Found problems: 85335
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)
1996 AMC 8, 14
Six different digits from the set
\[\{ 1,2,3,4,5,6,7,8,9\}\]
are placed in the squares in the figure shown so that the sum of the entries in the vertical column is 23 and the sum of the entries in the horizontal row is 12.
The sum of the six digits used is
[asy]
unitsize(18);
draw((0,0)--(1,0)--(1,1)--(4,1)--(4,2)--(1,2)--(1,3)--(0,3)--cycle);
draw((0,1)--(1,1)--(1,2)--(0,2));
draw((2,1)--(2,2));
draw((3,1)--(3,2));
label("$23$",(0.5,0),S);
label("$12$",(4,1.5),E);
[/asy]
$\text{(A)}\ 27 \qquad \text{(B)}\ 29 \qquad \text{(C)}\ 31 \qquad \text{(D)}\ 33 \qquad \text{(E)}\ 35$
2017 BMT Spring, 18
Consider the sequence $(k_n)$ defined by $k_{n+1} = n(k_n + k_{n-1})$ and $k_0 = 0$, $k_1 = 1$. What is $\lim
_{n\to \infty} \frac{k_n}{n!}$ ?
2004 Bulgaria Team Selection Test, 2
The edges of a graph with $2n$ vertices ($n \ge 4$) are colored in blue and red such that there is no blue triangle and there is no red complete subgraph with $n$ vertices. Find the least possible number of blue edges.
Estonia Open Senior - geometry, 2016.2.5
The circumcentre of an acute triangle $ABC$ is $O$. Line $AC$ intersects the circumcircle of $AOB$ at a point $X$, in addition to the vertex $A$. Prove that the line $XO$ is perpendicular to the line $BC$.
2021 Belarusian National Olympiad, 8.5
Let $f(x)$ be a linear function and $k,l,m$ - pairwise different real numbers. It is known that $f(k)=l^3+m^3$, $f(l)=m^3+k^3$ and $f(m)=k^3+l^3$.
Find the value of $k+l+m$.
1967 Polish MO Finals, 6
Given a sphere and a plane that has no common points with the sphere. Find the geometric locus of the centers of the circles of tangency with the sphere of those cones circumcribed on the sphere whose vertices lie on the given plane.
2022 Azerbaijan National Mathematical Olympiad, 4
Find all quadruplets $(x_1, x_2, x_3, x_4)$ of real numbers such that the next six equalities apply:
$$\begin{cases} x_1 + x_2 = x^2_3 + x^2_4 + 6x_3x_4\\
x_1 + x_3 = x^2_2 + x^2_4 + 6x_2x_4\\
x_1 + x_4 = x^2_2 + x^2_3 + 6x_2x_3\\
x_2 + x_3 = x^2_1 + x^2_4 + 6x_1x_4\\
x_2 + x_4 = x^2_1 + x^2_3 + 6x_1x_3 \\
x_3 + x_4 = x^2_1 + x^2_2 + 6x_1x_2 \end{cases}$$
2015 ASDAN Math Tournament, 6
Find all triples of integers $(x,y,z)$ which satisfy the equations
\begin{align*}
x^2-y-2z&=4\\
y^2-2z-3x&=-2\\
2z^2-3x-5y&=-22.\\
\end{align*}
1986 AMC 8, 7
How many whole numbers are between $ \sqrt{8}$ and $ \sqrt{80}$?
\[ \textbf{(A)}\ 5 \qquad
\textbf{(B)}\ 6 \qquad
\textbf{(C)}\ 7 \qquad
\textbf{(D)}\ 8 \qquad
\textbf{(E)}\ 9
\]
2007 Iran Team Selection Test, 2
Find all monic polynomials $f(x)$ in $\mathbb Z[x]$ such that $f(\mathbb Z)$ is closed under multiplication.
[i]By Mohsen Jamali[/i]
2003 Junior Tuymaada Olympiad, 1
A $2003\times 2004$ rectangle consists of unit squares. We consider rhombi formed by four diagonals of unit squares.
What maximum number of such rhombi can be arranged in this rectangle so that no two of them have any common points except vertices?
[i]Proposed by A. Golovanov[/i]
2019 Sharygin Geometry Olympiad, 24
Two unit cubes have a common center. Is it always possible to number the vertices of each cube from $1$ to $8$ so that the distance between each pair of identically numbered vertices would be at most $4/5$? What about at most $13/16$?
2020 Dutch IMO TST, 4
Given are two positive integers $k$ and $n$ with $k \le n \le 2k - 1$. Julian has a large stack of rectangular $k \times 1$ tiles. Merlin calls a positive integer $m$ and receives $m$ tiles from Julian to place on an $n \times n$ board. Julian first writes on every tile whether it should be a horizontal or a vertical tile. Tiles may be used the board should not overlap or protrude. What is the largest number $m$ that Merlin can call if he wants to make sure that he has all tiles according to the rule of Julian can put on the plate?
2017 239 Open Mathematical Olympiad, 5
Given a quadrilateral $ABCD$ in which$$\sqrt{2}(BC-BA)=AC.$$Let $X$ be the midpoint of $AC$. Prove that $2\angle BXD=\angle DAB - \angle DCB.$