Found problems: 85335
2020 ASDAN Math Tournament, 5
Two quadratic polynomials $A(x)$ and $B(x)$ have a leading term of $x^2$. For some real numbers $a$ and $b$, the roots of $A(x)$ are $1$ and $a$, and the roots of $B(x)$ are $6$ and $b$. If the roots of $A(x) + B(x)$ are $a + 3$ and $b + \frac1 2$ , then compute $a^2 + b^2$.
2010 Stanford Mathematics Tournament, 17
An equilateral triangle is inscribed inside of a circle of radius $R$. Find the side length of the triangle
2018 Bulgaria JBMO TST, 4
Each cell of an infinite table (infinite in all directions) is colored with one of $n$ given colors. All six cells of any $2\times 3$ (or $3 \times 2$) rectangle have different colors. Find the smallest possible value of $n$.
2021 AMC 12/AHSME Spring, 4
Ms. Blackwell gives an exam to two classes. The mean of the scores of the students in the morning class is $84$, and the afternoon class’s mean score is $70$. The ratio of the number of students in the morning class to the number of students in the afternoon class is $\frac{3}{4}$. What is the mean of the scores of all the students?
$\textbf{(A) }74 \qquad \textbf{(B) }75 \qquad \textbf{(C) }76 \qquad \textbf{(D) }77 \qquad \textbf{(E) }78$
2007 Singapore Team Selection Test, 3
Let $A,B,C$ be $3$ points on the plane with integral coordinates. Prove that there exists a point $P$ with integral coordinates distinct from $A,B$ and $C$ such that the interiors of the segments $PA,PB$ and $PC$ do not contain points with integral coordinates.
2022 Thailand TST, 3
Find all integers $n\geq 3$ for which every convex equilateral $n$-gon of side length $1$ contains an equilateral triangle of side length $1$. (Here, polygons contain their boundaries.)
2005 India Regional Mathematical Olympiad, 6
Determine all triples of positive integers $(a,b,c)$ such that $a \leq b \leq c$ and $a +b + c + ab+ bc +ca = abc +1$.
MOAA Gunga Bowls, 2023.12
Andy is planning to flip a fair coin 10 times. Among the 10 flips, Valencia randomly chooses one flip to exchange Andy's fair coin with her special coin which lands on heads with a probability of $\frac{1}{4}$. If the coin is exchanged in a certain flip, then that flip, along with all following flips will be performed with the special coin. The expected number of heads Andy flips can be expressed as $\frac{m}{n}$ where $m$ and $n$ are positive integers. Find $m+n$.
[i]Proposed by Andy Xu[/i]
2020 ASDAN Math Tournament, 1
Consider triangle $\vartriangle ABC$ with $\angle C = 90^o$. Let $P$ be the midpoint of $\overline{AC}$ so that $AP = PC = 1$, and suppose $\angle BAC = \angle CBP$. Compute $AB^2$.
2020 Online Math Open Problems, 8
Let $a>b$ be positive integers. Compute the smallest possible integer value of $\frac{a!+1}{b!+1}$.
[i]Proposed by Sean Li[/i]
2017 CMIMC Individual Finals, 3
Let $n=2017$ and $x_1,\dots,x_n$ be boolean variables. An \emph{$7$-CNF clause} is an expression of the form $\phi_1(x_{i_1})+\dots+\phi_7(x_{i_7})$, where $\phi_1,\dots,\phi_7$ are each either the function $f(x)=x$ or $f(x)=1-x$, and $i_1,i_2,\dots,i_7\in\{1,2,\dots,n\}$. For example, $x_1+(1-x_1)+(1-x_3)+x_2+x_4+(1-x_3)+x_{12}$ is a $7$-CNF clause. What's the smallest number $k$ for which there exist $7$-CNF clauses $f_1,\dots,f_k$ such that \[f(x_1,\dots,x_n):=f_1(x_1,\dots,x_n)\cdots f_k(x_1,\dots,x_n)\] is $0$ for all values of $(x_1,\dots,x_n)\in\{0,1\}^n$?
2001 All-Russian Olympiad, 4
Participants to an olympiad worked on $n$ problems. Each problem was worth a [color=#FF0000]positive [/color]integer number of points, determined by the jury. A contestant gets $0$ points for a wrong answer, and all points for a correct answer to a problem. It turned out after the olympiad that the jury could impose worths of the problems, so as to obtain any (strict) final ranking of the contestants. Find the greatest possible number of contestants.
