Found problems: 85335
1995 Moldova Team Selection Test, 6
On a spherical surface there is a set $M{}$ with $n{}$ points with the property: for every point $A{}$ from $M{}$ there exist points $B$ and $C$ from $M{}$ such that the triangle $ABC$ is equilateral. For every equilateral triangle with vertexes in $M{}$ the perpendicular on its plane that goes through the geometric center of the other points from $M{}$. Prove that all these perpendiculars are concurrent.
2001 JBMO ShortLists, 6
Find all integers $x$ and $y$ such that $x^3\pm y^3 =2001p$, where $p$ is prime.
2013 All-Russian Olympiad, 2
Acute-angled triangle $ABC$ is inscribed into circle $\Omega$. Lines tangent to $\Omega$ at $B$ and $C$ intersect at $P$. Points $D$ and $E$ are on $AB$ and $AC$ such that $PD$ and $PE$ are perpendicular to $AB$ and $AC$ respectively. Prove that the orthocentre of triangle $ADE$ is the midpoint of $BC$.
2019 AMC 10, 1
Alicia had two containers. The first was $\tfrac{5}{6}$ full of water and the second was empty. She poured all the water from the first container into the second container, at which point the second container was $\tfrac{3}{4}$ full of water. What is the ratio of the volume of the first container to the volume of the second container?
$\textbf{(A) } \frac{5}{8} \qquad \textbf{(B) } \frac{4}{5} \qquad \textbf{(C) } \frac{7}{8} \qquad \textbf{(D) } \frac{9}{10} \qquad \textbf{(E) } \frac{11}{12}$
2018 Mathematical Talent Reward Programme, SAQ: P 5
[list=1]
[*] Prove that, the sequence of remainders obtained when the Fibonacci numbers are divided by $n$ is periodic, where $n$ is a natural number.
[*] There exists no such non-constant polynomial with integer coefficients such that for every Fibonacci number $n,$ $ P(n)$ is a prime.
[/list]
2005 QEDMO 1st, 6 (U1)
Prove that for any four real numbers $a$, $b$, $c$, $d$, the inequality
\[ \left(a-b\right)\left(b-c\right)\left(c-d\right)\left(d-a\right)+\left(a-c\right)^2\left(b-d\right)^2\geq 0 \]
holds.
[hide="comment"]This is inequality (350) in: Mihai Onucu Drimbe, [i]Inegalitati, idei si metode[/i], Zalau: Gil, 2003.
Posted here only for the sake of completeness; in fact, it is more or less the same as http://www.mathlinks.ro/Forum/viewtopic.php?t=3152 .[/hide]
Darij
2022 JBMO Shortlist, N5
Find all pairs $(a, p)$ of positive integers, where $p$ is a prime, such that for any pair of positive integers $m$ and $n$ the remainder obtained when $a^{2^n}$ is divided by $p^n$ is non-zero and equals the remainder obtained when $a^{2^m}$ is divided by $p^m$.
2006 Purple Comet Problems, 1
Michael is celebrating his fifteenth birthday today. How many Sundays have there been in his lifetime?
2016 Indonesia TST, 1
Let $n \ge 3$ be a positive integer. We call a $3 \times 3$ grid [i]beautiful[/i] if the cell located at the center is colored white and all other cells are colored black, or if it is colored black and all other cells are colored white. Determine the minimum value of $a+b$ such that there exist positive integers $a$, $b$ and a coloring of an $a \times b$ grid with black and white, so that it contains $n^2 - n$ [i]beautiful[/i] subgrids.
2023 Pan-African, 3
Consider a sequence of real numbers defined by:
\begin{align*}
x_{1} & = c \\
x_{n+1} & = cx_{n} + \sqrt{c^{2} - 1}\sqrt{x_{n}^{2} - 1} \quad \text{for all } n \geq 1.
\end{align*}
Show that if $c$ is a positive integer, then $x_{n}$ is an integer for all $n \geq 1$. [i](South Africa)[/i]
1997 Tournament Of Towns, (533) 5
Prove that the number
(a) $97^{97}$
(b) $1997^{17}$
cannot be equal to a sum of cubes of several consecutive integers.
(AA Egorov)
2018 Purple Comet Problems, 26
Let $a, b$, and $c$ be real numbers. Let $u = a^2 + b^2 + c^2$ and $v = 2ab + 2bc + 2ca$. Suppose $2018u = 1001v + 1024$. Find the maximum possible value of $35a - 28b - 3c$.
