Found problems: 85335
2019 Sharygin Geometry Olympiad, 22
Let $AA_0$ be the altitude of the isosceles triangle $ABC~(AB = AC)$. A circle $\gamma$ centered at the midpoint of $AA_0$ touches $AB$ and $AC$. Let $X$ be an arbitrary point of line $BC$. Prove that the tangents from $X$ to $\gamma$ cut congruent segments on lines $AB$ and $AC$
1978 Putnam, A2
Let $a,b, p_1 ,p_2, \ldots, p_n$ be real numbers with $a \ne b$. Define $f(x)= (p_1 -x) (p_2 -x) \cdots (p_n -x)$. Show that
$$ \text{det} \begin{pmatrix} p_1 & a& a & \cdots & a \\
b & p_2 & a & \cdots & a\\
b & b & p_3 & \cdots & a\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
b & b& b &\cdots &p_n
\end{pmatrix}= \frac{bf(a) -af(b)}{b-a}.$$
2017 CHMMC (Fall), 6
The country of Claredena has $5$ cities, and is planning to build a road system so that each of its cities has exactly one outgoing (unidirectional) road to another city.
Two road systems are considered equivalent if we can get from one road system the other by just changing the names of the cities. That is, two road systems are considered the same if given a relabeling of the cities, if in the first configuration a road went from city $C$ to city $D$, then in the second configuration there is road that goes from the city now labeled $C$ to the city now labeled $D$.
How many distinct, nonequivalent possibilities are there for the road system Claredena builds?
1980 AMC 12/AHSME, 25
In the non-decreasing sequence of odd integers $\{a_1,a_2,a_3,\ldots \}=\{1,3,3,3,5,5,5,5,5,\ldots \}$ each odd positive integer $k$ appears $k$ times. It is a fact that there are integers $b$, $c$, and $d$ such that for all positive integers $n$,
\[ a_n=b\lfloor \sqrt{n+c} \rfloor +d, \]
where $\lfloor x \rfloor$ denotes the largest integer not exceeding $x$. The sum $b+c+d$ equals
$\text{(A)} \ 0 \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ 3 \qquad \text{(E)} \ 4$
2010 QEDMO 7th, 9
Let $p$ be an odd prime number and $c$ an integer for which $2c -1$ is divisible by $p$. Prove that
$$(-1)^{\frac{p+1}{2}}+\sum_{n=0}^{\frac{p-1}{2}} {2n \choose n}c^n$$ is divisible by $p$.
2014 NIMO Problems, 8
Triangle $ABC$ lies entirely in the first quadrant of the Cartesian plane, and its sides have slopes $63$, $73$, $97$. Suppose the curve $\mathcal V$ with equation $y=(x+3)(x^2+3)$ passes through the vertices of $ABC$. Find the sum of the slopes of the three tangents to $\mathcal V$ at each of $A$, $B$, $C$.
[i]Proposed by Akshaj[/i]
1950 Miklós Schweitzer, 10
Consider an arc of a planar curve such that the total curvature of the arc is less than $ \pi$. Suppose, further, that the curvature and its derivative with respect to the arc length exist at every point of the arc and the latter nowhere equals zero. Let the osculating circles belonging to the endpoints of the arc and one of these points be given. Determine the possible positions of the other endpoint.
2008 China Team Selection Test, 2
For a given integer $ n\geq 2,$ determine the necessary and sufficient conditions that real numbers $ a_{1},a_{2},\cdots, a_{n},$ not all zero satisfy such that there exist integers $ 0<x_{1}<x_{2}<\cdots<x_{n},$ satisfying $ a_{1}x_{1}\plus{}a_{2}x_{2}\plus{}\cdots\plus{}a_{n}x_{n}\geq 0.$
2008 Postal Coaching, 4
Let $n \in N$ and $k$ be such that $1 \le k \le n$. Find the number of ordered $k$-tuples $(a_1, a_2,...,a_k)$ of integers such the $1 \le a_j \le n$, for $1 \le j \le k$ and [u]either [/u] there exist $l,m \in \{1, 2,..., k\}$ such that $l < m$ but $a_l > a_m$ [u]or [/u] there exists $l \in \{1, 2,..., k\}$ such that $a_l - l$ is an odd number.
