Found problems: 85335
1994 Spain Mathematical Olympiad, 1
Prove that if an arithmetic progression contains a perfect square, then it contains infinitely many perfect squares.
2010 IFYM, Sozopol, 4
Let $x,y\in \mathbb{N}$ and $k=\frac{x^2+y^2}{2xy+1}$. Determine all natural values of $k$.
1975 Spain Mathematical Olympiad, 8
Two real numbers between $0$ and $1$ are randomly chosen. Calculate the probability that any one of them is less than the square of the other.
2010 Today's Calculation Of Integral, 586
Evaluate $ \int_0^1 \frac{x^{14}}{x^2\plus{}1}\ dx$.
2020 Dutch BxMO TST, 5
A set S consisting of $2019$ (different) positive integers has the following property:
[i]the product of every 100 elements of $S$ is a divisor of the product of the remaining $1919$ elements[/i].
What is the maximum number of prime numbers that $S$ can contain?
1987 Dutch Mathematical Olympiad, 4
On each side of a regular tetrahedron with edges of length $1$ one constructs exactly such a tetrahedron. This creates a dodecahedron with $8$ vertices and $18$ edges. We imagine that the dodecahedron is hollow. Calculate the length of the largest line segment that fits entirely within this dodecahedron.
2015 Saint Petersburg Mathematical Olympiad, 1
$x,y$ are real numbers such that $$x^2+y^2=1 , 20x^3-15x=3$$Find the value of $|20y^3-15y|$.(K. Tyshchuk)
2021 CMIMC Integration Bee, 2
$$\int\frac{\ln^2(x)}{x}\,dx$$
[i]Proposed by Connor Gordon[/i]
2001 China Team Selection Test, 3
For a given natural number $k > 1$, find all functions $f:\mathbb{R} \to \mathbb{R}$ such that for all $x, y \in \mathbb{R}$, $f[x^k + f(y)] = y +[f(x)]^k$.
2021 CMIMC, 13
Let $p=3\cdot 10^{10}+1$ be a prime and let $p_n$ denote the probability that $p\mid (k^k-1)$ for a random $k$ chosen uniformly from $\{1,2,\cdots,n\}$. Given that $p_n\cdot p$ converges to a value $L$ as $n$ goes to infinity, what is $L$?
[i]Proposed by Vijay Srinivasan[/i]
2010 Contests, 2
Let $P_1(x) = ax^2 - bx - c$, $P_2(x) = bx^2 - cx - a$, $P_3(x) = cx^2 - ax - b$ be three quadratic polynomials. Suppose there exists a real number $\alpha$ such that $P_1(\alpha) = P_2(\alpha) = P_3(\alpha)$. Prove that $a = b = c$.
2014 NZMOC Camp Selection Problems, 5
Let $ABC$ be an acute angled triangle. Let the altitude from $C$ to $AB$ meet $AB$ at $C'$ and have midpoint $M$, and let the altitude from $B$ to $AC$ meet $AC$ at $B'$ and have midpoint $N$. Let $P$ be the point of intersection of $AM$ and $BB'$ and $Q$ the point of intersection of $AN$ and $CC'$. Prove that the point $M, N, P$ and $Q$ lie on a circle.
PEN A Problems, 108
For each integer $n>1$, let $p(n)$ denote the largest prime factor of $n$. Determine all triples $(x, y, z)$ of distinct positive integers satisfying [list] [*] $x, y, z$ are in arithmetic progression, [*] $p(xyz) \le 3$. [/list]
2000 Iran MO (3rd Round), 1
Does there exist a natural number $N$ which is a power of$2$, such that one
can permute its decimal digits to obtain a different power of $2$?
2005 AMC 10, 15
How many positive integer cubes divide $ 3!\cdot 5!\cdot 7!$?
$ \textbf{(A)}\ 2\qquad
\textbf{(B)}\ 3\qquad
\textbf{(C)}\ 4\qquad
\textbf{(D)}\ 5\qquad
\textbf{(E)}\ 6$
2023 IMC, 6
Ivan writes the matrix $\begin{pmatrix} 2 & 3\\ 2 & 4\end{pmatrix}$ on the board. Then he performs the following operation on the matrix several times:
[b]1.[/b] he chooses a row or column of the matrix, and
[b]2.[/b] he multiplies or divides the chosen row or column entry-wise by the other row or column, respectively.
