Found problems: 85335
2013 China Team Selection Test, 3
Let $A$ be a set consisting of 6 points in the plane. denoted $n(A)$ as the number of the unit circles which meet at least three points of $A$. Find the maximum of $n(A)$
1999 National Olympiad First Round, 2
How many ordered integer pairs $ \left(x,y\right)$ are there such that $ xy \equal{} 4\left(y^{2} \plus{} x\right)$?
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 14 \qquad\textbf{(E)}\ \text{None}$
2017 QEDMO 15th, 7
Find all real solutions $x, y$ of the system of equations
$$\begin{cases} x + \dfrac{3x-y}{x^2 + y^2} = 3 \\ \\ y-\dfrac{x + 3y}{x^2 + y^2} = 0 \end{cases}$$
2013 IMO Shortlist, A4
Let $n$ be a positive integer, and consider a sequence $a_1 , a_2 , \dotsc , a_n $ of positive integers. Extend it periodically to an infinite sequence $a_1 , a_2 , \dotsc $ by defining $a_{n+i} = a_i $ for all $i \ge 1$. If \[a_1 \le a_2 \le \dots \le a_n \le a_1 +n \] and \[a_{a_i } \le n+i-1 \quad\text{for}\quad i=1,2,\dotsc, n, \] prove that \[a_1 + \dots +a_n \le n^2. \]
1983 National High School Mathematics League, 9
In $\triangle ABC,\sin A=\frac{3}{5},\cos B=\frac{5}{13}$, then $\cos C=$________.
2016 Serbia National Math Olympiad, 4
Let $ABC $be a triangle, and $I $ the incenter, $M $ midpoint of $ BC $, $ D $ the touch point of incircle and $ BC $. Prove that perpendiculars from $M, D, A $ to $AI, IM, BC $ respectively are concurrent
2008 Turkey Junior National Olympiad, 3
There are $24$ cups on a table. In the beginning, only three of them placed upside-down. At each step, we are turning four cups. Can we turn all the cups right-side up in at most $100$ steps?
2019 Tournament Of Towns, 6
Peter has several $100$ ruble notes and no other money. He starts buying books; each book costs a positive integer number of rubles, and he gets change in $1$ ruble coins. Whenever Peter is buying an expensive book for $100$ rubles or higher he uses only $100$ ruble notes in the minimum necessary number. Whenever he is buying a cheap one (for less than $100$ rubles) he uses his coins if he has enough, otherwise using a $100$ ruble note.
When the $100$ ruble notes have come to the end, Peter has expended exactly a half of his money. Is it possible that he has expended $5000$ rubles or more?
(Tatiana Kazitsina)
2023 LMT Fall, 13
Given that the base-$17$ integer $\overline{8323a02421_{17}}$ (where a is a base-$17$ digit) is divisible by $\overline{16_{10}}$, find $a$. Express your answer in base $10$.
[i]Proposed by Jonathan Liu[/i]
2021/2022 Tournament of Towns, P5
Consider the segment $[0; 1]$. At each step we may split one of the available segments into two new segments and write the product of lengths of these two new segments onto a blackboard. Prove that the sum of the numbers on the blackboard never will exceed $1/2$.
[i]Mikhail Lukin[/i]
Swiss NMO - geometry, 2016.5
Let $ABC$ be a right triangle with $\angle ACB = 90^o$ and M the center of $AB$. Let $G$ br any point on the line $MC$ and $P$ a point on the line $AG$, such that $\angle CPA = \angle BAC$ . Further let $Q$ be a point on the straight line $BG$, such that $\angle BQC = \angle CBA$ . Show that the circles of the triangles $AQG$ and $BPG$ intersect on the segment $AB$.
2011 Today's Calculation Of Integral, 724
Find $\lim_{n\to\infty}\left\{\left(1+n\right)^{\frac{1}{n}}\left(1+\frac{n}{2}\right)^{\frac{2}{n}}\left(1+\frac{n}{3}\right)^{\frac{3}{n}}\cdots\cdots 2\right\}^{\frac{1}{n}}$.
KoMaL A Problems 2020/2021, A. 782
Prove that the edges of a simple planar graph can always be oriented such that the outdegree of all vertices is at most three.
[i]UK Competition Problem[/i]
2005 France Pre-TST, 1
Let $I$ be the incenter of the triangle $ABC$, et let $A',B',C'$ be the symmetric of $I$ with respect to the lines $BC,CA,AB$ respectively. It is known that $B$ belongs to the circumcircle of $A'B'C'$.
Find $\widehat {ABC}$.
Pierre.
