Found problems: 85335
1999 Estonia National Olympiad, 1
Find all pairs of integers ($a, b$) such that $a^2 + b = b^{1999}$ .
2021 Ukraine National Mathematical Olympiad, 7
The sequence $a_1,a_2, ..., a_{2n}$ of integers is such that each number occurs in no more than $n$ times. Prove that there are two strictly increasing sequences of indices $b_1,b_2, ..., b_{n}$ and $c_1,c_2, ..., c_{n}$ are such that every positive integer from the set $\{1,2,...,2n\}$ occurs exactly in one of these two sequences, and for each $1\le i \le n$ is true the condition $a_{b_i} \ne a_{c_i}$
.
(Anton Trygub)
2004 Purple Comet Problems, 15
Find the prime number $p$ for which $p + 2500$ is a perfect square.
2004 Tournament Of Towns, 4
Vanya has chosen two positive numbers, x and y. He wrote the numbers x+y, x-y, x/y, and xy, and has shown these numbers to Petya. However, he didn't say which of the numbers was obtained from which operation. Show that Petya can uniquely recover x and y.
Kyiv City MO Juniors 2003+ geometry, 2018.8.3
In the isosceles triangle $ABC$ with the vertex at the point $B$, the altitudes $BH$ and $CL$ are drawn. The point $D$ is such that $BDCH$ is a rectangle. Find the value of the angle $DLH$.
(Bogdan Rublev)
2012 Romania National Olympiad, 4
[color=darkred]Let $n$ and $m$ be two natural numbers, $m\ge n\ge 2$ . Find the number of injective functions
\[f\colon\{1,2,\ldots,n\}\to\{1,2,\ldots,m\}\]
such that there exists a unique number $i\in\{1,2,\ldots,n-1\}$ for which $f(i)>f(i+1)\, .$[/color]
MMPC Part II 1996 - 2019, 2013
[b]p1.[/b] The number $100$ is written as a sum of distinct positive integers. Determine, with proof, the maximum number of terms that can occur in the sum.
[b]p2.[/b] Inside an equilateral triangle of side length $s$ are three mutually tangent circles of radius $1$, each one of which is also tangent to two sides of the triangle, as depicted below. Find $s$.
[img]https://cdn.artofproblemsolving.com/attachments/4/3/3b68d42e96717c83bd7fa64a2c3b0bf47301d4.png[/img]
[b]p3.[/b] Color a $4\times 7$ rectangle so that each of its $28$ unit squares is either red or green. Show that no matter how this is done, there will be two columns and two rows, so that the four squares occurring at the intersection of a selected row with a selected column all have the same color.
[b]p4.[/b] (a) Show that the $y$-intercept of the line through any two distinct points of the graph of $f(x) = x^2$ is $-1$ times the product of the $x$-coordinates of the two points.
(b) Find all real valued functions with the property that the $y$-intercept of the line through any two distinct points of its graph is $-1$ times the product of the $x$-coordinates. Prove that you have found all such functions and that all functions you have found have this property.
[b]p5.[/b] Let $n$ be a positive integer. We consider sets $A \subseteq \{1, 2,..., n\}$ with the property that the equation $x+y=z$ has no solution with $x\in A$, $y \in A$, $z \in A$.
(a) Show that there is a set $A$ as described above that contains $[(n + l)/2]$ members where $[x]$ denotes the largest integer less than or equal to $x$.
(b) Show that if $A$ has the property described above, then the number of members of $A$ is less than or equal to $[(n + l)/2]$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2017 Vietnamese Southern Summer School contest, Problem 1
A and B are friends at a summer school. When B asks A for his address, he answers: "My house is on XYZ street, and my house number is a 3-digit number with distinct digits, and if you permute its digits, you will have other 5 numbers. The interesting thing is that the sum of these 5 numbers is exactly 2017. That's all.". After a while, B can determine A's house number. And you, can you find his house number?
