Found problems: 85335
2024 China Team Selection Test, 24
Let $N=10^{2024}$. $S$ is a square in the Cartesian plane with side length $N$ and the sides parallel to the coordinate axes. Inside there are $N$ points $P_1$, $P_2$, $\dots$, $P_N$ all of which have different $x$ coordinates, and the absolute value of the slope of any connected line between these points is at most $1$. Prove that there exists a line $l$ such that at least $2024$ of these points is at most distance $1$ away from $l$.
2017 NIMO Problems, 4
For how many positive integers $100 < n \le 10000$ does $\lfloor \sqrt{n-100} \rfloor$ divide $n$?
[i]Proposed by Michael Tang[/i]
1984 IMO, 1
Prove that $0\le yz+zx+xy-2xyz\le{7\over27}$, where $x,y$ and $z$ are non-negative real numbers satisfying $x+y+z=1$.
2016 Brazil National Olympiad, 6
Lei it \(ABCD\) be a non-cyclical, convex quadrilateral, with no parallel sides.
The lines \(AB\) and \(CD\) meet in \(E\).
Let it \(M \not= E\) be the intersection of circumcircles of \(ADE\) and \(BCE\).
The internal angle bisectors of \(ABCD\) form an convex, cyclical quadrilateral with circumcenter \(I\).
The external angle bisectors of \(ABCD\) form an convex, cyclical quadrilateral with circumcenter \(J\).
Show that \(I,J,M\) are colinear.
2021/2022 Tournament of Towns, P3
The hypotenuse of a right triangle has length 1. Consider the line passing through the points of tangency of the incircle with the legs of the triangle. The circumcircle of the triangle cuts out a segment of this line. What is the possible length of this segment?
[i]Maxim Volchkevich[/i]
1990 National High School Mathematics League, 7
If $n\in\mathbb{Z_+}$, positive real numbers $a+b=2$, then the minumum value of $\frac{1}{1+a^n}+\frac{1}{1+b^n}$ is________.
2006 AMC 12/AHSME, 8
The lines $ x \equal{} \frac 14y \plus{} a$ and $ y \equal{} \frac 14x \plus{} b$ intersect at the point $ (1,2)$. What is $ a \plus{} b$?
$ \textbf{(A) } 0 \qquad \textbf{(B) } \frac 34 \qquad \textbf{(C) } 1 \qquad \textbf{(D) } 2 \qquad \textbf{(E) } \frac 94$
1982 IMO Longlists, 11
A rectangular pool table has a hole at each of three of its corners. The lengths of sides of the table are the real numbers $a$ and $b$. A billiard ball is shot from the fourth corner along its angle bisector. The ball falls in one of the holes. What should the relation between $a$ and $b$ be for this to happen?
1998 National Olympiad First Round, 7
Find the minimal value of integer $ n$ that guarantees:
Among $ n$ sets, there exits at least three sets such that any of them does not include any other; or there exits at least three sets such that any two of them includes the other.
$\textbf{(A)}\ 4 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ 8$
2007 Italy TST, 3
Find all $f: R \longrightarrow R$ such that
\[f(xy+f(x))=xf(y)+f(x)\]
for every pair of real numbers $x,y$.
1972 Polish MO Finals, 1
Polynomials $u_i(x) = a_ix+b_i$ ($a_i,b_i \in R$, $ i = 1,2,3$) satisfy
$$u_1(x)^n +u_2(x)^n = u_3(x)^n$$
for some integer $n \ge 2.$
Prove that there exist real numbers $A$,$B$,$c_1$,$c_2$,$c_3$ such that $u_i(x) = c_i(Ax+B)$ for $i = 1,2,3$.
2006 IMO Shortlist, 4
Let $P(x)$ be a polynomial of degree $n > 1$ with integer coefficients and let $k$ be a positive integer. Consider the polynomial $Q(x) = P(P(\ldots P(P(x)) \ldots ))$, where $P$ occurs $k$ times. Prove that there are at most $n$ integers $t$ such that $Q(t) = t$.
1949 Moscow Mathematical Olympiad, 158
a) Prove that $x^2 + y^2 + z^2 = 2xyz$ for integer $x, y, z$ only if $x = y = z = 0$.
b) Find integers $x, y, z, u$ such that $x^2 + y^2 + z^2 + u^2 = 2xyzu$.