2014 Harvard-MIT Mathematics Tournament, 6
Let $n$ be a positive integer. A sequence $(a_0,\ldots,a_n)$ of integers is $\textit{acceptable}$ if it satisfies the following conditions:
[list=a]
[*] $0=|a_0|<|a_1|<\cdots<|a_{n-1}|<|a_n|.$
[*]The sets $\{|a_1-a_0|,|a_2-a_1|,\ldots,|a_{n-1}-a_{n-2}|,|a_n-a_{n-1}|\}$ and $\{1,3,9,\ldots,3^{n-1}\}$ are equal.[/list]
Prove that the number of acceptable sequences of integers is $(n+1)!$.
2011 Singapore Junior Math Olympiad, 5
Initially, the number $10$ is written on the board. In each subsequent moves, you can either
(i) erase the number $1$ and replace it with a $10$, or
(ii) erase the number $10$ and replace it with a $1$ and a $25$ or
(iii) erase a $25$ and replace it with two $10$.
After sometime, you notice that there are exactly one hundred copies of $1$ on the board. What is the least possible sum of all the numbers on the board at that moment?
2024 Korea Winter Program Practice Test, Q8
Let $\omega$ be the incircle of triangle $ABC$. For any positive real number $\lambda$, let $\omega_{\lambda}$ be the circle concentric with $\omega$ that has radius $\lambda$ times that of $\omega$. Let $X$ be the intersection between a trisector of $\angle B$ closer to $BC$ and a trisector of $\angle C$ closer to $BC$. Similarly define $Y$ and $Z$. Let $\epsilon = \frac{1}{2024}$. Show that the circumcircle of triangle $XYZ$ lies inside $\omega_{1-\epsilon}$.
[i]Note. Weaker results with smaller $\epsilon$ may be awarded points depending on the value of the constant $\epsilon <\frac{1}{2024}$.[/i]
2020 Iranian Geometry Olympiad, 1
By a [i]fold[/i] of a polygon-shaped paper, we mean drawing a segment on the paper and folding the paper along that. Suppose that a paper with the following figure is given. We cut the paper along the boundary of the shaded region to get a polygon-shaped paper.
Start with this shaded polygon and make a rectangle-shaped paper from it with at most 5 number
of folds. Describe your solution by introducing the folding lines and drawing the shape after each fold on your solution sheet.
(Note that the folding lines do not have to coincide with the grid lines of the shape.)
[i]Proposed by Mahdi Etesamifard[/i]
2000 Harvard-MIT Mathematics Tournament, 38
What is the largest number you can write with three $3$’s and three $8$’s, using only symbols $+,-,/,\times$ and exponentiation?
2019 LIMIT Category C, Problem 11
Let
$$x=\frac1{1\cdot2}-\frac1{2\cdot3}+\frac1{3\cdot4}-\ldots$$Then $e^{x+1}$ is
2024 Yasinsky Geometry Olympiad, 2
Let \( O \) and \( H \) be the circumcenter and orthocenter of the acute triangle \( ABC \). On sides \( AC \) and \( AB \), points \( D \) and \( E \) are chosen respectively such that segment \( DE \) passes through point \( O \) and \( DE \parallel BC \). On side \( BC \), points \( X \) and \( Y \) are chosen such that \( BX = OD \) and \( CY = OE \). Prove that \( \angle XHY + 2\angle BAC = 180^\circ \).
[i]Proposed by Matthew Kurskyi[/i]
2006 Cuba MO, 6
Two concentric circles of radii $1$ and $2$ have centere the point $O$. The vertex $A$ of the equilateral triangle $ABC$ lies at the largest circle, while the midpoint of side $BC$ lies on the smaller circle. If$ B$,$O$ and $C$ are not collinear, what measure can the angle $\angle BOC$ have?
