Found problems: 85335
2007 Germany Team Selection Test, 1
The sequence of real numbers $a_0,a_1,a_2,\ldots$ is defined recursively by \[a_0=-1,\qquad\sum_{k=0}^n\dfrac{a_{n-k}}{k+1}=0\quad\text{for}\quad n\geq 1.\]Show that $ a_{n} > 0$ for all $ n\geq 1$.
[i]Proposed by Mariusz Skalba, Poland[/i]
Ukrainian From Tasks to Tasks - geometry, 2012.2
The triangle $ABC$ is equilateral. Find the locus of the points $M$ such that the triangles $ABM$ and $ACM$ are both isosceles.
2000 Mexico National Olympiad, 4
Let $a$ and $b$ be positive integers not divisible by $5$. A sequence of integers is constructed as follows: the first term is $5$, and every consequent term is obtained by multiplying its precedent by $a$ and adding $b$. (For example, if $a = 2$ and $b = 4$, the first three terms are $5,14,32$.) What is the maximum possible number of primes that can occur before encoutering the first composite term?
2018 AIME Problems, 3
Kathy has $5$ red cards and $5$ green cards. She shuffles the $10$ cards and lays out $5$ of the cards in a row in a random order. She will be happy if and only if all the red cards laid out are adjacent and all the green cards laid out are adjacent. For example, card orders RRGGG, GGGGR, or RRRRR will make Kathy happy, but RRRGR will not. The probability that Kathy will be happy is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2014 PUMaC Combinatorics A, 1
What is the largest $n$ such that a square cannot be partitioned into $n$ smaller, non-overlapping squares?
2009 Bulgarian Spring Mathematical Competition, Problem 10.3
On the side $BC$ of the triangle $\Delta ABC$ is choosen point $K$,such that $2\angle BAK=3\angle KAC$.Prove that $AB^2AC^3>AK^5$
2002 Manhattan Mathematical Olympiad, 2
Let us consider the sequence $1,2, 3, \ldots , 2002$. Somebody choses $1002$ numbers from the sequence. Prove that there are two of the chosen numbers which are relatively prime (i.e. do not have any common divisors except $1$).
2006 Vietnam National Olympiad, 2
Let $ABCD$ be a convex quadrilateral. Take an arbitrary point $M$ on the line $AB$, and let $N$ be the point of intersection of the circumcircles of triangles $MAC$ and $MBC$ (different from $M$). Prove that:
a) The point $N$ lies on a fixed circle;
b) The line $MN$ passes though a fixed point.
2007 Estonia Team Selection Test, 2
Let $D$ be the foot of the altitude of triangle $ABC$ drawn from vertex $A$. Let $E$ and $F$ be points symmetric to $D$ w.r.t. lines $AB$ and $AC$, respectively. Let $R_1$ and $R_2$ be the circumradii of triangles $BDE$ and $CDF$, respectively, and let $r_1$ and $r_2$ be the inradii of the same triangles. Prove that $|S_{ABD} - S_{ACD}| > |R_1r_1 - R_2r_2|$
2014 Bulgaria JBMO TST, 5
From the foot $D$ of the height $CD$ in the triangle $ABC,$ perpendiculars to $BC$ and $AC$ are drawn, which they intersect at points $M$ and $N.$ Let $\angle CAB = 60^{\circ} , \angle CBA = 45^{\circ} ,$ and $H$ be the orthocentre of $MNC.$ If $O$ is the midpoint of $CD,$ find $\angle COH.$
2006 National Olympiad First Round, 30
How many integer triples $(x,y,z)$ are there such that \[\begin{array}{rcl} x - yz^2&\equiv & 1 \pmod {13} \\ xz+y&\equiv& 4 \pmod {13} \end{array}\] where $0\leq x < 13$, $0\leq y <13$, and $0\leq z< 13$?
$
\textbf{(A)}\ 10
\qquad\textbf{(B)}\ 23
\qquad\textbf{(C)}\ 36
\qquad\textbf{(D)}\ 49
\qquad\textbf{(E)}\ \text{None of above}
$
2010 Harvard-MIT Mathematics Tournament, 2
Let $f$ be a function such that $f(0)=1$, $f^\prime (0)=2$, and \[f^{\prime\prime}(t)=4f^\prime(t)-3f(t)+1\] for all $t$. Compute the $4$th derivative of $f$, evaluated at $0$.
1970 Poland - Second Round, 6
If $ A $ is a subset of $ X $, then we take $ A^1 = A $, $ A^{-1} = X - A $. The subsets $ A_1, A_2, \ldots, A_k $ are called mutually independent if the product $ A_1^{\varepsilon_1} \cap A_2^{\varepsilon_2} \ldots A_k^{\varepsilon_k} $ is nonempty for every system of numbers $ \varepsilon_1 , \varepsilon_2, \ldots, \varepsilon_k $, such that $ |\varepsilon_2| = $1 for $ i = 1, 2, \ldots, k $.
What is the maximum number of mutually independent subsets of a $2^n $-element set?
2023 Mexican Girls' Contest, 7
Suppose $a$ and $b$ are real numbers such that $0 < a < b < 1$. Let
$$x= \frac{1}{\sqrt{b}} - \frac{1}{\sqrt{b+a}},\hspace{1cm}
y= \frac{1}{b-a} - \frac{1}{b}\hspace{0.5cm}\textrm{and}\hspace{0.5cm}
z= \frac{1}{\sqrt{b-a}} - \frac{1}{\sqrt{b}}.$$
Show that $x$, $y$, $z$ are always ordered from smallest to largest in the same way, regardless of the choice of $a$ and $b$. Find this order among $x$, $y$, $z$.
