Found problems: 85335
2014 USA TSTST, 4
Let $P(x)$ and $Q(x)$ be arbitrary polynomials with real coefficients, and let $d$ be the degree of $P(x)$. Assume that $P(x)$ is not the zero polynomial. Prove that there exist polynomials $A(x)$ and $B(x)$ such that:
(i) both $A$ and $B$ have degree at most $d/2$
(ii) at most one of $A$ and $B$ is the zero polynomial.
(iii) $\frac{A(x)+Q(x)B(x)}{P(x)}$ is a polynomial with real coefficients. That is, there is some polynomial $C(x)$ with real coefficients such that $A(x)+Q(x)B(x)=P(x)C(x)$.
2017 Vietnamese Southern Summer School contest, Problem 4
In a square board of size 1001 x 1001, we color some $m$ cells in such a way that:
i. Of any two cells that share an edge, at least one is colored.
ii. Of any 6 consecutive cells in a column or a row, at least 2 consecutive ones are colored.
Determine the smallest possible value of $m$.
2019 Teodor Topan, 1
Solve in the natural numbers the equation $ \log_{6n-19} (n!+1) =2. $
[i]Dragoș Crișan[/i]
2014 BMT Spring, 5
In a 100-dimensional hypercube, each edge has length $ 1$. The box contains $2^{100} + 1$ hyperspheres with the same radius $ r$. The center of one hypersphere is the center of the hypercube, and it touches all the other spheres. Each of the other hyperspheres is tangent to $100$ faces of the hypercube. Thus, the hyperspheres are tightly packed in the hypercube. Find $ r$.
2024 Regional Olympiad of Mexico Southeast, 2
Let \(ABC\) be an acute triangle with circumradius \(R\). Let \(D\) be the midpoint of \(BC\) and \(F\) the midpoint of \(AB\). The perpendicular to \(AC\) through \(F\) and the perpendicular to \(BC\) through \(B\) intersect at \(N\). Prove that \(ND = R\).
PEN A Problems, 90
Determine all pairs $(x, y)$ of positive integers with $y \vert x^2 +1$ and $x^2 \vert y^3 +1$.
Kyiv City MO 1984-93 - geometry, 1984.10.5
The vertices of a regular hexagon $A_1,A_2,...,A_6$ lie respectively on the sides $B_1B_2$, $B_2B_3$, $B_3B_4$, $B_4B_5$, $B_5B_6$, $B_6B_1$ of a convex hexagon $B_1B_2B_3B_4B_5B_6$. Prove that
$$S_{B_1B_2B_3B_4B_5B_6} \le \frac32 S_{A_1A_2A_3A_4A_5A_6}.$$
2000 Baltic Way, 20
For every positive integer $n$, let
\[x_n=\frac{(2n+1)(2n+3)\cdots (4n-1)(4n+1)}{(2n)(2n+2)\cdots (4n-2)(4n)}\]
Prove that $\frac{1}{4n}<x_n-\sqrt{2}<\frac{2}{n}$.
1971 IMO Longlists, 48
The diagonals of a convex quadrilateral $ABCD$ intersect at a point $O$. Find all angles of this quadrilateral if $\measuredangle OBA=30^{\circ},\measuredangle OCB=45^{\circ},\measuredangle ODC=45^{\circ}$, and $\measuredangle OAD=30^{\circ}$.
2014 Contests, 1
A basket is called "[i]Stuff Basket[/i]" if it includes $10$ kilograms of rice and $30$ number of eggs. A market is to distribute $100$ Stuff Baskets. We know that there is totally $1000$ kilograms of rice and $3000$ number of eggs in the baskets, but some of market's baskets include either more or less amount of rice or eggs. In each step, market workers can select two baskets and move an arbitrary amount of rice or eggs between selected baskets. Starting from an arbitrary situation, what's the minimum number of steps that workers provide $100$ Stuff Baskets?
2013 Tournament of Towns, 5
A quadratic trinomial with integer coefficients is called [i]admissible [/i] if its leading coeffi cient is $1$, its roots are integers and the absolute values of coefficients do not exceed $2013$. Basil has summed up all admissible quadratic trinomials. Prove that the resulting trinomial has no real roots.
1992 Romania Team Selection Test, 8
Let $m,n \ge 2$ be integers. The sides $A_{00}A_{0m}$ and $A_{nm}A_{n0}$ of a convex quadrilateral $A_{00}A_{0m}A_{nm}A_{n0}$ are divided into $m$ equal segments by points $A_{0j}$ and $A_{nj}$ respectively ($j = 1,...,m-1$). The other two sides are divided into $n$ equal segments by points $A_{i0}$ and $A_{im}$ ($i = 1,...,n -1$). Denote by $A_{ij}$ the intersection of lines $A_{0j}A{nj}$ and $A_{i0}A_{im}$, by $S_{ij}$ the area of quadrilateral $A_{ij}A_{i, j+1}A_{i+1, j+1}A_{i+1, j}$ and by $S$ the area of the big quadrilateral. Show that $S_{ij} +S_{n-1-i,m-1-j} =
\frac{2S}{mn}$
2012 Tournament of Towns, 5
Let $p$ be a prime number. A set of $p + 2$ positive integers, not necessarily distinct, is called [i]interesting [/i] if the sum of any $p$ of them is divisible by each of the other two. Determine all interesting sets.
2022 Assam Mathematical Olympiad, 9
What is the number formed by the last three digits of $1201^{1202}$?
