Found problems: 85335
1956 AMC 12/AHSME, 42
The equation $ \sqrt {x \plus{} 4} \minus{} \sqrt {x \minus{} 3} \plus{} 1 \equal{} 0$ has:
$ \textbf{(A)}\ \text{no root} \qquad\textbf{(B)}\ \text{one real root}$
$ \textbf{(C)}\ \text{one real root and one imaginary root}$
$ \textbf{(D)}\ \text{two imaginary roots} \qquad\textbf{(E)}\ \text{two real roots}$
1987 IMO Longlists, 53
Prove that there exists a four-coloring of the set $M = \{1, 2, \cdots, 1987\}$ such that any arithmetic progression with $10$ terms in the set $M$ is not monochromatic.
[b][i]Alternative formulation[/i][/b]
Let $M = \{1, 2, \cdots, 1987\}$. Prove that there is a function $f : M \to \{1, 2, 3, 4\}$ that is not constant on every set of $10$ terms from $M$ that form an arithmetic progression.
[i]Proposed by Romania[/i]
2024 All-Russian Olympiad Regional Round, 9.1
There are $2024$ rectangles $1 \times n$ for $n=1, 2, \ldots, 2024$. Is it possible to make a square using some of them, such that the side length of the square is greater than $1$?
2024 Sharygin Geometry Olympiad, 8.7
A convex quadrilateral $ABCD$ is given. A line $l \parallel AC$ meets the lines $AD$, $BC$, $AB$, $CD$ at points $X$, $Y$, $Z$, $T$ respectively. The circumcircles of triangles $XYB$ and $ZTB$ meet for the second time at point $R$. Prove that $R$ lies on $BD$.
2022 Paraguay Mathematical Olympiad, 3
From a list of integers from $1$ to $2022$, inclusive, delete all numbers in which at least one of its digits is a prime How many numbers remain without erasing?
2016 KOSOVO TST, 5
Let ABC be an acute triangle such that $|AB|=|AC|$ . Let D be a point on AB such that $<ACD = <CBD$. Let E be midpoint of BD and S be circumcenter of BCD. Prove that A,E,S,C are cyclic
1990 Tournament Of Towns, (244) 2
Two circles $c$ and $d$ are situated in the plane each outside the other. The points $C$ and $D$ are located on circles $c$ and $d$ respectively, so as to be as far apart as possible. Two smaller circles are constructed inside $c$ and $d$. Of these the first circle touches $c$ and the two tangents drawn from $C$ to $d$, while the second circle touches $d$ and the two tangents from $D$ to $c$. Prove that the small circles are equal.
(J. Tabov, Sofia)
2009 HMNT, 4
How many subsets $A$ of $ \{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \} $ have the property that no two elements of $A$ sum to $11$?
2017 ASDAN Math Tournament, 10
Triangle $ABC$ is inscribed in circle $\gamma_1$ with radius $r_1$. Let $\gamma_2$ (with radius $r_2$) be the circle internally tangent to $\gamma_1$ at $A$ and tangent to $BC$ at $D$. Let $I$ be the incenter of $ABC$, and $P$ and $Q$ be the intersection of $\gamma_2$ with $AB$ and $AC$ respectively. Given that $P$, $I$, and $Q$ are collinear, $AI=25$, and the circumradius of triangle $BIC$ is $24$, compute the ratio of the radii $\tfrac{r_2}{r_1}$.
2012 ELMO Shortlist, 5
Form the infinite graph $A$ by taking the set of primes $p$ congruent to $1\pmod{4}$, and connecting $p$ and $q$ if they are quadratic residues modulo each other. Do the same for a graph $B$ with the primes $1\pmod{8}$. Show $A$ and $B$ are isomorphic to each other.
[i]Linus Hamilton.[/i]
Oliforum Contest II 2009, 4
Let $ a,b,c$ be positive reals; show that $ \displaystyle a \plus{} b \plus{} c \leq \frac {bc}{b \plus{} c} \plus{} \frac {ca}{c \plus{} a} \plus{} \frac {ab}{a \plus{} b} \plus{} \frac {1}{2}\left(\frac {bc}{a} \plus{} \frac {ca}{b} \plus{} \frac {ab}{c}\right)$
[i](Darij Grinberg)[/i]
2019 IMO Shortlist, A3
Let $n \geqslant 3$ be a positive integer and let $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ be a strictly increasing sequence of $n$ positive real numbers with sum equal to 2. Let $X$ be a subset of $\{1,2, \ldots, n\}$ such that the value of
\[
\left|1-\sum_{i \in X} a_{i}\right|
\]
is minimised. Prove that there exists a strictly increasing sequence of $n$ positive real numbers $\left(b_{1}, b_{2}, \ldots, b_{n}\right)$ with sum equal to 2 such that
\[
\sum_{i \in X} b_{i}=1.
