Found problems: 85335
1977 All Soviet Union Mathematical Olympiad, 239
Given infinite sequence $a_n$. It is known that the limit of $$b_n=a_{n+1}-a_n/2$$ equals zero. Prove that the limit of $a_n$ equals zero.
2017 Harvard-MIT Mathematics Tournament, 22
Kelvin the Frog and $10$ of his relatives are at a party. Every pair of frogs is either [i]friendly[/i] or [i]unfriendly[/i]. When $3$ pairwise friendly frogs meet up, they will gossip about one another and end up in a [i]fight[/i] (but stay [i]friendly[/i] anyway). When $3$ pairwise unfriendly frogs meet up, they will also end up in a [i]fight[/i]. In all other cases, common ground is found and there is no fight. If all $\binom{11}{3}$ triples of frogs meet up exactly once, what is the minimum possible number of fights?
2001 Manhattan Mathematical Olympiad, 6
There are $n$ coins of the radius $r$ on a table which is a circle of the radius $R$. It is known that:
a) any two coins either touch each other or have no common points;
b) there is no place for one more coin on the table.
Prove that
\[ \dfrac12 \left(\dfrac{R}{r} - 1\right) < \sqrt{n} < \dfrac{R}{r}.\]
2000 Mongolian Mathematical Olympiad, Problem 2
Let $n\ge2$. For any two $n$-vectors $\vec x=(x_1,\ldots,x_n)$ and $\vec y=(y_1,\ldots,y_n)$, we define
$$f\left(\vec x,\vec y\right)=x_1\overline{y_1}-\sum_{i=2}^nx_i\overline{y_i}.$$Prove that if $f\left(\vec x,\vec x\right)\ge0$, and $f\left(\vec y,\vec y\right)\ge0$, then $\left|f\left(\vec x,\vec y\right)\right|^2\ge f\left(\vec x,\vec x\right)f\left(\vec y,\vec y\right)$.
2014 Online Math Open Problems, 2
Consider two circles of radius one, and let $O$ and $O'$ denote their centers. Point $M$ is selected on either circle. If $OO' = 2014$, what is the largest possible area of triangle $OMO'$?
[i]Proposed by Evan Chen[/i]
2020 AMC 8 -, 5
Three fourths of a pitcher is filled with pineapple juice. The pitcher is emptied by pouring an equal amount of juice into each of $5$ cups. What percent of the total capacity of the pitcher did each cup receive?
$\textbf{(A) }5 \qquad \textbf{(B) }10 \qquad \textbf{(C) }15 \qquad \textbf{(D) }20 \qquad \textbf{(E) }25$
2020 Dutch IMO TST, 1
Given are real numbers $a_1, a_2,..., a_{2020}$, not necessarily different.
For every $n \ge 2020$, define $a_{n + 1}$ as the smallest real zero of the polynomial $$P_n (x) = x^{2n} + a_1x^{2n - 2} + a_2x^{2n - 4} +... + a_{n -1}x^2 + a_n$$, if it exists. Assume that $a_{n + 1}$ exists for all $n \ge 2020$.
Prove that $a_{n + 1} \le a_n$ for all $n \ge 2021$.
1976 Dutch Mathematical Olympiad, 2
Given $\vartriangle ABC$ and a point $P$ inside that triangle. The parallelograms $CPBL$, $APCM$ and $BPAN$ are constructed. Prove that $AL$, $BM$ and $CN$ pass through one point $S$, and that $S$ is the midpoint of $AL$, $BM$ and $CN$.
2014 Ukraine Team Selection Test, 11
Find all functions $f: R \to R$ that satisfy the condition $(f (x) - f (y)) (u - v) = (f (u) - f (v)) (x -y)$ for arbitrary real $x, y, u, v$ such that $x + y = u + v$.
2012 National Olympiad First Round, 32
How many permutations $(a_1,a_2,\dots,a_{10})$ of $1,2,3,4,5,6,7,8,9,10$ satisfy $|a_1-1|+|a_2-2|+\dots+|a_{10}-10|=4$ ?
$ \textbf{(A)}\ 60 \qquad \textbf{(B)}\ 52 \qquad \textbf{(C)}\ 50 \qquad \textbf{(D)}\ 44 \qquad \textbf{(E)}\ 36$
2022 Tuymaada Olympiad, 5
Prove that a quadratic trinomial $x^2 + ax + b (a, b \in R)$ cannot attain at ten consecutive integral points values equal to powers of $2$ with non-negative integral exponent.
[i](F. Petrov )[/i]
2022 BMT, 1
To fold a paper airplane, Austin starts with a square paper $F OLD$ with side length $2$. First, he folds corners $L$ and $D$ to the square’s center. Then, he folds corner $F$ to corner $O$. What is the longest distance between two corners of the resulting figure?
2022 AIME Problems, 10
Find the remainder when $$\binom{\binom{3}{2}}{2} + \binom{\binom{4}{2}}{2} + \dots + \binom{\binom{40}{2}}{2}$$ is divided by $1000$.