2013 Switzerland - Final Round, 3
Let $ABCD$ be a cyclic quadrilateral with $\angle ADC = \angle DBA$. Furthermore, let $E$ be the projection of $A$ on $BD$. Show that $BC = DE - BE$ .
2024 AIME, 8
Eight circles of radius $34$ can be placed tangent to side $\overline{BC}$ of $\triangle ABC$ such that the first circle is tangent to $\overline{AB}$, subsequent circles are externally tangent to each other, and the last is tangent to $\overline{AC}$. Similarly, $2024$ circles of radius $1$ can also be placed along $\overline{BC}$ in this manner. The inradius of $\triangle ABC$ is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2020 CMIMC Geometry, 5
For every positive integer $k$, let $\mathbf{T}_k = (k(k+1), 0)$, and define $\mathcal{H}_k$ as the homothety centered at $\mathbf{T}_k$ with ratio $\tfrac{1}{2}$ if $k$ is odd and $\tfrac{2}{3}$ is $k$ is even. Suppose $P = (x,y)$ is a point such that
$$(\mathcal{H}_{4} \circ \mathcal{H}_{3} \circ \mathcal{H}_2 \circ \mathcal{H}_1)(P) = (20, 20).$$ What is $x+y$?
(A [i]homothety[/i] $\mathcal{H}$ with nonzero ratio $r$ centered at a point $P$ maps each point $X$ to the point $Y$ on ray $\overrightarrow{PX}$ such that $PY = rPX$.)
2014 Iran MO (3rd Round), 2
We say two sequence of natural numbers A=($a_1,...,a_n$) , B=($b_1,...,b_n$)are the exchange and we write $A\sim B$.
if $503\vert a_i - b_i$ for all $1\leq i\leq n$.
also for natural number $r$ : $A^r$ = ($a_1^r,a_2^r,...,a_n^r$).
Prove that there are natural number $k,m$ such that :
$i$)$250 \leq k $
$ii$)There are different permutations $\pi _1,...,\pi_k$ from {$1,2,3,...,502$} such that for $1\leq i \leq k-1$ we have $\pi _i^m\sim \pi _{i+1}$
(15 points)
2016 Indonesia MO, 2
Determine all triples of natural numbers $(a,b, c)$ with $b> 1$ such that $2^c + 2^{2016} = a^b$.
1999 National Olympiad First Round, 6
If $ a,b,c\in {\rm Z}$ and
\[ \begin{array}{l} {x\equiv a\, \, \, \pmod{14}} \\
{x\equiv b\, \, \, \pmod {15}} \\
{x\equiv c\, \, \, \pmod {16}} \end{array}
\]
, the number of integral solutions of the congruence system on the interval $ 0\le x < 2000$ cannot be
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{None}$
VMEO IV 2015, 11.1
On Cartesian plane, given a line defined by $y=x+\frac{1}{\sqrt{2}}$.
a) Prove that every circle has center $I\in d$ and radius is $\frac{1}{8}$ has no integral point inside.
b) Find the greatest $k>0$ such that the distance of every integral points to $d$ is greater or equal than $k$.
2001 AMC 10, 8
Wanda, Darren, Beatrice, and Chi are tutors in the school math lab. Their schedule is as follows: Darren works every third school day, Wanda works every fourth school day, Beatrice works every sixth school day, and Chi works every seventh school day. Today they are all working in the math lab. In how many school days from today will they next be together tutoring in the lab?
$ \textbf{(A) }42\qquad\textbf{(B) }84\qquad\textbf{(C) }126\qquad\textbf{(D) }178\qquad\textbf{(E) }252$
PEN G Problems, 17
Suppose that $p, q \in \mathbb{N}$ satisfy the inequality \[\exp(1)\cdot( \sqrt{p+q}-\sqrt{q})^{2}<1.\] Show that $\ln \left(1+\frac{p}{q}\right)$ is irrational.
2019 Polish Junior MO Second Round, 5.
The integer $n \geq 1$ does not contain digits: $1,\; 2,\; 9\;$ in its decimal notation. Prove that one of the digits: $1,\; 2,\; 9$ appears at least once in the decimal notation of the number $3n$.
2005 China Team Selection Test, 2
In acute angled triangle $ABC$, $BC=a$,$CA=b$,$AB=c$, and $a>b>c$. $I,O,H$ are the incentre, circumcentre and orthocentre of $\triangle{ABC}$ respectively. Point $D \in BC$, $E \in CA$ and $AE=BD$, $CD+CE=AB$. Let the intersectionf of $BE$ and $AD$ be $K$. Prove that $KH \parallel IO$ and $KH = 2IO$.