Found problems: 85335
1997 National High School Mathematics League, 1
Squence $(x_n)$ satisfies that $x_{n+1}=x_n-x_{n-1}(n\geq2)$. If $x_1=a,x_2=b$, $S_n=x_1+x_2+\cdots+x_n$. Wich one is correct?
$\text{(A)}x_{100}=-a,S_{100}=2b-a$
$\text{(B)}x_{100}=-b,S_{100}=2b-a$
$\text{(C)}x_{100}=-a,S_{100}=b-a$
$\text{(D)}x_{100}=-b,S_{100}=b-a$
2014 Estonia Team Selection Test, 6
Find all natural numbers $n$ such that the equation $x^2 + y^2 + z^2 = nxyz$ has solutions in positive integers
2011 Saudi Arabia Pre-TST, 4.4
In a triangle $ABC$, let $O$ be the circumcenter, $H$ the orthoÂcenter, and $M$ the midpoint of the segment $AH$. The perpendicular at $M$ onto $OM$ intersects lines $AB$ and $AC$ at $P$ and $Q$, respectively. Prove that $MP = MQ$.
2023 Romania Team Selection Test, P2
A [i]diagonal line[/i] of a (not necessarily convex) polygon with at least four sides is any line through two non-adjacent vertices of that polygon. Determine all polygons with at least four sides satisfying the following condition: The reflexion of each vertex in each diagonal line lies inside or on the boundary of the polygon.
[i]The Problem Selection Committee[/i]
2012 CHMMC Spring, 7
A positive integer $x$ is $k$-[i]equivocal [/i] if there exists two positive integers $b$, $b'$ such that when $x$ is represented in base $b$ and base $b'$, the two representations have digit sequences of length $k$ that are permutations of each other. The smallest $2$-equivocal number is $7$, since $7$ is $21$ in base $3$ and $12$ in base $5$. Find the smallest $3$-equivocal number.
2011 ISI B.Math Entrance Exam, 1
Given $a,x\in\mathbb{R}$ and $x\geq 0$,$a\geq 0$ . Also $\sin(\sqrt{x+a})=\sin(\sqrt{x})$ . What can you say about $a$??? Justify your answer.
2007 Iran Team Selection Test, 3
Find all solutions of the following functional equation: \[f(x^{2}+y+f(y))=2y+f(x)^{2}. \]
2010 Indonesia TST, 4
Given a positive integer $n$ and $I = \{1, 2,..., k\}$ with $k$ is a positive integer.
Given positive integers $a_1, a_2, ..., a_k$ such that for all $i \in I$: $1 \le a_i \le n$ and $$\sum_{i=1}^k a_i \ge 2(n!).$$
Show that there exists $J \subseteq I$ such that $$n! + 1 \ge \sum_{j \in J}a_j >\sqrt {n! + (n - 1)n}$$
2016 AMC 10, 25
Let $f(x)=\sum_{k=2}^{10}(\lfloor kx \rfloor -k \lfloor x \rfloor)$, where $\lfloor r \rfloor$ denotes the greatest integer less than or equal to $r$. How many distinct values does $f(x)$ assume for $x \ge 0$?
$\textbf{(A)}\ 32\qquad\textbf{(B)}\ 36\qquad\textbf{(C)}\ 45\qquad\textbf{(D)}\ 46\qquad\textbf{(E)}\ \text{infinitely many}$
1971 IMO Longlists, 44
Let $m$ and $n$ denote integers greater than $1$, and let $\nu (n)$ be the number of primes less than or equal to $n$. Show that if the equation $\frac{n}{\nu(n)}=m$ has a solution, then so does the equation $\frac{n}{\nu(n)}=m-1$.
2014 Estonia Team Selection Test, 2
Let $a, b$ and $c$ be positive real numbers for which $a + b + c = 1$. Prove that $$\frac{a^2}{b^3 + c^4 + 1}+\frac{b^2}{c^3 + a^4 + 1}+\frac{c^2}{a^3 + b^4 + 1} > \frac{1}{5}$$
1997 National High School Mathematics League, 7
Real numbers $x,y$ satisfy that $\begin{cases}
(x-1)^3+1997(x-1)=-1\\
(y-1)^3+1997(y-1)=1
\end{cases}$, then $x+y=$________.
1999 Tournament Of Towns, 1
The incentre of a triangle is joined by three segments to the three vertices of the triangle, thereby dividing it into three smaller triangles. If one of these three triangles is similar to the original triangle, find its angles.