\]
1995 IMO Shortlist, 3
Determine all integers $ n > 3$ for which there exist $ n$ points $ A_{1},\cdots ,A_{n}$ in the plane, no three collinear, and real numbers $ r_{1},\cdots ,r_{n}$ such that for $ 1\leq i < j < k\leq n$, the area of $ \triangle A_{i}A_{j}A_{k}$ is $ r_{i} \plus{} r_{j} \plus{} r_{k}$.
2016 Argentina National Olympiad Level 2, 6
There are $999$ black points marked on a circle, dividing it into $999$ arcs of length $1$. We need to place $d$ arcs of lengths $1, 2, \dots, d$ such that each arc starts and ends at two black points, and none of the $d$ arcs is contained within another. Find the maximum value of $d$ for which this construction is possible.
[b]Note:[/b] Two arcs can have one or more black points in common.
2014 Regional Olympiad of Mexico Center Zone, 6
In a school there are $n$ classes and $n$ students. The students are enrolled in classes, such that no two of them have exactly the same classes. Prove that we can close a class in a such way that there still are no two of them which have exactly the same classes.
2021 Greece JBMO TST, 3
Determine whether exists positive integer $n$ such that the number $A=8^n+47$ is prime.
2020 Harvard-MIT Mathematics Tournament, 1
How many ways can the vertices of a cube be colored red or blue so that the color of each vertex is the color of the majority of the three vertices adjacent to it?
[i]Proposed by Milan Haiman.[/i]
2023-IMOC, A3
Given positive reals $x,y,z$ satisfying $x+y+z=3$, prove that \[\sum_{cyc}\left( x^2+y^2+x^2y^2+\frac{y^2}{x^2}\right)\geq 4\sum_{cyc}\frac{y}{x}.\]
[i]Proposed by chengbilly.[/i]
2017 Dutch IMO TST, 3
Compute the product of all positive integers $n$ for which $3(n!+1)$ is divisible by $2n - 5$.
2024 Oral Moscow Geometry Olympiad, 3
An equilateral triangle $ABE$ is built inside the square $ABCD$ on the side $AB$, and an equilateral triangle $AFC$ is built on the diagonal $AC$ ($D$ is inside this triangle). The segment $EF$ intersects $CD$ at point $P$. Prove that the lines $AP$, $BE$ and $CF$ intersect at the same point.
2015 Romania Masters in Mathematics, 3
A finite list of rational numbers is written on a blackboard. In an [i]operation[/i], we choose any two numbers $a$, $b$, erase them, and write down one of the numbers \[
a + b, \; a - b, \; b - a, \; a \times b, \; a/b \text{ (if $b \neq 0$)}, \; b/a \text{ (if $a \neq 0$)}.
\] Prove that, for every integer $n > 100$, there are only finitely many integers $k \ge 0$, such that, starting from the list \[ k + 1, \; k + 2, \; \dots, \; k + n, \] it is possible to obtain, after $n - 1$ operations, the value $n!$.
2023 Indonesia Regional, 1
Let $ABCD$ be a square with side length $43$ and points $X$ and $Y$ lies on sides $AD$ and $BC$ respectively such that the ratio of the area of $ABYX$ to the area of $CDXY$ is $20 : 23$ . Find the maximum possible length of $XY$.
2021 BMT, 18
The equation $\sqrt[3]{\sqrt[3]{x - \frac38} - \frac38} = x^3+ \frac38$ has exactly two real positive solutions $r$ and $s$. Compute $r + s$.
2018 China Girls Math Olympiad, 3
Given a real sequence $\left \{ x_n \right \}_{n=1}^{\infty}$ with $x_1^2 = 1$. Prove that for each integer $n \ge 2$, $$\sum_{i|n}\sum_{j|n}\frac{x_ix_j}{\textup{lcm} \left ( i,j \right )} \ge \prod_{\mbox{\tiny$\begin{array}{c}
p \: \textup{is prime} \\ p|n \end{array}$} }\left ( 1-\frac{1}{p} \right ). $$
2019 Purple Comet Problems, 4
Of the students attending a school athletic event, $80\%$ of the boys were dressed in the school colors, $60\%$ of the girls were dressed in the school colors, and $45\% $ of the students were girls. Find the percentage of students attending the event who were wearing the school colors.