Found problems: 85335
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]
2017 ASDAN Math Tournament, 1
Suppose $(x+y)^2=25$ and $(x-y)^2=1$. Compute $xy$.
2024 LMT Fall, C5
Kanye West's favorite positive integer this year is $c$, and last year it was $c-t=20011$ (a prime), for some positive integer $t$ relatively prime to $c$. His two most streamed albums got $a$ and $b$ streams this year and $a-t$ and $b-t$ streams last year with $a > b > c$. Suppose $a \le 1.6 \times 10^9$ and his favorite integer in each year divides the number of streams for both albums in the corresponding year. Find the largest possible value of $c$.
2017 Peru Iberoamerican Team Selection Test, P1
Let $C_1$ and $C_2$ be tangent circles internally at point $A$, with $C_2$ inside of $C_1$. Let $BC$ be a chord of $C_1$ that is tangent to $C_2$. Prove that the ratio between the length $BC$ and the perimeter of the triangle $ABC$ is constant, that is, it does not depend of the selection of the chord $BC$ that is chosen to construct the trangle.
2022 German National Olympiad, 3
Let $M$ and $N$ be the midpoints of segments $BC$ and $AC$ of a triangle $ABC$, respectively. Let $Q$ be a point on the line through $N$ parallel to $BC$ such that $Q$ and $C$ are on opposite sides of $AB$ and $\vert QN\vert \cdot \vert BC\vert=\vert AB\vert \cdot \vert AC\vert$.
Suppose that the circumcircle of triangle $AQN$ intersects the segment $MN$ a second time in a point $T \ne N$.
Prove that there is a circle through points $T$ and $N$ touching both the side $BC$ and the incircle of triangle $ABC$.
2012 Belarus Team Selection Test, 3
Find all triples $(a,b, c)$ of real numbers for which there exists a non-zero function $f: R \to R$, such that $$af(xy + f(z)) + bf(yz + f(x)) + cf(zx + f(y)) = 0$$ for all real $x, y, z$.
(E. Barabanov)
2009 Princeton University Math Competition, 3
Let $x_1,x_2,\ldots, x_{10}$ be non-negative real numbers such that $\frac{x_1}{1}+ \frac{x_2}{2} +\cdots+ \frac{x_{10}}{10}$ $\leq9$. Find the maximum possible value of $\frac{{x_1}^2}{1}+\frac{{x_2}^2}{2}+\cdots+\frac{{x_{10}}^2}{10}$.
2021 Math Prize for Girls Problems, 5
Among all fractions (whose numerator and denominator are positive integers) strictly between $\tfrac{6}{17}$ and $\tfrac{9}{25}$, which one has the smallest denominator?
1992 IMO Shortlist, 1
Prove that for any positive integer $ m$ there exist an infinite number of pairs of integers $ (x, y)$ such that
[i](i)[/i] $ x$ and $ y$ are relatively prime;
[i](ii)[/i] $ y$ divides $ x^2 \plus{} m$;
[i](iii)[/i] $ x$ divides $ y^2 \plus{} m.$
[i](iv)[/i] $ x \plus{} y \leq m \plus{} 1\minus{}$ (optional condition)
1967 IMO Longlists, 15
Suppose $\tan \alpha = \dfrac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. Prove that the number $\tan \beta$ for which $\tan {2 \beta} = \tan {3 \alpha}$ is rational only when $p^2 + q^2$ is the square of an integer.
1975 Bundeswettbewerb Mathematik, 3
For $n$positive integers $ x_1,x2,...,x_n$, $a_n$ is their arithmetic and $g_n$ the geometric mean. Consider the statement $S_n$: If $a_n/g_n$ is a positive integer, then $x_1 = x_2 = ··· = x_n$. Prove $S_2$ and disprove $S_n$ for all even $n > 2$.
1985 AIME Problems, 6
As shown in the figure, triangle $ABC$ is divided into six smaller triangles by lines drawn from the vertices through a common interior point. The areas of four of these triangles are as indicated. Find the area of triangle $ABC$.
[asy]
size(200);
pair A=origin, B=(14,0), C=(9,12), D=foot(A, B,C), E=foot(B, A, C), F=foot(C, A, B), H=orthocenter(A, B, C);
draw(F--C--A--B--C^^A--D^^B--E);
label("$A$", A, SW);
label("$B$", B, SE);
label("$C$", C, N);
label("84", centroid(H, C, E), fontsize(9.5));
label("35", centroid(H, B, D), fontsize(9.5));
label("30", centroid(H, F, B), fontsize(9.5));
label("40", centroid(H, A, F), fontsize(9.5));[/asy]
2021 Belarusian National Olympiad, 9.2
A bug is walking on the surface of a Rubik's cube(cube $3 \times 3 \times 3$). It can go to the adjacent cell on the same face or on the adjacent face. One day the bug started walking from some cell and returned to it, and visited all other cells exactly once.
Prove that he made an even amount of moves that changed the face he is on.
2006 Petru Moroșan-Trident, 1
Let be four distinct complex numbers $ a,b,c,d $ chosen such that
$$ |a|=|b|=|c|=|d|=|b-c|=\frac{|c-d|}{2}=1, $$
and
$$ \min_{\lambda\in\mathbb{C}} |a-\lambda d -(1-\lambda )c| =\min_{\lambda\in\mathbb{C}} |b-\lambda d -(1-\lambda )c| . $$
Calculate $ |a-c| $ and $ |a-d|. $
[i]Carmen Botea[/i]
2002 AMC 10, 13
Find the value(s) of $ x$ such that $ 8xy\minus{}12y\plus{}2x\minus{}3\equal{}0$ is true for all values of $ y$.
$ \textbf{(A)}\ \frac{2}{3} \qquad
\textbf{(B)}\ \frac{3}{2}\text{ or }\minus{}\frac{1}{4} \qquad
\textbf{(C)}\ \minus{}\frac{2}{3}\text{ or }\minus{}\frac{1}{4} \qquad
\textbf{(D)}\ \frac{3}{2} \qquad
\textbf{(E)}\ \minus{}\frac{3}{2}\text{ or }\minus{}\frac{1}{4}$
PEN D Problems, 2
Suppose that $p$ is an odd prime. Prove that \[\sum_{j=0}^{p}\binom{p}{j}\binom{p+j}{j}\equiv 2^{p}+1\pmod{p^{2}}.\]
1988 Czech And Slovak Olympiad IIIA, 2
If for the coefficients of equation $x^3+ax^2+bx+c=0$ whose roots are all real, holds, $a^2= 2(b+1)$ then $|a-c|\le 2$. Prove it.
2011 Postal Coaching, 2
Let $x$ be a positive real number and let $k$ be a positive integer. Assume that $x^k+\frac{1}{x^k}$ and $x^{k+1}+\frac{1}{x^{k+1}}$ are both rational numbers. Prove that $x+\frac{1}{x}$ is also a rational number.