Found problems: 85335
1991 IberoAmerican, 5
Let $P(x,\, y)=2x^{2}-6xy+5y^{2}$. Let us say an integer number $a$ is a value of $P$ if there exist integer numbers $b$, $c$ such that $P(b,\, c)=a$.
a) Find all values of $P$ lying between 1 and 100.
b) Show that if $r$ and $s$ are values of $P$, then so is $rs$.
1990 AMC 12/AHSME, 17
How many of the numbers, $100,101,\ldots,999$, have three different digits in increasing order or in decreasing order?
$\text{(A)} \ 120 \qquad \text{(B)} \ 168 \qquad \text{(C)} \ 204 \qquad \text{(D)} \ 216 \qquad \text{(E)} \ 240$
2007 Peru IMO TST, 3
Let $T$ a set with 2007 points on the plane, without any 3 collinear points.
Let $P$ any point which belongs to $T$.
Prove that the number of triangles that contains the point $P$ inside and
its vertices are from $T$, is even.
1988 IMO Longlists, 22
In a triangle $ ABC,$ choose any points $ K \in BC, L \in AC, M \in AB, N \in LM, R \in MK$ and $ F \in KL.$ If $ E_1, E_2, E_3, E_4, E_5, E_6$ and $ E$ denote the areas of the triangles $ AMR, CKR, BKF, ALF, BNM, CLN$ and $ ABC$ respectively, show that
\[ E \geq 8 \cdot \sqrt [6]{E_1 E_2 E_3 E_4 E_5 E_6}.
\]
1986 Bundeswettbewerb Mathematik, 1
There are $n$ points on a circle ($n > 1$). Denote them with $P_1,P_2, P_3, ..., P_n$ such that the polyline $P_1P_2P_3... P_n$ does not intersect itself. In how many ways is this possible?
Kyiv City MO 1984-93 - geometry, 1991.9.3
The point $M$ is the midpoint of the median $BD$ of the triangle $ABC$, the area of which is $S$. The line $AM$ intersects the side $BC$ at the point $K$. Determine the area of the triangle $BKM$.
2019 Belarusian National Olympiad, 10.8
Call a polygon on a Cartesian plane to be[i]integer[/i] if all its vertices are integer. A convex integer $14$-gon is cut into integer parallelograms with areas not greater than $C$.
Find the minimal possible $C$.
[i](A. Yuran)[/i]
2024 New Zealand MO, 5
Determine the least real number $L$ such that $$\dfrac{1}{a}+\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{d}\leqslant L$$ for all quadruples $(a,b,c,d)$ of integers satisfying $1<a<b<c<d$.
1982 Miklós Schweitzer, 10
Let $ p_0,p_1,\ldots$ be a probability distribution on the set of nonnegative integers. Select a number according to this distribution and repeat the selection independently until either a zero or an already selected number is obtained. Write the selected numbers in a row in order of selection without the last one. Below this line, write the numbers again in increasing order. Let $ A_i$ denote the event that the number $ i$ has been selected and that it is in the same place in both lines. Prove that the events $ A_i \;(i\equal{}1,2,\ldots)$ are mutually independent, and $ P(A_i)\equal{}p_i$.
[i]T. F. Mori[/i]
2022 Balkan MO Shortlist, A6
Determine all functions $f : \mathbb{R}^2 \to\mathbb {R}$ for which \[f(A)+f(B)+f(C)+f(D)=0,\]whenever $A,B,C,D$ are the vertices of a square with side-length one.
[i]Ilir Snopce[/i]
2016 Korea - Final Round, 6
Let $U$ be a set of $m$ triangles. Prove that there exists a subset $W$ of $U$ which satisfies the following.
(i). The number of triangles in $W$ is at least $0.45m^{\frac{4}{5}}$
(ii) There are no points $A, B, C, D, E, F$ such that triangles $ABC$, $BCD$, $CDE$, $DEF$, $EFA$, $FAB$ are all in $W$.