2021-IMOC, G8
Let $P$ be an arbitrary interior point of $\triangle ABC$, and $AP$, $BP$, $CP$ intersect $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. Suppose that $M$ be the midpoint of $BC$, $\odot(AEF)$ and $\odot(ABC)$ intersect at $S$, $SD$ intersects $\odot(ABC)$ at $X$, and $XM$ intersects $\odot(ABC)$ at $Y$. Show that $AY$ is tangent to $\odot(AEF)$.
2012 USAMO, 1
Find all integers $n \geq 3$ such that among any $n$ positive real numbers $a_1, a_2, \hdots, a_n$ with $\text{max}(a_1,a_2,\hdots,a_n) \leq n \cdot \text{min}(a_1,a_2,\hdots,a_n)$, there exist three that are the side lengths of an acute triangle.
2020 AIME Problems, 9
While watching a show, Ayako, Billy, Carlos, Dahlia, Ehuang, and Frank sat in that order in a row of six chairs. During the break, they went to the kitchen for a snack. When they came back, they sat on those six chairs in such a way that if two of them sat next to each other before the break, then they did not sit next to each other after the break. Find the number of possible seating orders they could have chosen after the break.
2022 Baltic Way, 3
We call a two-variable polynomial $P(x, y)$ [i]secretly one-variable,[/i] if there exist polynomials $Q(x)$ and $R(x, y)$ such that $\deg(Q) \ge 2$ and $P(x, y) = Q(R(x, y))$ (e.g. $x^2 + 1$ and $x^2y^2 +1$ are [i]secretly one-variable[/i], but $xy + 1$ is not).
Prove or disprove the following statement: If $P(x, y)$ is a polynomial such that both $P(x, y)$ and $P(x, y) + 1$ can be written as the product of two non-constant polynomials, then $P$ is [i]secretly one-variable[/i].
[i]Note: All polynomials are assumed to have real coefficients. [/i]
Fractal Edition 1, P4
Let \( P(x) \) be a polynomial with natural coefficients. We denote by \( d(n) \) the number of positive divisors of the natural number \( n \), and by \( \sigma(n) \), the sum of these divisors. The sequence \( a_n \) is defined as follows:
\[
a_{n+1} \in \left\{
\begin{array}{ll}
\sigma(P(d(a_n))) \\
d(P(\sigma(a_n)))
\end{array}
\right.
\]
That is, \( a_{n+1} \) is one of the two terms above. Show that there exists a constant \( C \), depending on \( a_1 \) and \( P(x) \), such that for all \( i \), \( a_i < C \); in other words, show that the sequence \( a_n \) is bounded.
2022 CHMMC Winter (2022-23), 4
Let $ABC$ be a triangle with $AB = 4$, $BC = 5$, $CA = 6$. Triangles $APB$ and $CQA$ are erected outside $ABC$ such that $AP=PB$, $\overline{AP}\perp \overline{PB}$ and $CQ=QA$, $\overline{CQ}\perp \overline{QA}$. Pick a point $X$ uniformly at random from segment $\overline{BC}$. What is the expected value of the area of triangle $PXQ$?
2010 AMC 10, 22
Eight points are chosen on a circle, and chords are drawn connecting every pair of points. No three chords intersect in a single point inside the circle. How many triangles with all three vertices in the interior of the circle are created?
$ \textbf{(A)}\ 28 \qquad
\textbf{(B)}\ 56 \qquad
\textbf{(C)}\ 70 \qquad
\textbf{(D)}\ 84 \qquad
\textbf{(E)}\ 140$
2016 India IMO Training Camp, 3
Let $n$ be an odd natural number. We consider an $n\times n$ grid which is made up of $n^2$ unit squares and $2n(n+1)$ edges. We colour each of these edges either $\color{red} \textit{red}$ or $\color{blue}\textit{blue}$. If there are at most $n^2$ $\color{red} \textit{red}$ edges, then show that there exists a unit square at least three of whose edges are $\color{blue}\textit{blue}$.
2017 ASDAN Math Tournament, 3
Compute
$$\int_0^1\frac{x^{2017}-1}{\log x}dx.$$
2017 South East Mathematical Olympiad, 8
Given the positive integer $m \geq 2$, $n \geq 3$. Define the following set
$$S = \left\{(a, b) | a \in \{1, 2, \cdots, m\}, b \in \{1, 2, \cdots, n\} \right\}.$$
Let $A$ be a subset of $S$. If there does not exist positive integers $x_1, x_2, x_3, y_1, y_2, y_3$ such that $x_1 < x_2 < x_3, y_1 < y_2 < y_3$ and
$$(x_1, y_2), (x_2, y_1), (x_2, y_2), (x_2, y_3), (x_3, y_2) \in A.$$
Determine the largest possible number of elements in $A$.
2010 Today's Calculation Of Integral, 605
Let $f(x)$ be a differentiable function. Find the following limit value:
\[\lim_{n\to\infty} \dbinom{n}{k}\left\{f\left(\frac{x}{n}\right)-f(0)\right\}^k.\]
Especially, for $f(x)=(x-\alpha)(x-\beta)$ find the limit value above.
1956 Tokyo Institute of Technology entrance exam