Found problems: 85335
2023 Princeton University Math Competition, 12
12. What is the sum of all possible $\left(\begin{array}{l}i \\ j\end{array}\right)$ subject to the restrictions that $i \geq 10, j \geq 0$, and $i+j \leq 20$ ? Count different $i, j$ that yield the same value separately - for example, count both $\left(\begin{array}{c}10 \\ 1\end{array}\right)$ and $\left(\begin{array}{c}10 \\ 9\end{array}\right)$.
2009 India IMO Training Camp, 2
Let us consider a simle graph with vertex set $ V$. All ordered pair $ (a,b)$ of integers with $ gcd(a,b) \equal{} 1$, are elements of V.
$ (a,b)$ is connected to $ (a,b \plus{} kab)$ by an edge and to $ (a \plus{} kab,b)$ by another edge for all integer k.
Prove that for all $ (a,b)\in V$, there exists a path fromm $ (1,1)$ to $ (a,b)$.
II Soros Olympiad 1995 - 96 (Russia), 10.3
Each side of an acute triangle is multiplied by the cosine of the opposite angle.
a) Prove that a triangle can be formed from the resulting segments.
6) Find the radius of the circle circumscribed around the resulting triangle if the radius of the circle circumscribed around the original triangle is equal to $R$.
2012 India PRMO, 19
How many integer pairs $(x,y)$ satisfy $x^2+4y^2-2xy-2x-4y-8=0$?
2022 BMT, 10
Let $p, q,$ and $r$ be the roots of the polynomial $f(t) = t^3 - 2022t^2 + 2022t - 337.$ Given
$$x = (q-1)\left ( \frac{2022 - q}{r-1} + \frac{2022 - r}{p-1} \right )$$
$$y = (r-1)\left ( \frac{2022 - r}{p-1} + \frac{2022 - p}{q-1} \right )$$
$$z = (p-1)\left ( \frac{2022 - p}{q-1} + \frac{2022 - q}{r-1} \right )$$
compute $xyz - qrx - rpy - pqz.$
2016 Turkey EGMO TST, 3
Let $X$ be a variable point on the side $BC$ of a triangle $ABC$. Let $B'$ and $C'$ be points on the rays $[XB$ and $[XC$, respectively, satisfying $B'X=BC=C'X$. The line passing through $X$ and parallel to $AB'$ cuts the line $AC$ at $Y$ and the line passing through $X$ and parallel to $AC'$ cuts the line $AB$ at $Z$. Prove that all lines $YZ$ pass through a fixed point as $X$ varies on the line segment $BC$.
2018 Thailand TST, 3
Let $n > 1$ be a given integer. An $n \times n \times n$ cube is composed of $n^3$ unit cubes. Each unit cube is painted with one colour. For each $n \times n \times 1$ box consisting of $n^2$ unit cubes (in any of the three possible orientations), we consider the set of colours present in that box (each colour is listed only once). This way, we get $3n$ sets of colours, split into three groups according to the orientation.
It happens that for every set in any group, the same set appears in both of the other groups. Determine, in terms of $n$, the maximal possible number of colours that are present.
2009 Purple Comet Problems, 25
The polynomial $P(x)=a_0+a_1x+a_2x^2+...+a_8x^8+2009x^9$ has the property that $P(\tfrac{1}{k})=\tfrac{1}{k}$ for $k=1,2,3,4,5,6,7,8,9$. There are relatively prime positive integers $m$ and $n$ such that $P(\tfrac{1}{10})=\tfrac{m}{n}$. Find $n-10m$.
1963 Dutch Mathematical Olympiad, 2
The straight lines $k$ and $\ell$ intersect at right angles. A line intersects $k$ in $A$ and $\ell$ in $B$. Consider all straight line segments $PQ$ ($P$ on $k$ and $Q$ on $\ell$), which makes an angle of $45^o$ with $AB$.
(a) Determine the locus of the midpoints of the line segments $PQ$,
(b) If the perpendicular bisector of such a line segment $PQ$ intersects the line $k$ at $K$ and the line $\ell$ at $L$, then prove that $KL \ge PQ$.
[hide=original wording of second sentence]De loodrechte snijlijn van k en l snijdt k in A en t in B[/hide]
2015 ITAMO, 3
Let ABC a triangle, let K be the foot of the bisector relative to BC and J be the foot of the trisectrix relative to BC closer to the side AC (3* m(JAC)=m(CAB) ). Let C' and B' be two point on the line AJ on the side of J with respect to A, such that AC'=AC and AB=AB'. Prove that ABB'C is cyclic if and only if lines C'K and BB' are parallel.
