Found problems: 85335
2004 Junior Tuymaada Olympiad, 8
Zeroes and ones are arranged in all the squares of $n\times n$ table.
All the squares of the left column are filled by ones, and the sum of numbers in every figure of the form
[asy]size(50); draw((2,1)--(0,1)--(0,2)--(2,2)--(2,0)--(1,0)--(1,2));[/asy]
(consisting of a square and its neighbours from left and from below)
is even.
Prove that no two rows of the table are identical.
[i]Proposed by O. Vanyushina[/i]
2011 IFYM, Sozopol, 7
Find all function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that
$f(x+y)-2f(x-y)+f(x)-2f(y)=y-2,\forall x,y\in \mathbb{R}$.
2016 Harvard-MIT Mathematics Tournament, 3
In the below picture, $T$ is an equilateral triangle with a side length of $5$ and $\omega$ is a circle with a radius of $2$. The triangle and the circle have the same center. Let $X$ be the area of the shaded region, and let $Y$ be the area of the starred region. What is $X - Y$?
2022 Turkey Junior National Olympiad, 2
In a school with $101$ students, each student has at least one friend among the other students. Show that for every integer $1<n<101$, a group of $n$ students can be selected from this school in such a way that each selected student has at least one friend among the other selected students.
2013 Princeton University Math Competition, 6
Draw an equilateral triangle with center $O$. Rotate the equilateral triangle $30^\circ, 60^\circ, 90^\circ$ with respect to $O$ so there would be four congruent equilateral triangles on each other. Look at the diagram. If the smallest triangle has area $1$, the area of the original equilateral triangle could be expressed as $p+q\sqrt r$ where $p,q,r$ are positive integers and $r$ is not divisible by a square greater than $1$. Find $p+q+r$.
1966 IMO Longlists, 7
For which arrangements of two infinite circular cylinders does their intersection lie in a plane?
2014 Harvard-MIT Mathematics Tournament, 27
Suppose that $(a_1,\ldots,a_{20})$ and $(b_1,\ldots,b_{20})$ are two sequences of integers such that the sequence $(a_1,\ldots,a_{20},b_1,\ldots,b_{20})$ contains each of the numbers $1,\ldots,40$ exactly once. What is the maximum possible value of the sum \[\sum_{i=1}^{20}\sum_{j=1}^{20}\min(a_i,b_j)?\]
2017 Auckland Mathematical Olympiad, 2
The sum of the three nonnegative real numbers $ x_1, x_2, x_3$ is not greater than $\frac12$.
Prove that $(1 - x_1)(1 - x_2)(1 - x_3) \ge \frac12$
OMMC POTM, 2024 6
Find the remainder modulo $101$ of
$$\left\lfloor \dfrac{1}{(2 \cos \left(\frac{4\pi}{7} \right))^{103}}\right\rfloor$$
2024 Malaysian IMO Training Camp, 3
Given $n$ students in the plane such that the $\frac{n(n-1)}{2}$ distances are pairwise distinct. Each student gives a candy each to the $k$ students closest to him. Given that each student receives the same amount of candies, determine all possible values of $n$ in terms of $k$.
[i]Proposed by Wong Jer Ren[/i]
2024 Tuymaada Olympiad, 7
Given are two polynomial $f$ and $g$ of degree $100$ with real coefficients. For each positive integer $n$ there is an integer $k$ such that
\[\frac{f(k)}{g(k)}=\frac{n+1}{n}.\]
Prove that $f$ and $g$ have a common non-constant factor.
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]
2018 CCA Math Bonanza, L3.1
The number $16^4+16^2+1$ is divisible by four distinct prime numbers. Compute the sum of these four primes.
[i]2018 CCA Math Bonanza Lightning Round #3.1[/i]
2015 BMT Spring, P2
Suppose $k>3$ is a divisor of $2^p+1$, where $p$ is prime. Prove that $k\ge2p+1$.
2010 Stanford Mathematics Tournament, 6
A triangle has side lengths $7, 9,$ and $12$. What is the area of the triangle?
2003 District Olympiad, 4
Let $\alpha>1$ and $f:\left[\frac{1}{\alpha},\alpha\right]\rightarrow \left[\frac{1}{\alpha},\alpha\right]$, a bijective function. If $f^{-1}(x)=\frac{1}{f(x)},\ \forall x\in \left[\frac{1}{\alpha},\alpha\right]$, prove that:
a)$f$ has at least one point of discontinuity;
b)if $f$ is continuous in $1$, then $f$ has an infinity points of discontinuity;
c)there is a function $f$ which satisfies the conditions from the hypothesis and has a finite number of points of dicontinuity.
