Found problems: 85335
Let $p_1, p_2, p_3$, and $p$ be prime numbers. Prove that there exist $x,y\in \mathbb{Z}$ such that
$y^2\equiv p_1 x^4-p_1 p_2^2 p_3^2\, (mod\, p)$.
$g$ is a twice differentiable function over the positive reals such that \begin{align}g(x)+2x^3g^\prime(x)+x^4g^{\prime\prime}(x)&= 0 \qquad\text{ for all positive reals } x\\\lim_{x\to\infty}xg(x)&=1\end{align}
Find the real number $\alpha>1$ such that $g(\alpha)=1/2$.
a) Fold a $10 \times 10$ square from a $1 \times 118$ rectangular strip.
b) Fold a $10 \times 10$ square from a $1 \times (100+9\sqrt3)$ rectangular strip (approximately $1\times 115.58$).
The strip can be bent, but not torn.
Given four non-coplanar points, is it always possible to find a plane such that the orthogonal projections of the points onto the plane are the vertices of a parallelogram? How many such planes are there in general?
Suppose real numbers $A,B,C$ such that for all real numbers $x,y,z$ the following inequality holds:
\[A(x-y)(x-z) + B(y-z)(y-x) + C(z-x)(z-y) \geq 0.\]
Find the necessary and sufficient condition $A,B,C$ must satisfy (expressed by means of an equality or an inequality).
Let $P$ and $Q$ be the midpoints of sides $AB$ and $BC,$ respectively, of $\triangle ABC.$ Suppose $\angle A = 30^{\circ}$ and $\angle PQC = 110^{\circ}.$ Find $\angle B$ in degrees.
Find all reals $A,B,C$ such that there exists a real function $f$ satisfying $f(x+f(y))= Ax+By+C$ for all reals $x,y$.
A sum of $335$ pairwise distinct positive integers equals $100000$.
a) What is the least number of uneven integers in that sum?
b) What is the greatest number of uneven integers in that sum?
Evaluate $\frac{1}{2+\frac{3}{1+\frac{2}{3+x}}}$ if $x = 1$.
$ P(x)$ is a quadratic trinomial. What maximum number of terms equal to the sum of the two preceding terms can occur in the sequence $ P(1)$, $ P(2)$, $ P(3)$, $ \dots?$
[i]Proposed by A. Golovanov[/i]
Certain king of a certain state wants to build $n$ cities and $n-1$ roads, connecting them to provide a possibility to move from every city to every city. (Each road connects two cities, the roads do not intersect, and don't come through another city.) He wants also, to make the shortests distances between the cities, along the roads, to be $1,2,3,...,n(n-1)/2$ kilometres. Is it possible for
a) $n=6$
b) $n=1986$ ?
We are given $n \geq 3$ weights of masses $1, 2, 3, \ldots , n$ grams. Find all $n$ for which it is possible to divide these weights into three groups with the same mass.
Consider a $2023\times2023$ board split into unit squares. Two unit squares are called adjacent is they share at least one vertex. Mahler and Srecko play a game on this board. Initially, Mahler has one piece placed on the square marked [b]M[/b], and Srecko has a piece placed on the square marked by [b]S[/b] (see the attachment). The players alternate moving their piece, following three rules:
1. A piece can only be moved to a unit square adjacent to the one it is placed on.
2. A piece cannot be placed on a unit square on which a piece has been placed before (once used, a unit square can never be used again).
3. A piece cannot be moved to a unit square adjacent to the square occupied by the opponent’s piece.
A player wins the game if his piece gets to the corner diagonally opposite to its starting position (i.e. Srecko moves to $s_p$, Mahler moves to $m_p$) or if the opponent has to move but has no legal move. Mahler moves first. Which player has a winning strategy?
At a football tournament, each team plays with each of the remaining teams, winning three points for the win, one point for the draw score and zero points for the defeat. At the end of the tournament it turned out that the sum of the winning points of all teams was $50$.
(a) How many teams participated in this tournament?
(b) How big is the difference between the team with the highest number and the number of points won?