1973 Miklós Schweitzer, 4
Let $ f(n)$ be that largest integer $ k$ such that $ n^k$ divides $ n!$, and let $ F(n)\equal{} \max_{2 \leq m \leq n} f(m)$. Show that \[ \lim_{n\rightarrow \infty} \frac{F(n) \log n}{n \log \log n}\equal{}1.\]
[i]P. Erdos[/i]
2021 AMC 12/AHSME Fall, 5
Elmer the emu takes $44$ equal strides to walk between consecutive telephone poles on a rural road. Oscar the ostrich can cover the same distance in $12$ equal leaps. The telephone poles are evenly spaced, and the $41$st pole along this road is exactly one mile ($5280$ feet) from the first pole. How much longer, in feet, is Oscar's leap than Elmer's stride?
$\textbf{(A) }6\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }11\qquad\textbf{(E) }15$
1984 National High School Mathematics League, 10
All solutions of $\cos x=\cos\frac{x}{4}$ are________, it has_________different solutions in $(0,24\pi)$.
2023 Azerbaijan IMO TST, 1
Let $k\ge2$ be an integer. Find the smallest integer $n \ge k+1$ with the property that there exists a set of $n$ distinct real numbers such that each of its elements can be written as a sum of $k$ other distinct elements of the set.
2024 Irish Math Olympiad, P4
How many 4-digit numbers $ABCD$ are there with the property that $|A-B|= |B-C|= |C-D|$?
Note that the first digit $A$ of a four-digit number cannot be zero.
2025 China Team Selection Test, 15
Let \( X \) be a finite set of real numbers, \( d \) be a real number, and \(\lambda_1, \lambda_2, \cdots, \lambda_{2025}\) be 2025 non-zero real numbers. Define
\[A =
\left\{
(x_1, x_2, \cdots, x_{2025}) : x_1, x_2, \cdots, x_{2025} \in X \text{ and } \sum_{i=1}^{2025} \lambda_i x_i = d
\right\},\]
\[B =
\left\{
(x_1, x_2, \cdots, x_{2024}) : x_1, x_2, \cdots, x_{2024} \in X \text{ and } \sum_{i=1}^{2024} (-1)^i x_i = 0
\right\},\]
\[C =
\left\{
(x_1, x_2, \cdots, x_{2026}) : x_1, x_2, \cdots, x_{2026} \in X \text{ and } \sum_{i=1}^{2026} (-1)^i x_i = 0
\right\}.\]
Show that \( |A|^2 \leq |B| \cdot |C| \).
2003 Singapore MO Open, 3
For any given prime $p$, determine whether the equation $x^2 + y^2 + p^z = 2003$ always has integer solutions in $x, y, z$. Justify your answer
2015 ASDAN Math Tournament, 3
Place points $A$, $B$, $C$, $D$, $E$, and $F$ evenly spaced on a unit circle. Compute the area of the shaded $12$-sided region, where the region is bounded by line segments $AD$, $DF$, $FB$, $BE$, $EC$, and $CA$.
[center]<see attached>[/center]
2015 HMNT, 5
Kelvin the Frog is trying to hop across a river. The river has $10$ lilypads on it, and he must hop on them in a specific order (the order is unknown to Kelvin). If Kelvin hops to the wrong lilypad at any point, he will be thrown back to the wrong side of the river and will have to start over. Assuming Kelvin is infinitely intelligent, what is the minimum number of hops he will need to guarantee reaching the other side?
2007 Korea - Final Round, 4
Find all pairs $ (p, q)$ of primes such that $ {p}^{p}\plus{}{q}^{q}\plus{}1$ is divisible by $ pq$.
2011 Saudi Arabia BMO TST, 4
Let $p \ge 3$ be a prime. For $j = 1,2 ,... ,p - 1$, let $r_j$ be the remainder when the integer $\frac{j^{p-1}-1}{p}$ is divided by $p$.
Prove that $$r_1 + 2r_2 + ... + (p - 1)r_{p-1} \equiv \frac{p+1}{2} (\mod p)$$
2002 Turkey MO (2nd round), 3
Graph Airlines $ (GA)$ operates flights between some of the cities of the Republic of Graphia. There are at least three $ GA$ flights from each city, and it is possible to travel from any city in Graphia to any city in Graphia using $ GA$ flights. $ GA$ decides to discontinue some of its flights. Show that this can be done in such a way that it is still possible to travel between any two cities using $ GA$ flights, yet at least $ 2/9$ of the cities have only one flight.
2017 Estonia Team Selection Test, 3
Let $ABC$ be a triangle with $AB = AC \neq BC$ and let $I$ be its incentre. The line $BI$ meets $AC$ at $D$, and the line through $D$ perpendicular to $AC$ meets $AI$ at $E$. Prove that the reflection of $I$ in $AC$ lies on the circumcircle of triangle $BDE$.