Found problems: 85335
2007 ITest, 17
If $x$ and $y$ are acute angles such that $x+y=\pi/4$ and $\tan y=1/6$, find the value of $\tan x$.
$\textbf{(A) }\dfrac{27\sqrt2-18}{71}\hspace{11.5em}\textbf{(B) }\dfrac{35\sqrt2-6}{71}\hspace{11.2em}\textbf{(C) }\dfrac{35\sqrt3+12}{33}$
$\textbf{(D) }\dfrac{37\sqrt3+24}{33}\hspace{11.5em}\textbf{(E) }1\hspace{15em}\textbf{(F) }\dfrac57$
$\textbf{(G) }\dfrac37\hspace{15.4em}\textbf{(H) }6\hspace{15em}\textbf{(I) }\dfrac16$
$\textbf{(J) }\dfrac12\hspace{15.7em}\textbf{(K) }\dfrac67\hspace{14.8em}\textbf{(L) }\dfrac47$
$\textbf{(M) }\sqrt3\hspace{14.5em}\textbf{(N) }\dfrac{\sqrt3}3\hspace{14em}\textbf{(O) }\dfrac56$
$\textbf{(P) }\dfrac23\hspace{15.4em}\textbf{(Q) }\dfrac1{2007}$
2018 BMT Spring, 5
If ri are integers such that $0 \le r_i < 31$ and $r_i$ satis fies the polynomial $x^4 + x^3 + x^2 + x \equiv 30$ (mod $31$), find $$\sum^4_{i=1}(r^2_i + 1)^{-1} \,\,\,\, (mod \,\,\,\, 31)$$ where $x^{-1}$ is the modulo inverse of $x$, that is, it is the unique integer $y$ such that $0 < y < 31$ and $xy -1$ is divisible by $31$.
2017 Australian MO, 3
Anna and Berta play a game in which they take turns in removing marbles from a table. Anna takes the first turn. When at the beginning of the turn there are $n\geq 1$ marbles on the table, then the player whose turn it is removes $k$ marbles, where $k\geq 1$ either is an even number with $k\leq \frac{n}{2}$ or an odd number with $\frac{n}{2}\leq k\leq n$. A player win the game if she removes the last marble from the table.
Determine the smallest number $N\geq 100000$ such that Berta can enforce a victory if there are exactly $N$ marbles on the tale in the beginning.
2016 Romania Team Selection Tests, 3
Given a positive integer $n$, show that for no set of integers modulo $n$, whose size exceeds $1+\sqrt{n+4}$, is it possible that the pairwise sums of unordered pairs be all distinct.
2010 LMT, 16
Determine the number of three digit integers that are equal to $19$ times the sum of its digits.
2015 Peru Cono Sur TST, P9
Let $m$ and $n$ be positive integers. A child walks the Cartesian plane taking a few steps.
The child begins its journey at the point $(0, n)$ and ends at the point $(m, 0)$ in such a way that:
$\bullet$ Each step has length $1$ and is parallel to either the $X$ or $Y$ axis.
$\bullet$ For each point $(x, y)$ of its path it is true that $x\ge 0$ and $y\ge 0$.
For each step of the child, the distance between the child and the axis to which said step is parallel is calculated. If the step causes the child to be further from the point $(0, 0)$ than before, we consider that distance as positive, otherwise, we consider that distance as negative. Prove that at the end of the boy's journey, the sum of all the distances is $0$.
2001 All-Russian Olympiad Regional Round, 11.8
Prove that in any set consisting of $117$ pairwise distinct three-digit numbers, you can choose $4$ pairwise disjoint subsets in which the sums of numbers are equal.
2012 District Olympiad, 3
Let be a natural number $ n, $ and two matrices $ A,B\in\mathcal{M}_n\left(\mathbb{C}\right) $ with the property that
$$ AB^2=A-B. $$
[b]a)[/b] Show that the matrix $ I_n+B $ is inversable.
[b]b)[/b] Show that $ AB=BA. $
2005 Purple Comet Problems, 8
Find $x$ if\[\cfrac{1}{\cfrac{1}{\cfrac{1}{\cfrac{1}{x}+\cfrac12}+\cfrac{1}{\cfrac{1}{x}+\cfrac12}}+\cfrac{1}{\cfrac{1}{\cfrac{1}{x}+\cfrac12}+\cfrac{1}{\cfrac{1}{x}+\cfrac12}}}=\frac{x}{36}.\]
2000 Miklós Schweitzer, 4
Let $a_1<a_2<a_3$ be positive integers. Prove that there are integers $x_1,x_2,x_3$ such that $\sum_{i=1}^3 |x_i | >0$, $\sum_{i=1}^3 a_ix_i= 0$ and
$$\max_{1\le i\le 3} | x_i|<\frac{2}{\sqrt{3}}\sqrt{a_3}+1$$.
