Found problems: 85335
2017 Mathematical Talent Reward Programme, MCQ: P 10
Let $f:\mathbb{R}\to \mathbb{R}$ be a differentiable function such that $\lim \limits_{x\to \infty}f'(x)=1$, then
[list=1]
[*] $f$ is increasing
[*] $f$ is unbounded
[*] $f'$ is bounded
[*] All of these
[/list]
1987 Greece National Olympiad, 3
There is no sequence $x_n$ strictly increasing with terms natural numbers such that : $$ x_n+x_{k}=x_{nk}, \ \ for \, any \,\,\, n, k \in \mathbb{N}^*$$
2024 ELMO Shortlist, A2
Let $n$ be a positive integer. Find the number of sequences $a_0,a_1,a_2,\dots,a_{2n}$ of integers in the range $[0,n]$ such that for all integers $0\leq k\leq n$ and all nonnegative integers $m$, there exists an integer $k\leq i\leq 2k$ such that $\lfloor k/2^m\rfloor=a_i.$
[i]Andrew Carratu[/i]
2024 Canada National Olympiad, 4
Treasure was buried in a single cell of an $M\times N$ ($2\le M$, $N$) grid. Detectors were brought to find the cell with the treasure. For each detector, you can set it up to scan a specific subgrid $[a,b]\times[c,d]$ with $1\le a\le b\le M$ and $1\le c\le d\le N$. Running the detector will tell you whether the treasure is in the region or not, though it cannot say where in the region the treasure was detected. You plan on setting up $Q$ detectors, which may only be run simultaneously after all $Q$ detectors are ready.
In terms of $M$ and $N$, what is the minimum $Q$ required to gaurantee to determine the location of the treasure?
2024 OMpD, 2
Let \( n \) be a positive integer, and let \( A \) and \( B \) be \( n \times n \) matrices with real coefficients such that
\[
ABBA - BAAB = A - B.
\]
(a) Prove that \( \text{Tr}(A) = \text{Tr}(B) \) and that \( \text{Tr}(A^2) = \text{Tr}(B^2) \).
(b) If \(BA^2B= A^2B^2\) and \(AB^2A= B^2A^2\), prove that \( \det A = \det B \).
Note: \( \text{Tr}(X) \) denotes the trace of \( X \), which is the sum of the elements on its main diagonal, and \( \det X \) denotes the determinant of \( X \).
2024 Baltic Way, 20
Positive integers $a$, $b$ and $c$ satisfy the system of equations
\begin{align*}
(ab-1)^2&=c(a^2+b^2)+ab+1,\\
a^2+b^2&=c^2+ab.
\end{align*}
a) Prove that $c+1$ is a perfect square.
b) Find all such triples $(a,b,c)$.
Durer Math Competition CD 1st Round - geometry, 2016.C+3
Let $ABCD$ be a square with unit sides. Which interior point $P$ will the expression $\sqrt2 \cdot AP + BP + CP$ have a minimum value, and what is this minimum?
2011 AMC 12/AHSME, 6
The players on a basketball team made some three-point shots, some two-point shots, and some one-point free throws. They scored as many points with two-point shots as with three-point shots. Their number of successful free throws was one more than their number of successful two-point shots. The team's total score was 61 points. How many free throws did they make?
$ \textbf{(A)}\ 13 \qquad
\textbf{(B)}\ 14 \qquad
\textbf{(C)}\ 15 \qquad
\textbf{(D)}\ 16 \qquad
\textbf{(E)}\ 17
$
2023 AMC 12/AHSME, 25
A regular pentagon with area $\sqrt{5}+1$ is printed on paper and cut out. The five vertices of the pentagon are folded into the center of the pentagon, creating a smaller pentagon. What is the area of the new pentagon?
$\textbf{(A)}~4-\sqrt{5}\qquad\textbf{(B)}~\sqrt{5}-1\qquad\textbf{(C)}~8-3\sqrt{5}\qquad\textbf{(D)}~\frac{\sqrt{5}+1}{2}\qquad\textbf{(E)}~\frac{2+\sqrt{5}}{3}$
2010 Swedish Mathematical Competition, 5
Consider the number of triangles where the side lengths $a,b,c$ satisfy $(a + b + c) (a + b -c) = 2b^2$.
Determine the angles in the triangle for which the angle opposite to the side with the length $a$ is as big as possible.
2023 Harvard-MIT Mathematics Tournament, 7
Let $ABC$ be a triangle. Point $D$ lies on segment $BC$ such that $\angle BAD = \angle DAC$. Point $X$ lies on the opposite side of line $BC$ as $A$ and satisfies $XB=XD$ and $\angle BXD = \angle ACB$. The point $Y$ is defined similarly. Prove that the lines $XY$ and $AD$ are perpendicular.
