Found problems: 85335
2001 China Team Selection Test, 1
For a given natural number $n > 3$, the real numbers $x_1, x_2, \ldots, x_n, x_{n + 1}, x_{n + 2}$ satisfy the conditions $0
< x_1 < x_2 < \cdots < x_n < x_{n + 1} < x_{n + 2}$. Find the minimum possible value of
\[\frac{(\sum _{i=1}^n \frac{x_{i + 1}}{x_i})(\sum _{j=1}^n \frac{x_{j + 2}}{x_{j +
1}})}{(\sum _{k=1}^n \frac{x_{k + 1} x_{k + 2}}{x_{k + 1}^2 + x_k
x_{k + 2}})(\sum _{l=1}^n \frac{x_{l + 1}^2 + x_l x_{l + 2}}{x_l
x_{l + 1}})}\] and find all $(n + 2)$-tuplets of real numbers $(x_1, x_2, \ldots, x_n, x_{n + 1}, x_{n + 2})$ which gives this value.
2014 Singapore Senior Math Olympiad, 2
Find, with justification, all positive real numbers $a,b,c$ satisfying the system of equations:
\[a\sqrt{b}=a+c,b\sqrt{c}=b+a,c\sqrt{a}=c+b.\]
1986 China Team Selection Test, 1
If $ABCD$ is a cyclic quadrilateral, then prove that the incenters of the triangles $ABC$, $BCD$, $CDA$, $DAB$ are the vertices of a rectangle.
2022 Taiwan TST Round 1, 5
Let $H$ be the orthocenter of a given triangle $ABC$. Let $BH$ and $AC$ meet at a point $E$, and $CH$ and $AB$ meet at $F$. Suppose that $X$ is a point on the line $BC$. Also suppose that the circumcircle of triangle $BEX$ and the line $AB$ intersect again at $Y$, and the circumcircle of triangle $CFX$ and the line $AC$ intersect again at $Z$.
Show that the circumcircle of triangle $AYZ$ is tangent to the line $AH$.
[i]Proposed by usjl[/i]
2023 USA IMO Team Selection Test, 6
Let $\mathbb{N}$ denote the set of positive integers. Fix a function $f: \mathbb{N} \rightarrow \mathbb{N}$ and for any $m,n \in \mathbb{N}$ define $$\Delta(m,n)=\underbrace{f(f(\ldots f}_{f(n)\text{ times}}(m)\ldots))-\underbrace{f(f(\ldots f}_{f(m)\text{ times}}(n)\ldots)).$$ Suppose $\Delta(m,n) \neq 0$ for any distinct $m,n \in \mathbb{N}$. Show that $\Delta$ is unbounded, meaning that for any constant $C$ there exists $m,n \in \mathbb{N}$ with $\left|\Delta(m,n)\right| > C$.
2022 Moscow Mathematical Olympiad, 3
In a convex $12$-gon, all angles are equal. It is known that the lengths of some $10$ of its sides are equal to $1$, and the length of one more equals $2$.
What can be the area of ​​this $12$-gon?
2005 Gheorghe Vranceanu, 2
$ 15 $ minors of order $ 3 $ of a $ 4\times 4 $ real matrix whose determinant is a nonzero rational number, are rational.
Prove that this matrix is rational.
2021 Iran Team Selection Test, 5
Call a triple of numbers [b]Nice[/b] if one of them is the average of the other two. Assume that we have $2k+1$ distinct real numbers with $k^2$ [b] Nice[/b] triples. Prove that these numbers can be devided into two arithmetic progressions with equal ratios
Proposed by [i]Morteza Saghafian[/i]
2012 CHMMC Spring, 2
In the diagram below, $A$ and $B$ trisect $DE$, $C$ and $A$ trisect $F G$, and $B$ and $C$ trisect $HI$. Given that $DI = 5$, $EF = 6$, $GH = 7$, find the area of $\vartriangle ABC$.
[img]https://cdn.artofproblemsolving.com/attachments/d/5/90334e1bf62c99433be41f3b5e03c47c4d4916.png[/img]
2013 Vietnam Team Selection Test, 3
Given a number $n\in\mathbb{Z}^+$ and let $S$ denotes the set $\{0,1,2,...,2n+1\}$. Consider the function $f:\mathbb{Z}\times S\to [0,1]$ satisfying two following conditions simultaneously:
i) $f(x,0)=f(x,2n+1)=0\forall x\in\mathbb{Z}$;
ii) $f(x-1,y)+f(x+1,y)+f(x,y-1)+f(x,y+1)=1$ for all $x\in\mathbb{Z}$ and $y\in\{1,2,3,...,2n\}$.
Let $F$ be the set of such functions. For each $f\in F$, let $v(f)$ be the set of values of $f$.
a) Proof that $|F|=\infty$.
b) Proof that for each $f\in F$ then $|v(f)|<\infty$.
c) Find the maximum value of $|v(f)|$ for $f\in F$.
2018 Argentina National Olympiad Level 2, 5
A positive integer is called [i]pretty[/i] if it is equal to the sum of the fourth powers of five distinct divisors.
[list=a]
[*]Prove that every pretty number is divisible by $5$.
[*]Determine if there are infinitely many beautiful numbers.
[/list]
KoMaL A Problems 2017/2018, A. 722
The Hawking Space Agency operates $n-1$ space flights between the $n$ habitable planets of the Local Galaxy Cluster. Each flight has a fixed price which is the same in both directions, and we know that using these flights, we can travel from any habitable planet to any habitable planet.