1970 All Soviet Union Mathematical Olympiad, 131
How many sides of the convex polygon can equal its longest diagonal?
2006 AMC 10, 4
A digital watch displays hours and minutes with $ \text c{AM}$ and $ \text c{PM}$. What is the largest possible sum of the digits in the display?
$ \textbf{(A) } 17\qquad \textbf{(B) } 19\qquad \textbf{(C) } 21\qquad \textbf{(D) } 22\qquad \textbf{(E) } 23$
2006 Purple Comet Problems, 13
An equilateral triangle with side length $6$ has a square of side length $6$ attached to each of its edges as shown. The distance between the two farthest vertices of this figure (marked $A$ and $B$ in the figure) can be written as $m + \sqrt{n}$ where $m$ and $n$ are positive integers. Find $m + n$.
[asy]
draw((0,0)--(1,0)--(1/2,sqrt(3)/2)--cycle);
draw((1,0)--(1+sqrt(3)/2,1/2)--(1/2+sqrt(3)/2,1/2+sqrt(3)/2)--(1/2,sqrt(3)/2));
draw((0,0)--(-sqrt(3)/2,1/2)--(-sqrt(3)/2+1/2,1/2+sqrt(3)/2)--(1/2,sqrt(3)/2));
dot((-sqrt(3)/2+1/2,1/2+sqrt(3)/2));
label("A", (-sqrt(3)/2+1/2,1/2+sqrt(3)/2), N);
draw((1,0)--(1,-1)--(0,-1)--(0,0));
dot((1,-1));
label("B", (1,-1), SE);
[/asy]
1971 Canada National Olympiad, 1
$DEB$ is a chord of a circle such that $DE=3$ and $EB=5$. Let $O$ be the centre of the circle. Join $OE$ and extend $OE$ to cut the circle at $C$. (See diagram). Given $EC=1$, find the radius of the circle.
[asy]
size(6cm);
pair O = (0,0), B = dir(110), D = dir(30), E = 0.4 * B + 0.6 * D, C = intersectionpoint(O--2*E, unitcircle);
draw(unitcircle);
draw(O--C);
draw(B--D);
dot(O);
dot(B);
dot(C);
dot(D);
dot(E);
label("$B$", B, B);
label("$C$", C, C);
label("$D$", D, D);
label("$E$", E, dir(280));
label("$O$", O, dir(270));
[/asy]
2008 Irish Math Olympiad, 4
Given $ k \in [0,1,2,3]$ and a positive integer $ n$, let $ f_k(n)$ be the number of sequences $ x_1,...,x_n,$ where $ x_i \in [\minus{}1,0,1]$ for $ i\equal{}1,...,n,$ and
$ x_1\plus{}...\plus{}x_n \equiv k$ mod 4
a) Prove that $ f_1(n) \equal{} f_3(n)$ for all positive integers $ n$.
(b) Prove that
$ f_0(n) \equal{} [{3^n \plus{} 2 \plus{} [\minus{}1]^n}] / 4$
for all positive integers $ n$.
2015 South East Mathematical Olympiad, 3
Can you make $2015$ positive integers $1,2, \ldots , 2015$ to be a certain permutation which can be ordered in the circle such that the sum of any two adjacent numbers is a multiple of $4$ or a multiple of $7$?
2020 ELMO Problems, P1
Let $\mathbb{N}$ be the set of all positive integers. Find all functions $f : \mathbb{N} \to \mathbb{N}$ such that $$f^{f^{f(x)}(y)}(z)=x+y+z+1$$ for all $x,y,z \in \mathbb{N}$.
[i]Proposed by William Wang.[/i]
2024 Iran Team Selection Test, 2
For a right angled triangle $\triangle ABC$ with $\angle A=90$ we have $AC=2AB$. Point $M$ is the midpoint of side $BC$ and $I$ is incenter of triangle $\triangle ABC$. The line passing trough $M$ and perpendicular to $BI$ intersect with lines $BI$ and $AC$ at points $H$ and $K$ respectively. If the semi-line $IK$ cuts circumcircle of triangle $\triangle ABC$ at $F$ and $S$ be the second intersection point of line $FH$ with circumcircle of triangle $\triangle ABC$ , then prove that $SM$ is tangent to the incircle of triangle $\triangle ABC$.