A model car is tested on some closed circuit $600$ meters long, consisting of flat stretches, uphill and downhill. All uphill and downhill have the same slope. The test highlights the following facts:
[list]
(a) The velocity of the car depends only on the fact that the car is driving along a stretch of uphill, plane or downhill; calling these three velocities $v_s, v_p$ and $v_d$ respectively, we have $v_s <v_p <v_d$;
(b) $v_s,v_p$ and $v_d$, expressed in meter per second, are integers.
(c) Whatever may be the structure of the circuit, the time taken to complete the circuit is always $50$ seconds.
[/list]
Find all possible values of $v_s, v_p$ and $v_d$.
Let $ABC$ be an equilateral triangle with $AB=1.$ Let $M$ be the midpoint of $BC,$ and let $P$ be on segment $AM$ such that $AM/MP=4.$ Find $BP.$
Let $H$ be a regular hexagon with side length one. Peter picks a point $P$ uniformly and at random within $H$, then draws the largest circle with center $P$ that is contained in $H$. What is this probability that the radius of this circle is less than $1/2$ ?
$A_0B_0C_0$ and $A_1B_1C_1$ are acute-angled triangles. Describe, and prove, how to construct the triangle $ABC$ with the largest possible area which is circumscribed about $A_0B_0C_0$ (so $BC$ contains $B_0, CA$ contains $B_0$, and $AB$ contains $C_0$) and similar to $A_1B_1C_1.$
Suppose that in a group of $6$ people, if $A$ is friends with $B$, then $B$ is friends with $A$. If each of the $6$ people draws a graph of the friendships between the other $5$ people, we get these $6$ graphs, where edges represent
friendships and points represent people.
[img]https://cdn.artofproblemsolving.com/attachments/5/5/7265067f585e3dfe77ba94ac6261b4462cd015.png[/img]
If Sue drew the first graph, how many friends does she have?
Let $n$ be a positive integer. An $n\times n$ board consisting of $n^2$ cells, each being a unit square colored either black or white, is called [i]convex[/i] if for every black colored cell, both the cell directly to the left of it and the cell directly above it are also colored black. We define the [i]beauty[/i] of a board as the number of pairs of its cells $(u,v)$ such that $u$ is black, $v$ is white, and $u$ and $v$ are in the same row or column. Determine the maximum possible beauty of a convex $n\times n$ board.
[i]Proposed by Ivan Novak[/i]
In parallelogram $ABCD$, $AB = 10$, and $AB = 2BC$. Let $M$ be the midpoint of $CD$, and suppose that $BM = 2AM$. Compute $AM$.
See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url]
Let $x,y>0$ and $xy\ge1.$ Prove that $x^3+y^3+4xy\ge x^2+y^2+x+y+2.$
Let $x,y>0$ and $xy\ge1.$ Prove that $2(x^3+y^3+xy+x+y)\ge5(x^2+y^2).$
Let $G$ be a directed graph with infinitely many vertices. It is known that for each vertex the outdegree is greater than the indegree. Let $O$ be a fixed vertex of $G$. For an arbitrary positive number $n$, let $V_{n}$ be the number of vertices which can be reached from $O$ passing through at most $n$ edges ( $O$ counts). Find the smallest possible value of $V_{n}$.
Let the [i]square decomposition[/i] of a number be defined as the sequence of numbers given by the following algorithm. Given a positive integer $n$, add the largest possible perfect square that is less than or equal to $n$ to a sequence, and then subtract that number from $n$. Repeat as many times as necessary until your current $n$ is $0$. So for example, the square decomposition of $60$ would be $49, 9, 1, 1$. Define the size of a square decomposition to be the number of numbers in the sequence. Say that the maximal size of a square decomposition of a number in the range $[1, 2020]$ is $m$. Find the largest number in the range $[1, 2020]$ that has a square decomposition of size $m$.
[i]Proposed by Timothy Qian[/i]
The Fibonacci numbers are defined by $F_1=F_2=1$ and $F_n=F_{n-1}+F_{n-2}$ for $n>2$. It is well-known that the sum of any $10$ consecutive Fibonacci numbers is divisible by $11$. Determine the smallest integer $N$ so that the sum of any $N$ consecutive Fibonacci numbers is divisible by $12$.