2015 Math Prize for Girls Problems, 14
Let $C$ be a three-dimensional cube with edge length 1. There are 8 equilateral triangles whose vertices are vertices of $C$. The 8 planes that contain these 8 equilateral triangles divide $C$ into several nonoverlapping regions. Find the volume of the region that contains the center of $C$.
2016 CIIM, Problem 1
Find all functions $f:(0,+\infty) \to (0,+\infty)$ that satisfy
$(i)$ $f(xf(y))=yf(x), \forall x,y > 0,$
$(ii)$ $\displaystyle\lim_{x\to+\infty} f(x) = 0.$
2017 AMC 12/AHSME, 15
Let $ABC$ be an equilateral triangle. Extend side $\overline{AB}$ beyond $B$ to a point $B'$ so that $BB' = 3AB$. Similarly, extend side $\overline{BC}$ beyond $C$ to a point $C'$ so that $CC' = 3BC$, and extend side $\overline{CA}$ beyond $A$ to a point $A'$ so that $AA' = 3CA$. What is the ratio of the area of $\triangle A'B'C'$ to the area of $\triangle ABC$?
$\textbf{(A) }9:1\qquad\textbf{(B) }16:1\qquad\textbf{(C) }25:1\qquad\textbf{(D) }36:1\qquad\textbf{(E) }37:1$
2017 NIMO Summer Contest, 4
The square $BCDE$ is inscribed in circle $\omega$ with center $O$. Point $A$ is the reflection of $O$ over $B$. A "hook" is drawn consisting of segment $AB$ and the major arc $\widehat{BE}$ of $\omega$ (passing through $C$ and $D$). Assume $BCDE$ has area $200$. To the nearest integer, what is the length of the hook?
[i]Proposed by Evan Chen[/i]
2007 IMO Shortlist, 2
Let $b,n > 1$ be integers. Suppose that for each $k > 1$ there exists an integer $a_k$ such that $b - a^n_k$ is divisible by $k$. Prove that $b = A^n$ for some integer $A$.
[i]Author: Dan Brown, Canada[/i]
2007 Pre-Preparation Course Examination, 2
Let $\{A_{1},\dots,A_{k}\}$ be matrices which make a group under matrix multiplication. Suppose $M=A_{1}+\dots+A_{k}$. Prove that each eigenvalue of $M$ is equal to $0$ or $k$.
2010 IberoAmerican Olympiad For University Students, 7
(a) Prove that, for any positive integers $m\le \ell$ given, there is a positive integer $n$ and positive integers $x_1,\cdots,x_n,y_1,\cdots,y_n$ such that the equality \[ \sum_{i=1}^nx_i^k=\sum_{i=1}^ny_i^k\] holds for every $k=1,2,\cdots,m-1,m+1,\cdots,\ell$, but does not hold for $k=m$.
(b) Prove that there is a solution of the problem, where all numbers $x_1,\cdots,x_n,y_1,\cdots,y_n$ are distinct.
[i]Proposed by Ilya Bogdanov and Géza Kós.[/i]
2010 N.N. Mihăileanu Individual, 2
If at least one of the integers $ a,b $ is not divisible by $ 3, $ then the polynom $ X^2-abX+a^2+b^2 $ is irreducible over the integers.
[i]Ion Cucurezeanu[/i]
1999 Moldova Team Selection Test, 7
Let $ABC$ be an equilateral triangle and $n{}, n>1$ an integer. Let $S{}$ be the set of the $n-1$ lines parallel with $BC$ that cut $ABC$ in $n{}$ figures with equal areas and $S^{'}$ be the set of the $n-1$ lines parallel with $BC$ that cut $ABC$ in $n{}$ figures with equal perimeters. Show that $S{}$ and $S^{'}$ are disjunctive.
1991 IMTS, 4
Let $n$ points with integer coordinates be given in the $xy$-plane. What is the minimum value of $n$ which will ensure that three of the points are the vertices of a triangel with integer (possibly, 0) area?
2000 IMC, 2
Let $f$ be continuous and nowhere monotone on $[0,1]$. Show that the set of points on which $f$ obtains a local minimum is dense.
1998 Slovenia National Olympiad, Problem 3
A point $E$ on side $CD$ of a rectangle $ABCD$ is such that $\triangle DBE$ is isosceles and $\triangle ABE$ is right-angled. Find the ratio between the side lengths of the rectangle.