(A Shapovalov)
MBMT Guts Rounds, 2015.4
Find the fourth-smallest positive integer that can be expressed as the product of two different prime numbers.
2021 Girls in Math at Yale, 4
Cat and Claire are having a conversation about Cat's favorite number.
Cat says, "My favorite number is a two-digit positive integer that is the product of three distinct prime numbers!"
Claire says, "I don't know your favorite number yet, but I do know that among four of the numbers that might be your favorite number, you could start with any one of them, add a second, subtract a third, and get the fourth!"
Cat says, "That's cool! My favorite number is not among those four numbers, though."
Claire says, "Now I know your favorite number!"
What is Cat's favorite number?
[i]Proposed by Andrew Wu and Andrew Milas[/i]
2008 Germany Team Selection Test, 1
Let $ ABC$ be an acute triangle, and $ M_a$, $ M_b$, $ M_c$ be the midpoints of the sides $ a$, $ b$, $ c$. The perpendicular bisectors of $ a$, $ b$, $ c$ (passing through $ M_a$, $ M_b$, $ M_c$) intersect the boundary of the triangle again in points $ T_a$, $ T_b$, $ T_c$. Show that if the set of points $ \left\{A,B,C\right\}$ can be mapped to the set $ \left\{T_a, T_b, T_c\right\}$ via a similitude transformation, then two feet of the altitudes of triangle $ ABC$ divide the respective triangle sides in the same ratio. (Here, "ratio" means the length of the shorter (or equal) part divided by the length of the longer (or equal) part.) Does the converse statement hold?
2017 Purple Comet Problems, 5
Find the greatest odd divisor of $160^3$.
2019 India PRMO, 30
Let $E$ denote the set of all natural numbers $n$ such that $3 < n < 100$ and the set $\{ 1, 2, 3, \ldots , n\}$ can be partitioned in to $3$ subsets with equal sums. Find the number of elements of $E$.
2016 Brazil Team Selection Test, 3
Let $ABC$ be a triangle with $\angle{C} = 90^{\circ}$, and let $H$ be the foot of the altitude from $C$. A point $D$ is chosen inside the triangle $CBH$ so that $CH$ bisects $AD$. Let $P$ be the intersection point of the lines $BD$ and $CH$. Let $\omega$ be the semicircle with diameter $BD$ that meets the segment $CB$ at an interior point. A line through $P$ is tangent to $\omega$ at $Q$. Prove that the lines $CQ$ and $AD$ meet on $\omega$.
2025 Korea Winter Program Practice Test, P8
Determine all triplets of positive integers $(p,m,n)$ such that $p$ is a prime, $m \neq n < 2p$ and $2 \nmid n$. Also, the following polynomial is reducible in $\mathbb{Z}[x]$
$$x^{2p} - 2px^m - p^2x^n - 1$$
2017 Polish Junior Math Olympiad Finals, 3.
Positive integers $a$ and $b$ are given such that each of the numbers $ab$ and $(a+1)(b+1)$ is a perfect square. Prove that there exists an integer $n>1$ such that the number $(a+n)(b+n)$ is a perfect square.
2025 Kyiv City MO Round 2, Problem 3
In a school, \( n \) different languages are taught. It is known that for any subset of these languages (including the empty set), there is exactly one student who knows these and only these languages (there are \( 2^n \) students in total). Each day, the students are divided into pairs and teach each other the languages that only one of them knows. If students are not allowed to be in the same pair twice, what is the minimum number of days the school administration needs to guarantee that all their students know all \( n \) languages?
[i]Proposed by Oleksii Masalitin[/i]
2014 Contests, 1
Prove that for $n\ge 2$ the following inequality holds:
$$\frac{1}{n+1}\left(1+\frac{1}{3}+\ldots +\frac{1}{2n-1}\right) >\frac{1}{n}\left(\frac{1}{2}+\ldots+\frac{1}{2n}\right).$$
2021 Balkan MO, 2
Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$, such that $f(x+f(x)+f(y))=2f(x)+y$ for all positive reals $x,y$.
[i]Proposed by Athanasios Kontogeorgis, Greece[/i]
2017 Thailand TSTST, 5
Let $\omega_1, \omega_2$ be two circles with different radii, and let $H$ be the exsimilicenter of the two circles. A point X outside both circles is given. The tangents from $X$ to $\omega_1$ touch $\omega_1$ at $P, Q$, and the tangents from $X$ to $\omega_2$ touch $\omega_2$ at $R, S$. If $PR$ passes through $H$ and is not a common tangent line of $\omega_1, \omega_2$, prove that $QS$ also passes through $H$.