$ and $\Gamma_3$; $\Gamma_3$ and $\Gamma_4$; $\Gamma_4$ and $\Gamma_1$ meet at $A$, $B$, $C$, $D$, respectively, and that all these points are different from $P$. Prove that
\[
\frac{AB\cdot BC}{AD\cdot DC}=\frac{PB^2}{PD^2}.
\]