2010 Bosnia and Herzegovina Junior BMO TST, 1
Prove that number $2^{2008}\cdot2^{2010}+5^{2012}$ is not prime
2004 Serbia Team Selection Test, 2
Let $a$, $b$ and $c$ be real numbers such that $abc=1$. Prove that the most two of numbers
$$2a-\frac{1}{b},\ 2b-\frac{1}{c},\ 2c-\frac{1}{a}$$
are greater than $1$.
2020 AMC 8 -, 7
How many integers between $2020$ and $2400$ have four distinct digits arranged in increasing order? (For example, $2357$ is one such integer.)
$\textbf{(A) }9 \qquad \textbf{(B) }10 \qquad \textbf{(C) }15 \qquad \textbf{(D) }21 \qquad \textbf{(E) }28$
2023-24 IOQM India, 15
Let $A B C D$ be a unit square. Suppose $M$ and $N$ are points on $B C$ and $C D$ respectively such that the perimeter of triangle $M C N$ is 2 . Let $O$ be the circumcentre of triangle $M A N$, and $P$ be the circumcentre of triangle $M O N$. If $\left(\frac{O P}{O A}\right)^2=\frac{m}{n}$ for some relatively prime positive integers $m$ and $n$, find the value of $m+n$.
2012 Online Math Open Problems, 17
Each pair of vertices of a regular 10-sided polygon is connected by a line segment. How many unordered pairs of distinct parallel line segments can be chosen from these segments?
[i]Author: Ray Li[/i]
2020 China Team Selection Test, 5
Let $a_1,a_2,\cdots,a_n$ be a permutation of $1,2,\cdots,n$. Among all possible permutations, find the minimum of $$\sum_{i=1}^n \min \{ a_i,2i-1 \}.$$
2014 Harvard-MIT Mathematics Tournament, 1
There are $100$ students who want to sign up for the class Introduction to Acting. There are three class sections for Introduction to Acting, each of which will fit exactly $20$ students. The $100$ students, including Alex and Zhu, are put in a lottery, and 60 of them are randomly selected to fill up the classes. What is the probability that Alex and Zhu end up getting into the same section for the class?
2024 PErA, P1
Let $n$ be a positive integer, and let $[n]=\{1,2,\dots,n\}$. Find the maximum posible cardinality of a subset $S$ of $[n]$ with the property that there aren't any distinct $a,b,c\in S$ such that $a+b=c$.
2007 USA Team Selection Test, 3
Let $ \theta$ be an angle in the interval $ (0,\pi/2)$. Given that $ \cos \theta$ is irrational, and that $ \cos k \theta$ and $ \cos[(k \plus{} 1)\theta ]$ are both rational for some positive integer $ k$, show that $ \theta \equal{} \pi/6$.
2007 Stanford Mathematics Tournament, 22
Katie begins juggling five balls. After every second elapses, there is a chance she will drop a ball. If she is currently juggling $ k$ balls, this probability is $ \frac{k}{10}$. Find the expected number of seconds until she has dropped all the balls.
2014 JHMMC 7 Contest, 4
$27$ students in a school take French. $32$ students in a school take Spanish. $5$ students take both courses. How many of these students in total take only $1$ language course?
2006 MOP Homework, 4
Let $ABCD$ be a tetrahedron and let $H_{a},H_{b},H_{c},H_{d}$ be the orthocenters of triangles $BCD,CDA,DAB,ABC$, respectively.
Prove that lines $AH_{a},BH_{b},CH_{c}, DH_{d}$ are concurrent
if and only if
$AB^2 + CD^2 = AC^2 + BD^2 = AD^2 + BC^2$
2009 IMO Shortlist, 4
Let $a$, $b$, $c$ be positive real numbers such that $ab+bc+ca\leq 3abc$. Prove that
\[\sqrt{\frac{a^2+b^2}{a+b}}+\sqrt{\frac{b^2+c^2}{b+c}}+\sqrt{\frac{c^2+a^2}{c+a}}+3\leq \sqrt{2}\left(\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}\right)\]
[i]Proposed by Dzianis Pirshtuk, Belarus[/i]
2015 District Olympiad, 3
Determine the perfect squares $ \overline{aabcd} $ of five digits such that $ \overline{dcbaa} $ is a perfect square of five digits.