In the headquarters of the Agency, there is a clearly visible board on a wall, with a portrait, containing all the pairs of different habitable planets with the total price of the cheapest possible sequence of flights connecting them. Suppose that these prices are precisely $1,2, ... , \binom{n}{2}$ monetary units in some order. prove that $n$ or $n-2$ is a square number.
2023 MOAA, 10
Let $S$ be a set of integers such that if $a$ and $b$ are in $S$ then $3a-2b$ is also in $S$. How many ways are there to construct $S$ such that $S$ contains exactly $4$ elements in the interval $[0,40]$?
[i]Proposed by Harry Kim[/i]
2013 China Team Selection Test, 1
The quadrilateral $ABCD$ is inscribed in circle $\omega$. $F$ is the intersection point of $AC$ and $BD$. $BA$ and $CD$ meet at $E$. Let the projection of $F$ on $AB$ and $CD$ be $G$ and $H$, respectively. Let $M$ and $N$ be the midpoints of $BC$ and $EF$, respectively. If the circumcircle of $\triangle MNG$ only meets segment $BF$ at $P$, and the circumcircle of $\triangle MNH$ only meets segment $CF$ at $Q$, prove that $PQ$ is parallel to $BC$.
2020 Junior Balkan Team Selection Tests - Moldova, 8
Find the pairs of real numbers $(a,b)$ such that the biggest of the numbers $x=b^2-\frac{a-1}{2}$ and $y=a^2+\frac{b+1}{2}$ is less than or equal to $\frac{7}{16}$
2006 QEDMO 2nd, 2
There are $N$ cities in the country. Any two of them are connected either by a road or by an airway. A tourist wants to visit every city exactly once and return to the city at which he started the trip. Prove that he can choose a starting city and make a path, changing means of transportation at most once.
2021 Balkan MO Shortlist, C2
Let $K$ and $N > K$ be fixed positive integers. Let $n$ be a positive integer and let $a_1, a_2, ..., a_n$ be distinct integers. Suppose that whenever $m_1, m_2, ..., m_n$ are integers, not all equal to $0$, such that $\mid{m_i}\mid \le K$ for each $i$, then the sum
$$\sum_{i = 1}^{n} m_ia_i$$
is not divisible by $N$. What is the largest possible value of $n$?
[i]Proposed by Ilija Jovcevski, North Macedonia[/i]
2017 Dutch IMO TST, 4
Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that
$$(y + 1)f(x) + f(xf(y) + f(x + y))= y$$
for all $x, y \in \mathbb{R}$.
2018 Regional Olympiad of Mexico Center Zone, 2
Let $\vartriangle ABC$be a triangle and let $\Gamma$ its circumscribed circle. Let $M$ be the midpoint of the side $BC$ and let $D$ be the point of intersection of the line $AM$ with $\Gamma$. By $D$ a straight line is drawn parallel to $BC$, which intersects $\Gamma$ at a point $E$. Let $N$ be the midpoint of the segment $AE$ and let $P$ be the point of intersection of $CN$ with $AM$. Show that $AP = PC$.
2011 National Chemistry Olympiad, 29
Introduction of two drops of concentrated sulfuric acid, $\text{H}_2\text{SO}_4$, speeds up an esterification reaction. Introduction of a piece of platinum metal, $\text{Pt}$, speeds up the reaction of $\text{H}_2$ and $\text{O}_2$ gas. Which of the following statements is true?
$ \textbf{(A)}\ \text{Pt is a homogeneous catalyst; sulfuric acid is a heterogeneous catalyst}\qquad$
$\textbf{(B)}\ \text{Pt is a heterogeneous catalyst; sulfuric acid is a homogeneous catalyst}\qquad$
$\textbf{(C)}\ \text{Pt and sulfuric acid are both heterogeneous catalysts}\qquad$
$\textbf{(D)}\ \text{Pt and sulfuric acid are both homogeneous catalysts}\qquad$
2019 Saudi Arabia Pre-TST + Training Tests, 3.2
Find all triples of real numbers $(x, y,z)$ such that
$$\begin{cases} x^4 + y^2 + 4 = 5yz \\ y^4 + z^2 + 4 = 5zx \\ z^4 + x^2 + 4 = 5xy\end{cases}$$
2025 Harvard-MIT Mathematics Tournament, 6
Let $\triangle{ABC}$ be an equilateral triangle. Point $D$ is on segment $\overline{BC}$ such that $BD=1$ and $DC=4.$ Points $E$ and $F$ lie on rays $\overrightarrow{AC}$ and $\overrightarrow{AB},$ respectively, such that $D$ is the midpoint of $\overline{EF}.$ Compute $EF.$
Novosibirsk Oral Geo Oly VIII, 2021.4
Angle bisectors $AD$ and $BE$ are drawn in triangle $ABC$. It turned out that $DE$ is the bisector of triangle $ADC$. Find the angle $BAC$.
1998 Irish Math Olympiad, 2
Prove that if $ a,b,c$ are positive real numbers, then:
$ \frac{9}{a\plus{}b\plus{}c} \le 2 \left( \frac{1}{a\plus{}b}\plus{}\frac{1}{b\plus{}c}\plus{}\frac{1}{c\plus{}a} \right) \le \frac{1}{a}\plus{}\frac{1}{b}\plus{}\frac{1}{c}.$
1998 All-Russian Olympiad, 2
A convex polygon is partitioned into parallelograms. A vertex of the polygon is called [i]good[/i] if it belongs to exactly one parallelogram. Prove that there are more than two good vertices.