This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 12

2006 Swedish Mathematical Competition, 6
Tags: algebra , Exponents , Sweden
Determine all positive integers $a,b,c$ satisfying $a^{(b^c)}=(b^a)^c$

2006 Swedish Mathematical Competition, 5
Tags: combinatorics , symmetry , Sweden
In each square of an $m \times n$ rectangular board there is a nought or a cross. Let $f(m,n)$ be the number of such arrangements that contain a row or a column consisting of noughts only. Let $g(m,n)$ be the number of arrangements that contain a row consisting of noughts only, or a column consisting of crosses only. Which of the numbers $f(m,n)$ and $g(m,n)$ is larger?

2005 Swedish Mathematical Competition, 6
Tags: geometry , 3D geometry , tetrahedron , Sweden
A regular tetrahedron of edge length $1$ is orthogonally projected onto a plane. Find the largest possible area of its image.

2005 Swedish Mathematical Competition, 5
Tags: combinatorics unsolved , combinatorics , Sweden
Every cell of a $2005 \times 2005$ square board is colored white or black so that every $2 \times 2$ subsquare contains an odd number of black cells. Show that among the corner cells there is an even number of black ones. How many ways are there to color the board in this manner?

2005 Swedish Mathematical Competition, 4
Tags: arithmetic sequence , algebra , Sweden , derivative
The zeroes of a fourth degree polynomial $f(x)$ form an arithmetic progression. Prove that the three zeroes of the polynomial $f'(x)$ also form an arithmetic progression.

2005 Swedish Mathematical Competition, 1
Tags: number theory , Diophantine Equations , Sweden
Find all integer solutions $x$,$y$ of the equation $(x+y^2)(x^2+y)=(x+y)^3$.

2006 Swedish Mathematical Competition, 2
Tags: geometry , incenter , Sweden
In a triangle $ABC$, point $P$ is the incenter and $A'$, $B'$, $C'$ its orthogonal projections on $BC$, $CA$, $AB$, respectively. Show that $\angle B'A'C'$ is acute.

2005 Swedish Mathematical Competition, 2
Tags: combinatorics , Sweden
There are 12 people in a line in a bank. When the desk closes, the people form a new line at a newly opened desk. In how many ways can they do this in such a way that none of the 12 people changes his/her position in the line by more than one?

2006 Swedish Mathematical Competition, 4
Tags: combinatorics , Sweden
Saskia and her sisters have been given a large number of pearls. The pearls are white, black and red, not necessarily the same number of each color. Each white pearl is worth $5$ Ducates, each black one is worth $7$, and each red one is worth $12$. The total worth of the pearls is $2107$ Ducates. Saskia and her sisters split the pearls so that each of them gets the same number of pearls and the same total worth, but the color distribution may vary among the sisters. Interestingly enough, the total worth in Ducates that each of the sisters holds equals the total number of pearls split between the sisters. Saskia is particularly fond of the red pearls, and therefore makes sure that she has as many of those as possible. How many pearls of each color has Saskia?

2006 Swedish Mathematical Competition, 3
Tags: analytic geometry , graphing lines , slope , calculus , derivative , algebra , Sweden
A cubic polynomial $f$ with a positive leading coefficient has three different positive zeros. Show that $f'(a)+ f'(b)+ f'(c) > 0$.

2006 Swedish Mathematical Competition, 1
Tags: Sweden , number theory
If positive integers $a$ and $b$ have 99 and 101 different positive divisors respectively (including 1 and the number itself), can the product $ab$ have exactly 150 positive divisors?

2005 Swedish Mathematical Competition, 3
Tags: Sweden , geometry , angle bisector
In a triangle $ABC$ the bisectors of angles $A$ and $C$ meet the opposite sides at $D$ and $E$ respectively. Show that if the angle at $B$ is greater than $60^\circ$, then $AE +CD <AC$.