[i]Proposed by Mahdi Etesami Fard[/i]
1994 Czech And Slovak Olympiad IIIA, 5
In an acute-angled triangle $ABC$, the altitudes $AA_1,BB_1,CC_1$ intersect at point $V$. If the triangles $AC_1V, BA_1V, CB_1V$ have the same area, does it follow that the triangle $ABC$ is equilateral?
2004 Gheorghe Vranceanu, 3
Consider the function $ f:(-\infty,1]\longrightarrow\mathbb{R} $ defined as
$$ f(x)=\left\{ \begin{matrix} \frac{5}{2} +2^x-\frac{1}{2^x} ,& \quad x<-1 \\ 3^{\sqrt{1-x^2}} ,& \quad x\in [-1,1] \end{matrix} \right. . $$
[b]a)[/b] For a fixed parameter, find the roots of $ f-m. $
[b]b)[/b] Study the inversability of the restrictions of $ f $ to $ (-\infty,-1] $ and $ [-1,1] $ and find the inverses of these that admit them.
[i]D. Zaharia[/i]
2008 ITest, 58
Finished with rereading Isaac Asimov's $\textit{Foundation}$ series, Joshua asks his father, "Do you think somebody will build small devices that run on nuclear energy while I'm alive?"
"Honestly, Josh, I don't know. There are a lot of very different engineering problems involved in designing such devices. But technology moves forward at an amazing pace, so I won't tell you we can't get there in time for you to see it. I $\textit{did}$ go to a graduate school with a lady who now works on $\textit{portable}$ nuclear reactors. They're not small exactly, but they aren't nearly as large as most reactors. That might be the first step toward a nuclear-powered pocket-sized video game.
Hannah adds, "There are already companies designing batteries that are nuclear in the sense that they release energy from uranium hydride through controlled exoenergetic processes. This process is not the same as the nuclear fission going on in today's reactors, but we can certainly call it $\textit{nuclear energy}$."
"Cool!" Joshua's interest is piqued.
Hannah continues, "Suppose that right now in the year $2008$ we can make one of these nuclear batteries in a battery shape that is $2$ meters $\textit{across}$. Let's say you need that size to be reduced to $2$ centimeters $\textit{across}$, in the same proportions, in order to use it to run your little video game machine. If every year we reduce the necessary volume of such a battery by $1/3$, in what year will the batteries first get small enough?"
Joshua asks, "The battery shapes never change? Each year the new batteries are similar in shape - in all dimensions - to the bateries from previous years?"
"That's correct," confirms Joshua's mother. "Also, the base $10$ logarithm of $5$ is about $0.69897$ and the base $10$ logarithm of $3$ is around $0.47712$." This makes Joshua blink. He's not sure he knows how to use logarithms, but he does think he can compute the answer. He correctly notes that after $13$ years, the batteries will already be barely more than a sixth of their original width.
Assuming Hannah's prediction of volume reduction is correct and effects are compounded continuously, compute the first year that the nuclear batteries get small enough for pocket video game machines. Assume also that the year $2008$ is $7/10$ complete.
1998 Romania Team Selection Test, 2
Let $ n \ge 3$ be a prime number and $ a_{1} < a_{2} < \cdots < a_{n}$ be integers. Prove that $ a_{1}, \cdots,a_{n}$ is an arithmetic progression if and only if there exists a partition of $ \{0, 1, 2, \cdots \}$ into sets $ A_{1},A_{2},\cdots,A_{n}$ such that
\[ a_{1} \plus{} A_{1} \equal{} a_{2} \plus{} A_{2} \equal{} \cdots \equal{} a_{n} \plus{} A_{n},\]
where $ x \plus{} A$ denotes the set $ \{x \plus{} a \vert a \in A \}$.
2018 Canadian Open Math Challenge, A4
Source: 2018 Canadian Open Math Challenge Part A Problem 4
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In the sequence of positive integers, starting with $2018, 121, 16, ...$ each term is the square of the sum of digits of the previous term. What is the $2018^{\text{th}}$ term of the sequence?
2022 Brazil National Olympiad, 2
The nonzero real numbers $a, b, c$ satisfy the following system: $$\begin{cases} a+ab=c\\ b+bc=a\\ c+ca=b \end{cases}$$ Find all possible values of the $abc$.
1970 IMO Longlists, 56
A square hole of depth $h$ whose base is of length $a$ is given. A dog is tied to the center of the square at the bottom of the hole by a rope of length $L >\sqrt{2a^2+h^2}$, and walks on the ground around the hole. The edges of the hole are smooth, so that the rope can freely slide along it. Find the shape and area of the territory accessible to the dog (whose size is neglected).