1992 Canada National Olympiad, 1
Prove that the product of the first $ n$ natural numbers is divisible by the sum of the first $ n$ natural numbers if and only if $ n\plus{}1$ is not an odd prime.
2017 USAMO, 3
Let $ABC$ be a scalene triangle with circumcircle $\Omega$ and incenter $I$. Ray $AI$ meets $\overline{BC}$ at $D$ and meets $\Omega$ again at $M$; the circle with diameter $\overline{DM}$ cuts $\Omega$ again at $K$. Lines $MK$ and $BC$ meet at $S$, and $N$ is the midpoint of $\overline{IS}$. The circumcircles of $\triangle KID$ and $\triangle MAN$ intersect at points $L_1$ and $L_2$. Prove that $\Omega$ passes through the midpoint of either $\overline{IL_1}$ or $\overline{IL_2}$.
[i]Proposed by Evan Chen[/i]
2008 Teodor Topan, 4
Let $ (a_n)_{n \in \mathbb{N}^*}$ be a sequence of real positive numbers such that $ a_n>a_0,n\in \mathbb{N}$. Prove that $ \displaystyle\lim_{n\to\infty}\displaystyle\sum_{k\equal{}0}^{n}(\frac{a_k}{a_{n\minus{}k}})^k\equal{}\infty$.
2003 AMC 8, 6
Given the areas of the three squares in the figure, what is the area of the interior triangle?
[asy]
real r=22.61986495;
pair A=origin, B=(12,0), C=(12,5);
draw(A--B--C--cycle);
markscalefactor=0.1;
draw(rightanglemark(C, B, A));
draw(scale(12)*shift(0,-1)*unitsquare);
draw(scale(5)*shift(12/5,0)*unitsquare);
draw(scale(13)*rotate(r)*unitsquare);
pair P=shift(0,-1)*(13/sqrt(2) * dir(r+45)), Q=(14.5,1.2), R=(6, -7);
label("169", P, N);
label("25", Q, N);
label("144", R, N);
[/asy]
$ \textbf{(A)}\ 13\qquad\textbf{(B)}\ 30\qquad\textbf{(C)}\ 60\qquad\textbf{(D)}\ 300\qquad\textbf{(E)}\ 1800$
2017 Balkan MO Shortlist, G2
Let $ABC$ be an acute triangle and $D$ a variable point on side $AC$ . Point $E$ is on $BD$ such that $BE =\frac{BC^2-CD\cdot CA}{BD}$ . As $D$ varies on side $AC$ prove that the circumcircle of $ADE$ passes through a fixed point other than $A$ .
2024 Malaysia IMONST 2, 3
Ivan claims that for all positive integers $n$, $$\left\lfloor\sqrt[2]{\frac{n}{1^3}}\right\rfloor + \left\lfloor\sqrt[2]{\frac{n}{2^3}}\right\rfloor + \left\lfloor\sqrt[2]{\frac{n}{3^3}}\right\rfloor + \cdots = \left\lfloor\sqrt[3]{\frac{n}{1^2}}\right\rfloor + \left\lfloor\sqrt[3]{\frac{n}{2^2}}\right\rfloor + \left\lfloor\sqrt[3]{\frac{n}{3^2}}\right\rfloor + \cdots$$ Why is he correct? (Note: $\lfloor x \rfloor$ denotes the floor function.)
2021 Alibaba Global Math Competition, 3
Given positive integers $k \ge 2$ and $m$ sufficiently large. Let $\mathcal{F}_m$ be the infinite family of all the (not necessarily square) binary matrices which contain exactly $m$ 1's. Denote by $f(m)$ the maximum integer $L$ such that for every matrix $A \in \mathcal{F}_m$, there always exists a binary matrix $B$ of the same dimension such that (1) $B$ has at least $L$ 1-entries; (2) every entry of $B$ is less or equal to the corresponding entry of $A$; (3) $B$ does not contain any $k \times k$ all-1 submatrix. Show the equality
\[\lim_{m \to \infty} \frac{\ln f(m)}{\ln m}=\frac{k}{k+1}.\]
1991 Flanders Math Olympiad, 2
(a) Show that for every $n\in\mathbb{N}$ there is exactly one $x\in\mathbb{R}^+$ so that $x^n+x^{n+1}=1$. Call this $x_n$.
(b) Find $\lim\limits_{n\rightarrow+\infty}x_n$.
2013 Pan African, 1
Let $ABCD$ be a convex quadrilateral with $AB$ parallel to $CD$. Let $P$ and $Q$ be the midpoints of $AC$ and $BD$, respectively. Prove that if $\angle ABP=\angle CBD$, then $\angle BCQ=\angle ACD$.
2014 Saudi Arabia IMO TST, 1
Tarik and Sultan are playing the following game. Tarik thinks of a number that is greater than $100$. Then Sultan is telling a number greater than $1$. If Tarik’s number is divisible by Sultan’s number, Sultan wins, otherwise Tarik subtracts Sultan’s number from his number and Sultan tells his next number. Sultan is forbidden to repeat his numbers. If Tarik’s number becomes negative, Sultan loses. Does Sultan have a winning strategy?
2020 BMT Fall, 18
Let $T$ be the answer to question $17$, and let $N =\frac{24}{T}$. Leanne flips a fair coin $N$ times. Let $X$ be the number of times that within a series of three consecutive flips, there were exactly two heads or two tails. What is the expected value of $X$?