2021 Brazil National Olympiad, 3
Find all positive integers \(k\) for which there is an irrational \(\alpha>1\) and a positive integer \(N\) such that \(\left\lfloor\alpha^{n}\right\rfloor\) is a perfect square minus \(k\) for every integer \(n\) with \(n>N\).
2001 Stanford Mathematics Tournament, 3
Find the 2000th positive integer that is not the difference between any two integer squares.
2020 CHKMO, 2
Let $S={1,2,\ldots,100}$. Consider a partition of $S$ into $S_1,S_2,\ldots,S_n$ for some $n$, i.e. $S_i$ are nonempty, pairwise disjoint and $\displaystyle S=\bigcup_{i=1}^n S_i$. Let $a_i$ be the average of elements of the set $S_i$. Define the score of this partition by
\[\dfrac{a_1+a_2+\ldots+a_n}{n}.\]
Among all $n$ and partitions of $S$, determine the minimum possible score.
2006 Princeton University Math Competition, 3
Find the fifth root of $14348907$.
2012 Today's Calculation Of Integral, 842
Let $S_n=\int_0^{\pi} \sin ^ n x\ dx\ (n=1,\ 2,\ ,\ \cdots).$ Find $\lim_{n\to\infty} nS_nS_{n+1}.$
2018 Yasinsky Geometry Olympiad, 5
The point $M$ lies inside the rhombus $ABCD$. It is known that $\angle DAB=110^o$, $\angle AMD=80^o$, $\angle BMC= 100^o$. What can the angle $\angle AMB$ be equal?
2008 Bulgarian Autumn Math Competition, Problem 10.1
For which values of the parameter $a$ does the equation
\[(2x-a)\sqrt{ax^2-(a^2+a+2)x+2(a+1)}=0\]
has three different real roots.
2015 ASDAN Math Tournament, 13
A three-digit number $x$ in base $10$ has a units-digit of $6$. When $x$ is written is base $9$, the second digit of the number is $4$, and the first and third digit are equal in value. Compute $x$ in base $10$.
1996 All-Russian Olympiad, 6
In the isosceles triangle $ABC$ ($AC = BC$) point $O$ is the circumcenter, $I$ the incenter, and $D$ lies on $BC$ so that lines $OD$ and $BI$ are perpendicular. Prove that $ID$ and $AC$ are parallel.
[i]M. Sonkin[/i]
2021 Vietnam National Olympiad, 5
Let the polynomial $P(x)=a_{21}x^{21}+a_{20}x^{20}+\cdots +a_1x+a_0$ where $1011\leq a_i\leq 2021$ for all $i=0,1,2,...,21.$ Given that $P(x)$ has an integer root and there exists an positive real number$c$ such that $|a_{k+2}-a_k|\leq c$ for all $k=0,1,...,19.$
a) Prove that $P(x)$ has an only integer root.
b) Prove that $$\sum_{k=0}^{10}(a_{2k+1}-a_{2k})^2\leq 440c^2.$$
1984 Putnam, A5
Putnam 1984/A5) Let $R$ be the region consisting of all triples $(x,y,z)$ of nonnegative real numbers satisfying $x+y+z\leq 1$. Let $w=1-x-y-z$. Express the value of the triple integral
\[\iiint_{R}xy^{9}z^{8}w^{4}\ dx\ dy\ dz\]
in the form $a!b!c!d!/n!$ where $a,b,c,d$ and $n$ are positive integers.
[hide="A solution"]\[\iiint_{R}xy^{9}z^{8}w^{4}\ dx dy dz = 4\iiint_{R}\int_{0}^{1-x-y-z}xy^{9}z^{8}w^{3}\ dw dx dy dz = 4\iiiint_{Q}xy^{9}z^{8}w^{3}\ dw dx dy dz\]
where $Q=\left\{ (x,y,z,w)\in\mathbb{R}^{4}|\ x,y,z,w\geq 0, x+y+z+w\leq 1\right\}$, which is a Dirichlet integral giving
\[4\iiiint_{Q}x^{1}y^{9}z^{8}w^{3}\ dw dx dy dz = 4\cdot\frac{1!9!8!3!}{(2+10+9+4)!}= \frac{1!9!8!4!}{25!}\][/hide]
1989 Tournament Of Towns, (210) 4
Prove that if $k$ is an even positive integer then it is possible to write the integers from $1$ to $k-1$ in such an order that the sum of no set of successive numbers is divisible by $k$ .
2020 Durer Math Competition Finals, 2
What number should we put in place of the question mark such that the following statement becomes true?
$$11001_? = 54001_{10}$$
A number written in the subscript means which base the number is in.
2004 Gheorghe Vranceanu, 1
Define a finite sequence $ \left( s_i \right)_{1\le i\le 2004} $ with $ s_0+2=s_1+1=s_2=2 $ and the recurrence relation
$$ s_n=1+s_{n-1} +s_{n-2} -s_{n-3} . $$
Calculate its last element.