Can Ivan end up with the matrix $\begin{pmatrix} 2 & 4\\ 2 & 3\end{pmatrix}$ after finitely many steps?
2021 Bosnia and Herzegovina Junior BMO TST, 4
Let $n$ be a nonzero natural number and let $S = \{1, 2, . . . , n\}$.
A $3 \times n$ board is called [i]beautiful [/i] if it can be completed with numbers from the set $S$ like this as long as the following conditions are met:
$\bullet$ on each line, each number from the set S appears exactly once,
$\bullet$ on each column the sum of the products of two numbers on that column is divisible by $n$ (that is, if the numbers $a, b, c$ are written on a column, it must be $ab + bc + ca$ be divisible by $n$).
For which values of the natural number $n$ are there beautiful tables ¸and for which values do not exist? Justify your answer.
1950 AMC 12/AHSME, 42
The equation $ x^{x^{x}}...\equal{}2$ is satisfied when $x$ is equal to:
$\textbf{(A)}\ \infty \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ \sqrt[4]{2} \qquad
\textbf{(D)}\ \sqrt{2} \qquad
\textbf{(E)}\ \text{None of these}$
1990 China Team Selection Test, 3
In set $S$, there is an operation $'' \circ ''$ such that $\forall a,b \in S$, a unique $a \circ b \in S$ exists. And
(i) $\forall a,b,c \in S$, $(a \circ b) \circ c = a \circ (b \circ c)$.
(ii) $a \circ b \neq b \circ a$ when $a \neq b$.
Prove that:
a.) $\forall a,b,c \in S$, $(a \circ b) \circ c = a \circ c$.
b.) If $S = \{1,2, \ldots, 1990\}$, try to define an operation $'' \circ ''$ in $S$ with the above properties.
2016 Belarus Team Selection Test, 1
a) Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ such that\[f(x-f(y))=f(f(x))-f(y)-1\]holds for all $x,y\in\mathbb{Z}$. (It is [url=https://artofproblemsolving.com/community/c6h1268817p6621849]2015 IMO Shortlist A2 [/url])
b) The same question for if \[f(x-f(y))=f(f(x))-f(y)-2\] for all integers $x,y$
2019 Saint Petersburg Mathematical Olympiad, 1
For a non-constant arithmetic progression $(a_n)$ there exists a natural $n$ such that $a_{n}+a_{n+1} = a_{1}+…+a_{3n-1}$ . Prove that there are no zero terms in this progression.
2015 IMC, 3
Let $F(0)=0$, $F(1)=\frac32$, and $F(n)=\frac{5}{2}F(n-1)-F(n-2)$
for $n\ge2$.
Determine whether or not $\displaystyle{\sum_{n=0}^{\infty}\,
\frac{1}{F(2^n)}}$ is a rational number.
(Proposed by Gerhard Woeginger, Eindhoven University of Technology)
2017 AMC 12/AHSME, 11
Claire adds the degree measures of the interior angles of a convex polygon and arrives at a sum of $2017$. She then discovers that she forgot to include one angle. What is the degree measure of the forgotten angle?
$\textbf{(A)}\ 37\qquad\textbf{(B)}\ 63\qquad\textbf{(C)}\ 117\qquad\textbf{(D)}\ 143\qquad\textbf{(E)}\ 163$
1995 Singapore Team Selection Test, 3
In a dance, a group $S$ of $1994$ students stand in a big circle. Each student claps the hands of each of his two neighbours a number of times. For each student $x,$ let $f(x)$ be the total number of times $x$ claps the hands of his neighbours. As an example, suppose there are $3$ students $A, B$ and $C$. A claps hand with $B$ two times, $B$ claps hand with $C$ three times and $C$ claps hand with $A$ five times. Then $f(A) = 7, f(B) = 5$ and $f(C) = 8.$
(i) Prove that $\{f(x) | x \in S\}\ne\{n | n$ is an integer, $2 \le n \le 1995\}$.
(ii) Find an example in which $\{f(x) | x \in S\} = \{n | n$ is an integer, $n \ne 3, 2 \le n \le 1996\}$
1968 German National Olympiad, 5
Prove that for all real numbers $x$ of the interval $0 < x <\pi$ the inequality
$$\sin x +\frac12 \sin 2x +\frac13 \sin 3x > 0$$
holds.