[i]Radu Mortici [/i]
2017 Online Math Open Problems, 19
For each integer $1\le j\le 2017$, let $S_j$ denote the set of integers $0\le i\le 2^{2017} - 1$ such that $\left\lfloor \frac{i}{2^{j-1}} \right\rfloor$ is an odd integer. Let $P$ be a polynomial such that
\[P\left(x_0, x_1, \ldots, x_{2^{2017} - 1}\right) = \prod_{1\le j\le 2017} \left(1 - \prod_{i\in S_j} x_i\right).\]
Compute the remainder when
\[ \sum_{\left(x_0, \ldots, x_{2^{2017} - 1}\right)\in\{0, 1\}^{2^{2017}}} P\left(x_0, \ldots, x_{2^{2017} - 1}\right)\]
is divided by $2017$.
[i]Proposed by Ashwin Sah[/i]
2024 ELMO Shortlist, N5
Let $T$ be a finite set of squarefree integers.
(a) Show that there exists an integer polynomial $P(x)$ such that the set of squarefree numbers in the range of $P(n)$ across all $n \in \mathbb{Z}$ is exactly $T$.
(b) Suppose that $T$ is allowed to be infinite. Is it still true that for all choices of $T$, such an integer polynomial $P(x)$ exists?
[i]Allen Wang[/i]
2013 AMC 10, 16
In $\triangle ABC$, medians $\overline{AD}$ and $\overline{CE}$ intersect at $P$, $PE=1.5$, $PD=2$, and $DE=2.5$. What is the area of $AEDC?$
[asy]
unitsize(75);
pathpen = black; pointpen=black;
pair A = MP("A", D((0,0)), dir(200));
pair B = MP("B", D((2,0)), dir(-20));
pair C = MP("C", D((1/2,1)), dir(100));
pair D = MP("D", D(midpoint(B--C)), dir(30));
pair E = MP("E", D(midpoint(A--B)), dir(-90));
pair P = MP("P", D(IP(A--D, C--E)), dir(150)*2.013);
draw(A--B--C--cycle);
draw(A--D--E--C);
[/asy]
$\textbf{(A)}\ 13 \qquad
\textbf{(B)}\ 13.5 \qquad
\textbf{(C)}\ 14 \qquad
\textbf{(D)}\ 14.5 \qquad
\textbf{(E)}\ 15 $
1955 Moscow Mathematical Olympiad, 295
Which convex domains (figures) on a plane can contain an entire straight line?
It is assumed that the figure is flat and does not degenerate into a straight line and is closed, that is, it contains all its boundary points.
2012 AMC 10, 3
A bug crawls along a number line, starting at $-2$. It crawls to $-6$, then turns around and crawls to $5$. How many units does the bug crawl altogether?
$ \textbf{(A)}\ 9
\qquad\textbf{(B)}\ 11
\qquad\textbf{(C)}\ 13
\qquad\textbf{(D)}\ 14
\qquad\textbf{(E)}\ 15
$
2020 Princeton University Math Competition, 8
Let there be a tiger, William, at the origin. William leaps $ 1$ unit in a random direction, then leaps $2$ units in a random direction, and so forth until he leaps $15$ units in a random direction to celebrate PUMaC’s 15th year.
There exists a circle centered at the origin such that the probability that William is contained in the circle (assume William is a point) is exactly $1/2$ after the $15$ leaps. The area of that circle can be written as $A\pi$. What is $A$?
2024 CMIMC Integration Bee, 4
\[\int_0^1 (x^6+6x^5+15x^4+15x^2+6x+1)\mathrm dx\]
[i]Proposed by Robert Trosten[/i]
2014 Estonia Team Selection Test, 1
In Wonderland, the government of each country consists of exactly $a$ men and $b$ women, where $a$ and $b$ are fixed natural numbers and $b > 1$. For improving of relationships between countries, all possible working groups consisting of exactly one government member from each country, at least $n$ among whom are women, are formed (where $n$ is a fixed non-negative integer). The same person may belong to many working groups. Find all possibilities how many countries can be in Wonderland, given that the number of all working groups is prime.
IV Soros Olympiad 1997 - 98 (Russia), 11.12
Find how many different solutions depending on $a$ has the system of equations :
$$\begin{cases} x+z=2a
\\ y+u+xz=a-3
\\ yz+xu=2a
\\ yu=1
\end{cases}$$