Found problems: 85335
2002 AIME Problems, 7
It is known that, for all positive integers $k,$
\[1^{2}+2^{2}+3^{2}+\cdots+k^{2}=\frac{k(k+1)(2k+1)}{6}. \]Find the smallest positive integer $k$ such that $1^{2}+2^{2}+3^{2}+\cdots+k^{2}$ is a multiple of $200.$
1997 Romania National Olympiad, 3
Let $\mathcal{F}$ be the set of the differentiable functions $f: \mathbb{R} \to \mathbb{R}$ satisfying $f(x) \ge f(x+ \sin x)$ for any $x \in \mathbb{R}.$
a) Prove that there exist nonconstant functions in $\mathcal{F}.$
b) Prove that if $f \in \mathcal{F},$ then the set of solutions of the equation $f'(x)=0$ is infinite.
2009 Belarus Team Selection Test, 2
Given trapezoid $ ABCD$ with parallel sides $ AB$ and $ CD$, assume that there exist points $ E$ on line $ BC$ outside segment $ BC$, and $ F$ inside segment $ AD$ such that $ \angle DAE \equal{} \angle CBF$. Denote by $ I$ the point of intersection of $ CD$ and $ EF$, and by $ J$ the point of intersection of $ AB$ and $ EF$. Let $ K$ be the midpoint of segment $ EF$, assume it does not lie on line $ AB$. Prove that $ I$ belongs to the circumcircle of $ ABK$ if and only if $ K$ belongs to the circumcircle of $ CDJ$.
[i]Proposed by Charles Leytem, Luxembourg[/i]
2017 Taiwan TST Round 1, 2
The leader of an IMO team chooses positive integers $n$ and $k$ with $n > k$, and announces them to the deputy leader and a contestant. The leader then secretly tells the deputy leader an $n$-digit binary string, and the deputy leader writes down all $n$-digit binary strings which differ from the leader’s in exactly $k$ positions. (For example, if $n = 3$ and $k = 1$, and if the leader chooses $101$, the deputy leader would write down $001, 111$ and $100$.) The contestant is allowed to look at the strings written by the deputy leader and guess the leader’s string. What is the minimum number of guesses (in terms of $n$ and $k$) needed to guarantee the correct answer?
Bangladesh Mathematical Olympiad 2020 Final, #2
Consider rectangle $ABCD$.$ E$ is the mid-point of $AD$ and $F$ is the mid-point of $ED$. $CE$ cuts $AB$ in $G$ and $BF$ cuts $CD$ in $H$ point. We can write ratio of areas of $BCG$ and $BCH$ triangles as $\frac{m}{n}$. Find the value of $10m + 10n + mn$.
2011 Estonia Team Selection Test, 6
On a square board with $m$ rows and $n$ columns, where $m\le n$, some squares are colored black in such a way that no two rows are alike. Find tha biggest integer $k$ such that, for every possible coloring to start with, one can always color $k$ columns entirely red in such a way that still no two rows are alike.
2024 District Olympiad, P2
Consider the sequence $(a_n)_{n\geqslant 1}$ defined by $a_1=1/2$ and $2n\cdot a_{n+1}=(n+1)a_n.$[list=a]
[*]Determine the general formula for $a_n.$
[*]Let $b_n=a_1+a_2+\cdots+a_n.$ Prove that $\{b_n\}-\{b_{n+1}\}\neq \{b_{n+1}\}-\{b_{n+2}\}.$
[/list]
2001 Romania Team Selection Test, 4
Show that the set of positive integers that cannot be represented as a sum of distinct perfect squares is finite.
2024 HMNT, 27
For any positive integer $n,$ let $f(n)$ be the number of ordered triples $(a,b,c)$ of positive integers such that
[list]
[*] $\max(a,b,c)$ divides $n$ and
[*] $\gcd(a,b,c)=1.$
[/list]
Compute $f(1)+f(2)+\cdots+f(100).$
2020 HMNT (HMMO), 7
While waiting for their food at a restaurant in Harvard Square, Ana and Banana draw $3$ squares $\square_1, \square_2, \square_3$ on one of their napkins. Starting with Ana, they take turns filling in the squares with integers from the set $\{1,2,3,4,5\}$ such that no integer is used more than once. Ana's goal is to minimize the minimum value that the polynomial $a_1x^2 + a_2x + a_3$ attains over all real $x$, where $a_1, a_2, a_3$ are the integers written in $\square_1, \square_2, \square_3$ respectively. Banana aims to maximize $M$. Assuming both play optimally, compute the final value of $100a_1+10a_2+a_3$.
2018 PUMaC Number Theory A, 5
Find the remainder when
$$\prod_{i = 1}^{1903} (2^i + 5)$$
is divided by $1000$.
2016-2017 SDML (Middle School), 2
On a Cartesian coordinate plane, points $(1, 2)$ and $(7, 4)$ are opposite vertices of a square. What is the area of the square?
