Found problems: 85335
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.
LMT Team Rounds 2010-20, A2 B6
$1001$ marbles are drawn at random and without replacement from a jar of $2020$ red marbles and $n$ blue marbles. Find the smallest positive integer $n$ such that the probability that there are more blue marbles chosen than red marbles is strictly greater than $\frac{1}{2}$.
[i]Proposed by Taiki Aiba[/i]
Oliforum Contest I 2008, 3
Let $ C_1,C_2$ and $ C_3$ be three pairwise disjoint circles. For each pair of disjoint circles, we define their internal tangent lines as the two common tangents which intersect in a point between the two centres. For each $ i,j$, we define $ (r_{ij},s_{ij})$ as the two internal tangent lines of $ (C_i,C_j)$. Let $ r_{12},r_{23},r_{13},s_{12},s_{13},s_{23}$ be the sides of $ ABCA'B'C'$.
Prove that $ AA',BB'$ and $ CC'$ are concurrent.
[img]https://cdn.artofproblemsolving.com/attachments/1/2/5ef098966fc9f48dd06239bc7ee803ce4701e2.png[/img]
2013 Saudi Arabia IMO TST, 3
A Saudi company has two offices. One office is located in Riyadh and the other in Jeddah. To insure the connection between the two offices, the company has designated from each office a number of correspondents so that :
(a) each pair of correspondents from the same office share exactly one common correspondent from the other office.
(b) there are at least $10$ correspondents from Riyadh.
(c) Zayd, one of the correspondents from Jeddah, is in contact with exactly $8$ correspondents from Riyadh.
What is the minimum number of correspondents from Jeddah who are in contact with the correspondent Amr from Riyadh?