2021 Czech-Polish-Slovak Junior Match, 1
You are given a $2 \times 2$ array with a positive integer in each field. If we add the product of the numbers in the first column, the product of the numbers in the second column, the product of the numbers in the first row and the product of the numbers in the second row, we get $2021$.
a) Find possible values for the sum of the four numbers in the table.
b) Find the number of distinct arrays that satisfy the given conditions that contain four pairwise distinct numbers in arrays.
2014 CHMMC (Fall), 7
Let
$$P(x) = \sum^n_{k=1}(x^{3^k}+ x^{-3^k}- 1), Q(x) = \sum^n_{k=1}(x^{3^k}+ x^{-3^k}+ 1).$$
Given that
$$P(x)Q(x) =\sum^{2\cdot 3^n}_{k=-2\cdot 3^n} a_kx^k,$$
Compute $\sum^{3^n}_{k=0}a_k$ in terms of $n$.
2022 IMO Shortlist, N3
Let $a > 1$ be a positive integer and $d > 1$ be a positive integer coprime to $a$. Let $x_1=1$, and for $k\geq 1$, define
$$x_{k+1} = \begin{cases}
x_k + d &\text{if } a \text{ does not divide } x_k \\
x_k/a & \text{if } a \text{ divides } x_k
\end{cases}$$
Find, in terms of $a$ and $d$, the greatest positive integer $n$ for which there exists an index $k$ such that $x_k$ is divisible by $a^n$.
2014 Turkey EGMO TST, 6
For a given integer $n\ge3$, let $S_1, S_2,\ldots,S_m$ be distinct three-element subsets of the set $\{1,2,\ldots,n\}$ such that for each $1\le i,j\le m; i\neq j$ the sets $S_i\cap S_j$ contain exactly one element. Determine the maximal possible value of $m$ for each $n$.
2020 Cono Sur Olympiad, 3
Let $ABC$ be an acute triangle such that $AC<BC$ and $\omega$ its circumcircle. $M$ is the midpoint of $BC$. Points $F$ and $E$ are chosen in $AB$ and $BC$, respectively, such that $AC=CF$ and $EB=EF$. The line $AM$ intersects $\omega$ in $D\neq A$. The line $DE$ intersects the line $FM$ in $G$. Prove that $G$ lies on $\omega$.
2011 Junior Balkan Team Selection Tests - Moldova, 4
In the Cartesian $xOy$ coordinate system the points $A (36, 0)$, $A_1 (10, 0)$ are given, $B (0, 36)$, $B_1 (0, 10)$, $C (-36, 0)$, $C_1 (-10, 0)$, $D (0, -36)$, $D_1 (0, -10)$. A point of the plane is called [i]lattice[/i] if it has integer coordinates. Determine the number of lattice points that are located inside the square $ABCD$, but outside the square $A_1B_1C_1D_1$
2014 PUMaC Number Theory B, 5
Find the sum of all positive integers $x$ such that $3 \times 2^x = n^2 -1$ for some positive integer $n$.
1977 IMO Shortlist, 4
Describe all closed bounded figures $\Phi$ in the plane any two points of which are connectable by a semicircle lying in $\Phi$.
2022 Harvard-MIT Mathematics Tournament, 3
Michel starts with the string HMMT. An operation consists of either replacing an occurrence of H with HM, replacing an occurrence of MM with MOM, or replacing an occurrence of T with MT. For example, the two strings that can be reached after one operation are HMMMT and HMOMT. Compute the number of distinct strings Michel can obtain after exactly $10$ operations.
JBMO Geometry Collection, 2001
Let $ABC$ be a triangle with $\angle C = 90^\circ$ and $CA \neq CB$. Let $CH$ be an altitude and $CL$ be an interior angle bisector. Show that for $X \neq C$ on the line $CL$, we have $\angle XAC \neq \angle XBC$. Also show that for $Y \neq C$ on the line $CH$ we have $\angle YAC \neq \angle YBC$.
[i]Bulgaria[/i]
2000 Mongolian Mathematical Olympiad, Problem 4
In a country with $n$ towns, the distance between the towns numbered $i$ and $j$ is denoted by $x_{ij}$. Suppose that the total length of every cyclic route which passes through every town exactly once is the same. Prove that there exist numbers $a_i,b_i$ ($i=1,\ldots,n$) such that $x_{ij}=a_i+b_j$ for all distinct $i,j$.
1957 AMC 12/AHSME, 27
The sum of the reciprocals of the roots of the equation $ x^2 \plus{} px \plus{} q \equal{} 0$ is:
$ \textbf{(A)}\ \minus{}\frac{p}{q} \qquad
\textbf{(B)}\ \frac{q}{p}\qquad
\textbf{(C)}\ \frac{p}{q}\qquad
\textbf{(D)}\ \minus{}\frac{q}{p}\qquad
\textbf{(E)}\ pq$