2017 Harvard-MIT Mathematics Tournament, 4
Find the number of ordered triples of nonnegative integers $(a, b, c)$ that satisfy
\[(ab + 1)(bc + 1)(ca + 1) = 84.\]
1972 IMO Longlists, 20
Let $n_1, n_2$ be positive integers. Consider in a plane $E$ two disjoint sets of points $M_1$ and $M_2$ consisting of $2n_1$ and $2n_2$ points, respectively, and such that no three points of the union $M_1 \cup M_2$ are collinear. Prove that there exists a straightline $g$ with the following property: Each of the two half-planes determined by $g$ on $E$ ($g$ not being included in either) contains exactly half of the points of $M_1$ and exactly half of the points of $M_2.$
2024 ISI Entrance UGB, P6
Let $x_1 , \dots , x_{2024}$ be non negative real numbers with $\displaystyle{\sum_{i=1}^{2024}}x_i = 1$. Find, with proof, the minimum and maximum possible values of the following expression \[\sum_{i=1}^{1012} x_i + \sum_{i=1013}^{2024} x_i^2 .\]
2006 AMC 12/AHSME, 14
Elmo makes $ N$ sandwiches for a fundraiser. For each sandwich he uses $ B$ globs of peanut butter at 4 cents per glob and $ J$ blobs of jam at 5 cents per glob. The cost of the peanut butter and jam to make all the sandwiches is $ \$$2.53. Assume that $ B, J,$ and $ N$ are all positive integers with $ N > 1$. What is the cost of the jam Elmo uses to make the sandwiches?
$ \textbf{(A) } \$1.05 \qquad \textbf{(B) } \$1.25 \qquad \textbf{(C) } \$1.45 \qquad \textbf{(D) } \$1.65 \qquad \textbf{(E) } \$1.85$
2020 SAFEST Olympiad, 1
You are given a set of $n$ blocks, each weighing at least $1$; their total weight is $2n$. Prove that for every real number $r$ with $0 \leq r \leq 2n-2$ you can choose a subset of the blocks whose total weight is at least $r$ but at most $r + 2$.
1997 Tournament Of Towns, (543) 4
A convex polygon $G$ is placed inside a convex polygon $ F$ so that their boundaries have no common points. A segment $s$ joining two points on the boundary of $F$ is called a support chord for $G$ if s contains a side or only a vertex of $G$. Prove that
(a) there exists a support chord for $G$ such that its midpoint lies on the boundary of $G$,
(b) there exist at least two such chords.
(P Pushkar)
2015 AoPS Mathematical Olympiad, 7
Let $ABC$ be a right triangle with $\angle C = 90^\circ$. Let $P_A$, $P_B$, and $P_C$ be regular pentagons with side lengths $BC$, $CA$, and $AB$, respectively. Prove that $[P_A]+[P_B]=[P_C]$.
[i]Proposed by CaptainFlint[/i]
2008 Iran MO (3rd Round), 4
Let $ u$ be an odd number. Prove that $ \frac{3^{3u}\minus{}1}{3^u\minus{}1}$ can be written as sum of two squares.
2014 Online Math Open Problems, 4
A crazy physicist has discovered a new particle called an emon. He starts with two emons in the plane, situated a distance $1$ from each other. He also has a crazy machine which can take any two emons and create a third one in the plane such that the three emons lie at the vertices of an equilateral triangle. After he has five total emons, let $P$ be the product of the $\binom 52 = 10$ distances between the $10$ pairs of emons. Find the greatest possible value of $P^2$.
[i]Proposed by Yang Liu[/i]
2004 All-Russian Olympiad Regional Round, 10.5
Equation $$x^n + a_1x^{n-1} + a_2x^{n-2} +...+ a_{n-1}x + a_n = 0$$ with integer non-zero coefficients $a_1$, $a_2$, $...$ , $a_n$ has $n$ different integer roots. Prove that if any two roots are relatively prime, then the numbers $a_{n-1}$ and $a_n$ are coprime.
1997 IMC, 3
Show that $\sum^{\infty}_{n=1}\frac{(-1)^{n-1}\sin(\log n)}{n^\alpha}$ converges iff $\alpha>0$.
2007 ITest, 28
The space diagonal (interior diagonal) of a cube has length $6$. Find the $\textit{surface area}$ of the cube.
2003 Estonia Team Selection Test, 3
Let $N$ be the set of all non-negative integers and for each $n \in N$ denote $n'= n +1$. The function $A : N^3 \to N$ is defined as follows:
(i) $A(0, m, n) = m'$ for all $m, n \in N$
(ii) $A(k', 0, n) =\left\{ \begin{array}{ll}
n & if \, \, k = 0 \\
0 & if \, \,k = 1, \\
1 & if \, \, k > 1 \end{array} \right.$ for all $k, n \in N$
(iii) $A(k', m', n) = A(k, A(k',m,n), n)$ for all $k,m, n \in N$.
Compute $A(5, 3, 2)$.
(H. Nestra)