1987 AMC 8, 25
Ten balls numbered $1$ to $10$ are in a jar. Jack reaches into the jar and randomly removes one of the balls. Then Jill reaches into the jar and randomly removes a different ball. The probability that the sum of the two numbers on the balls removed is even is
$\text{(A)}\ \frac{4}{9} \qquad \text{(B)}\ \frac{9}{19} \qquad \text{(C)}\ \frac{1}{2} \qquad \text{(D)}\ \frac{10}{19} \qquad \text{(E)}\ \frac{5}{9}$
1981 Putnam, A3
Find
$$ \lim_{t\to \infty} e^{-t} \int_{0}^{t} \int_{0}^{t} \frac{e^x -e^y }{x-y} \,dx\,dy,$$
or show that the limit does not exist.
1972 Bulgaria National Olympiad, Problem 1
Prove that there are don't exist integers $a,b,c$ such that for every integer $x$ the number $A=(x+a)(x+b)(x+c)-x^3-1$ is divisible by $9$.
[i]I. Tonov[/i]
2021 BMT, 15
Compute
$$\frac{\cos \left(\frac{\pi}{12}\right)\cos \left(\frac{\pi}{24}\right)\cos \left(\frac{\pi}{48}\right)\cos \left(\frac{\pi}{96}\right)...}{\cos \left(\frac{\pi}{4}\right)\cos \left(\frac{\pi}{8}\right)\cos \left(\frac{\pi}{16}\right)\cos \left(\frac{\pi}{32}\right)...}$$
2016 Lusophon Mathematical Olympiad, 4
$8$ CPLP football teams competed in a championship in which each team played one and only time with each of the other teams. In football, each win is worth $3$ points, each draw is worth $1$ point and the defeated team does not score. In that championship four teams were in first place with $15$ points and the others four came in second with $N$ points each. Knowing that there were $12$ draws throughout the championship, determine $N$.
2016 Fall CHMMC, 15
In a $5 \times 5$ grid of squares, how many nonintersecting pairs rectangles of rectangles are there? (Note sharing a vertex or edge still means the rectangles intersect.)
2022 IMO, 2
Let $\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that for each $x \in \mathbb{R}^+$, there is exactly one $y \in \mathbb{R}^+$ satisfying $$xf(y)+yf(x) \leq 2$$
2018 Latvia Baltic Way TST, P7
Let $n \ge 3$ points be given in the plane, no three of which lie on the same line. Determine whether it is always possible to draw an $n$-gon whose vertices are the given points and whose sides do not intersect.
[i]Remark.[/i] The $n$-gon can be concave.
1964 Vietnam National Olympiad, 2
Draw the graph of the functions $y = | x^2 - 1 |$ and $y = x + | x^2 -1 |$. Find the number of roots of the equation $x + | x^2 - 1 | = k$, where $k$ is a real constant.
2014 IMS, 7
Let $G$ be a finite group such that for every two subgroups of it like $H$ and $K$, $H \cong K$ or $H \subseteq K$ or $K \subseteq H$. Prove that we can produce each subgroup of $G$ with 2 elements at most.
2010 Princeton University Math Competition, 5
Let $f(x)=3x^3-5x^2+2x-6$. If the roots of $f$ are given by $\alpha$, $\beta$, and $\gamma$, find
\[
\left(\frac{1}{\alpha-2}\right)^2+\left(\frac{1}{\beta-2}\right)^2+\left(\frac{1}{\gamma-2}\right)^2.
\]
2021 USMCA, 7
Find the expected value of $\max(\min(a,b),\min(c,d),\min(e,f))$ over all permutations $(a,b,c,d,e,f)$ of $(1,2,3,4,5,6)$.
2010 IFYM, Sozopol, 4
Find all integers $x,y,z$ such that:
$7^x+13^y=2^z$
2001 IMO Shortlist, 2
Consider the system \begin{align*}x + y &= z + u,\\2xy & = zu.\end{align*} Find the greatest value of the real constant $m$ such that $m \leq x/y$ for any positive integer solution $(x,y,z,u)$ of the system, with $x \geq y$.
2010 Romanian Master of Mathematics, 5
Let $n$ be a given positive integer. Say that a set $K$ of points with integer coordinates in the plane is connected if for every pair of points $R, S\in K$, there exists a positive integer $\ell$ and a sequence $R=T_0,T_1, T_2,\ldots ,T_{\ell}=S$ of points in $K$, where each $T_i$ is distance $1$ away from $T_{i+1}$. For such a set $K$, we define the set of vectors
\[\Delta(K)=\{\overrightarrow{RS}\mid R, S\in K\}\]
What is the maximum value of $|\Delta(K)|$ over all connected sets $K$ of $2n+1$ points with integer coordinates in the plane?
[i]Grigory Chelnokov, Russia[/i]