2006 Switzerland Team Selection Test, 2
Let $D$ be inside $\triangle ABC$ and $E$ on $AD$ different to $D$. Let $\omega_1$ and $\omega_2$ be the circumscribed circles of $\triangle BDE$ and $\triangle CDE$ respectively. $\omega_1$ and $\omega_2$ intersect $BC$ in the interior points $F$ and $G$ respectively. Let $X$ be the intersection between $DG$ and $AB$ and $Y$ the intersection between $DF$ and $AC$. Show that $XY$ is $\|$ to $BC$.
2021 Israel TST, 3
What is the smallest value of $k$ for which the inequality
\begin{align*}
ad-bc+yz&-xt+(a+c)(y+t)-(b+d)(x+z)\leq \\
&\leq k\left(\sqrt{a^2+b^2}+\sqrt{c^2+d^2}+\sqrt{x^2+y^2}+\sqrt{z^2+t^2}\right)^2
\end{align*}
holds for any $8$ real numbers $a,b,c,d,x,y,z,t$?
Edit: Fixed a mistake! Thanks @below.
2006 Tournament of Towns, 3
Let $a$ be some positive number. Find the number of integer solutions $x$ of inequality $100 < xa < 1000$ given that inequality $10 < xa < 100$ has exactly $5$ integer solutions. Consider all possible cases.
[i](4 points)[/i]
2022 Sharygin Geometry Olympiad, 5
Let the diagonals of cyclic quadrilateral $ABCD$ meet at point $P$. The line passing through $P$ and perpendicular to $PD$ meets $AD$ at point $D_1$, a point $A_1$ is defined similarly. Prove that the tangent at $P$ to the circumcircle of triangle $D_1PA_1$ is parallel to $BC$.
2019 LIMIT Category A, Problem 5
$64$ numbers (not necessarily distinct) are placed on the squares of a chessboard such that the sum of the numbers in every $2\times2$ square is $7$. What is the sum of the four numbers in the corners of the board?
2006 District Olympiad, 3
We say that a prism is [i]binary[/i] if there exists a labelling of the vertices of the prism with integers from the set $\{-1,1\}$ such that the product of the numbers assigned to the vertices of each face (base or lateral face) is equal to $-1$.
a) Prove that any [i]binary[/i] prism has the number of total vertices divisible by 8;
b) Prove that any prism with 2000 vertices is [i]binary[/i].
2023 CCA Math Bonanza, TB4
Charlotte the cat is placed at the origin of the coordinate plane, such that the positive $x$ direction is pointing east, and the positive $y$ direction is pointing north. Then, every single lattice square (a unit square with vertices all on lattice points) in the first quadrant whose southwest vertex is a lattice point with both odd coordinates is completely removed. Charlotte can traverse the coordinate plane by drawing a segment between two valid vertices, given that they do not intersect a lattice square that has not been removed. Let $P_1$ and $P_2$ denote the distances of the first and second shortest paths Charlotte can take to $(5,7),$ respectively. Find $P_1-P_2.$
[i]Tiebreaker #4[/i]
2013 Stanford Mathematics Tournament, 20
Ben is throwing darts at a circular target with diameter 10. Ben never misses the target when he throws a dart, but he is equally likely to hit any point on the target. Ben gets $\lceil 5-x \rceil$ points for having the dart land $x$ units away from the center of the target. What is the expected number of points that Ben can earn from throwing a single dart? (Note that $\lceil y \rceil$ denotes the smallest integer greater than or equal to $y$.)
2018 CHKMO, 2
Suppose $ABCD$ is a cyclic quadrilateral. Extend $DA$ and $DC$ to $P$ and $Q$ respectively such that $AP=BC$ and $CQ=AB$. Let $M$ be the midpoint of $PQ$. Show that $MA\perp MC$.
1955 AMC 12/AHSME, 44
In circle $ O$ chord $ AB$ is produced so that $ BC$ equals a radius of the circle. $ CO$ is drawn and extended to $ D$. $ AO$ is drawn. Which of the following expresses the relationship between $ x$ and $ y$?