1994 Balkan MO, 3
Let $a_1,a_2,\ldots,a_n$ be a permutation of the numbers $1,2,\ldots,n$, with $n\geq 2$. Determine the largest possible value of the sum \[ S(n)=|a_2-a_1|+ |a_3-a_2| + \cdots + |a_n-a_{n-1}| . \]
[i]Romania[/i]
2021 Tuymaada Olympiad, 2
The bisector of angle $B$ of a parallelogram $ABCD$ meets its diagonal $AC$ at $E$,
and the external bisector of angle $B$ meets line $AD$ at $F$. $M$ is the midpoint of $BE$.
Prove that $CM // EF$.
2018 Macedonia National Olympiad, Problem 3
Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that:$$f(\max \left\{ x, y \right\} + \min \left\{ f(x), f(y) \right\}) = x+y $$ for all real $x,y \in \mathbb{R}$
[i]Proposed by Nikola Velov[/i]
2012 Indonesia MO, 2
Let $n\ge 3$ be an integer, and let $a_2,a_3,\ldots ,a_n$ be positive real numbers such that $a_{2}a_{3}\cdots a_{n}=1$. Prove that
\[(1 + a_2)^2 (1 + a_3)^3 \dotsm (1 + a_n)^n > n^n.\]
[i]Proposed by Angelo Di Pasquale, Australia[/i]
1978 IMO Longlists, 17
Prove that for any positive integers $x, y, z$ with $xy-z^2 = 1$ one can find non-negative integers $a, b, c, d$ such that $x = a^2 + b^2, y = c^2 + d^2, z = ac + bd$.
Set $z = (2q)!$ to deduce that for any prime number $p = 4q + 1$, $p$ can be represented as the sum of squares of two integers.
2018 Iranian Geometry Olympiad, 1
Two circles $\omega_1,\omega_2$ intersect each other at points $A,B$. Let $PQ$ be a common tangent line of these two circles with $P \in \omega_1$ and $Q \in \omega_2$. An arbitrary point $X$ lies on $\omega_1$. Line $AX$ intersects $ \omega_2$ for the second time at $Y$ . Point $Y'\ne Y$ lies on $\omega_2$ such that $QY = QY'$. Line $Y'B$ intersects $ \omega_1$ for the second time at $X'$. Prove that $PX = PX'$.
Proposed by Morteza Saghafian
2023 Turkey Team Selection Test, 3
For all $n>1$, let $f(n)$ be the biggest divisor of $n$ except itself. Does there exists a positive integer $k$ such that the equality $n-f(n)=k$ has exactly $2023$ solutions?
2024 Indonesia TST, G
Let $ABCD$ be a cyclic quadrilateral with $\angle BAD < \angle ADC$. Let $M$ be the midpoint of the arc $CD$ not containing $A$. Suppose there is a point $P$ inside $ABCD$ such that $\angle ADB = \angle CPD$ and $\angle ADP = \angle PCB$.
Prove that lines $AD, PM$, and $BC$ are concurrent.
2013 Math Prize For Girls Problems, 18
Ranu starts with one standard die on a table. At each step, she rolls all the dice on the table: if all of them show a 6 on top, then she places one more die on the table; otherwise, she does nothing more on this step. After 2013 such steps, let $D$ be the number of dice on the table. What is the expected value (average value) of $6^D$?
2008 JBMO Shortlist, 8
Let $a, b, c, d, e, f$ are nonzero digits such that the natural numbers $\overline{abc}, \overline{def}$ and $\overline{abcdef }$ are squares.
a) Prove that $\overline{abcdef}$ can be represented in two different ways as a sum of three squares of natural numbers.
b) Give an example of such a number.
2015 European Mathematical Cup, 4
A group of mathematicians is attending a conference. We say that a mathematician is $k-$[i]content[/i] if he is in a room with at least $k$ people he admires or if he is admired by at least $k$ other people in the room. It is known that when all participants are in a same room then they are all at least $3k + 1$-content. Prove that you can assign everyone into one of $2$ rooms in a way that everyone is at least $k$-content in his room and neither room is empty. [i]Admiration is not necessarily mutual and no one admires himself.[/i]
[i]Matija Bucić[/i]
2008 Harvard-MIT Mathematics Tournament, 8
Let $ S$ be the smallest subset of the integers with the property that $ 0\in S$ and for any $ x\in S$, we have $ 3x\in S$ and $ 3x \plus{} 1\in S$. Determine the number of non-negative integers in $ S$ less than $ 2008$.
1966 IMO Longlists, 9
Find $x$ such that trigonometric
\[\frac{\sin 3x \cos (60^\circ -4x)+1}{\sin(60^\circ - 7x) - \cos(30^\circ + x) + m}=0\]
where $m$ is a fixed real number.