1976 Putnam, 2
Let $P(x,y)=x^2y+xy^2$ and $Q(x,y)=x^2+xy+y^2.$ For $n=1,2,3,\dots,$ let \begin{align*}F_n(x,y)=(x+y)^n-x^n-y^n \text{ and,}\\ G_n(x,y)=(x+y)^n+x^n+y^n. \end{align*} One observes that $$G_2=2Q, F_3=3P, G_4=2Q^2, F_5=5PQ, G_6=2Q^3+3P^2.$$ Prove that, in fact, for each $n$ either $F_n$ or $G_n$ is expressible as a polynomial in $P$ and $Q$ with integer coefficients.
1986 IMO Longlists, 3
A line parallel to the side $BC$ of a triangle $ABC$ meets $AB$ in $F$ and $AC$ in $E$. Prove that the circles on $BE$ and $CF$ as diameters intersect in a point lying on the altitude of the triangle $ABC$ dropped from $A$ to $BC.$
2023 ELMO Shortlist, A6
Let \(\mathbb R_{>0}\) denote the set of positive real numbers and \(\mathbb R_{\ge0}\) the set of nonnegative real numbers. Find all functions \(f:\mathbb R\times \mathbb R_{>0}\to \mathbb R_{\ge0}\) such that for all real numbers \(a\), \(b\), \(x\), \(y\) with \(x,y>0\), we have \[f(a,x)+f(b,y)=f(a+b,x+y)+f(ay-bx,xy(x+y)).\]
[i]Proposed by Luke Robitaille[/i]
1999 Gauss, 25
In a softball league, after each team has played every other team 4 times, the total accumulated points are: Lions 22, Tigers 19, Mounties 14, and Royals 12. If each team received 3 points for a win, 1 point for a tie and no points for a loss, how many games ended in a tie?
$\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 10$
2013 IFYM, Sozopol, 5
Determine all increasing sequences $\{a_n\}_{n=1}^\infty$ of natural numbers with the following property: for each two natural numbers $i$ and $j$ (not necessarily different), the numbers $i+j$ and $a_i+a_j$ have an equal number of distinct natural divisors.
2020 HK IMO Preliminary Selection Contest, 15
How many ten-digit positive integers consist of ten different digits and are divisible by $99$?
2011 AMC 12/AHSME, 6
Two tangents to a circle are drawn from a point $A$. The points of contact $B$ and $C$ divide the circle into arcs with lengths in the ratio $2:3$. What is the degree measure of $\angle BAC$?
$ \textbf{(A)}\ 24 \qquad
\textbf{(B)}\ 30 \qquad
\textbf{(C)}\ 36 \qquad
\textbf{(D)}\ 48 \qquad
\textbf{(E)}\ 60 $
1988 Irish Math Olympiad, 2
2. Let $x_1, . . . , x_n$ be $n$ integers, and let $p$ be a positive integer, with $p < n$. Put
$$S_1 = x_1 + x_2 + . . . + x_p$$
$$T_1 = x_{p+1} + x_{p+2} + . . . + x_n$$
$$S_2 = x_2 + x_3 + . . . + x_{p+1}$$
$$T_2 = x_{p+2} + x_{p+3} + . . . + x_n + x_1$$
$$...$$
$$S_n=x_n+x_1+...+x_{p-1}$$
$$T_n=x_p+x_{p+1}+...+x_{n-1}$$
For $a = 0, 1, 2, 3$, and $b = 0, 1, 2, 3$, let $m(a, b)$ be the number of numbers $i$, $1 \leq i \leq n$, such that $S_i$ leaves remainder $a$ on division by $4$ and $T_i$ leaves remainder $b$ on division by $4$. Show that $m(1, 3)$ and $m(3, 1)$ leave the same remainder when divided by $4$ if, and only if, $m(2, 2)$ is even.