[asy]size(200);defaultpen(linewidth(0.7)+fontsize(10));
pair O=origin, D=dir(195), A=dir(150), B=dir(30), C=B+1*dir(0);
draw(O--A--C--D);
dot(A^^B^^C^^D^^O);
pair point=O;
label("$A$", A, dir(point--A));
label("$B$", B, dir(point--B));
label("$C$", C, dir(point--C));
label("$D$", D, dir(point--D));
label("$O$", O, dir(285));
label("$x$", O+0.1*dir(172.5), dir(172.5));
label("$y$", C+0.4*dir(187.5), dir(187.5));
draw(Circle(O,1));
[/asy]
$ \textbf{(A)}\ x\equal{}3y \\
\textbf{(B)}\ x\equal{}2y \\
\textbf{(C)}\ x\equal{}60^\circ \\
\textbf{(D)}\ \text{there is no special relationship between }x\text{ and }y \\
\textbf{(E)}\ x\equal{}2y \text{ or }x\equal{}3y\text{, depending upon the length of }AB$
1999 Nordic, 4
Let $a_1, a_2, . . . , a_n$ be positive real numbers and $n \ge 1$. Show that
$n (\frac{1}{a_1}+...+\frac{1}{a_n}) \ge (\frac{1}{1+a_1}+...+\frac{1}{1+a_n})(n+\frac{1}{a_1}+...+\frac{1}{a_n})$
When does equality hold?
1998 Romania National Olympiad, 1
Find the integer numbers $a, b, c$ such that the function $f: R \to R$, $f(x) = ax^2 +bx + c$ satisfies the equalities : $$f(f(1) ))= f (f(2 ) )= f(f (3 ))$$
2000 Saint Petersburg Mathematical Olympiad, 11.5
Let $AA_1$, $BB_1$, $CC_1$ be the altitudes of an acute angled triangle $ABC$. On the side $BC$ point $K$ is taken such that $\angle BB_1K=\angle A$. On the side $AB$ a point $M$ is taken such that $\angle BB_1M\angle C$. Let $L$ be the intersection of $BB_1$ and $A_1C_1$. Prove that the quadrilateral $B_1KLM$ is circumscribed.
[I]Proposed by A. Khrabrov, D. Rostovski[/i]
2020 Macedonian NationŠ°l Olympiad, 2
Let $x_1, ..., x_n$ ($n \ge 2$) be real numbers from the interval $[1, 2]$. Prove that
$|x_1 - x_2| + ... + |x_n - x_1| \le \frac{2}{3}(x_1 + ... + x_n)$,
with equality holding if and only if $n$ is even and the $n$-tuple $(x_1, x_2, ..., x_{n - 1}, x_n)$ is equal to $(1, 2, ..., 1, 2)$ or $(2, 1, ..., 2, 1)$.
2011 Tuymaada Olympiad, 1
Red, blue, and green children are arranged in a circle. When a teacher asked the red children that have a green neighbor to raise their hands, $20$ children raised their hands. When she asked the blue children that have a green neighbor to raise their hands, $25$ children raised their hands. Prove that some child that raised her hand had two green neighbors.
2022 Moldova Team Selection Test, 7
Let $f:\mathbb{N} \rightarrow \mathbb{N},$ $f(n)=n^2-69n+2250$ be a function. Find the prime number $p$, for which the sum of the digits of the number $f(p^2+32)$ is as small as possible.
2022 China Team Selection Test, 6
Let $m$ be a positive integer, and $A_1, A_2, \ldots, A_m$ (not necessarily different) be $m$ subsets of a finite set $A$. It is known that for any nonempty subset $I$ of $\{1, 2 \ldots, m \}$,
\[ \Big| \bigcup_{i \in I} A_i \Big| \ge |I|+1. \]
Show that the elements of $A$ can be colored black and white, so that each of $A_1,A_2,\ldots,A_m$ contains both black and white elements.