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: 85335

2015 Taiwan TST Round 3, 1
Tags: function , number theory , algebra , induction
Let $\mathbb{Q}^+$ be the set of all positive rational numbers. Find all functions $f:\mathbb{Q}^+\rightarrow \mathbb{Q}^+$ satisfying $f(1)=1$ and \[ f(x+n)=f(x)+nf(\frac{1}{x}) \forall n\in\mathbb{N},x\in\mathbb{Q}^+\]

2018 Azerbaijan JBMO TST, 3
Tags: number theory , TST , AZE JBMO TST
Determine the integers $x$ such that $2^x + x^2 + 25$ is the cube of a prime number

2022 MIG, 1
Tags: 2022 B Individual
In a certain store, all pencils cost the same amount of money. If three pencils can be bought for six dollars, what is the price of two pencils? $\textbf{(A) }\$ 3\qquad\textbf{(B) }\$ 3.5\qquad\textbf{(C) }\$ 4\qquad\textbf{(D) }\$4.5\qquad\textbf{(E) }\$ 5$

2020 JBMO Shortlist, 1
Tags: Junior , Balkan , shortlist , 2020 , geometry
Let $\triangle ABC$ be an acute triangle. The line through $A$ perpendicular to $BC$ intersects $BC$ at $D$. Let $E$ be the midpoint of $AD$ and $\omega$ the the circle with center $E$ and radius equal to $AE$. The line $BE$ intersects $\omega$ at a point $X$ such that $X$ and $B$ are not on the same side of $AD$ and the line $CE$ intersects $\omega$ at a point $Y$ such that $C$ and $Y$ are not on the same side of $AD$. If both of the intersection points of the circumcircles of $\triangle BDX$ and $\triangle CDY$ lie on the line $AD$, prove that $AB = AC$.

1977 All Soviet Union Mathematical Olympiad, 240
Tags: combinatorics
There are direct routes from every city of a certain country to every other city. The prices are known in advance. Two tourists (they do not necessary start from one city) have decided to visit all the cities, using only direct travel lines. The first always chooses the cheapest ticket to the city, he has never been before (if there are several -- he chooses arbitrary destination among the cheapests). The second -- the most expensive (they do not return to the first city). Prove that the first will spend not more money for the tickets, than the second.

2009 National Olympiad First Round, 27
Tags:
$ f\left( x \right) \equal{} \frac {x^5}{5x^4 \minus{} 10x^3 \plus{} 10x^2 \minus{} 5x \plus{} 1}$. $ \sum_{i \equal{} 1}^{2009} f\left( \frac {i}{2009} \right) \equal{} ?$ $\textbf{(A)}\ 1000 \qquad\textbf{(B)}\ 1005 \qquad\textbf{(C)}\ 1010 \qquad\textbf{(D)}\ 2009 \qquad\textbf{(E)}\ 2010$

2017 Saudi Arabia IMO TST, 1
Tags: combinatorics
In the garden of Wonderland, there are $2016$ apples, $2017$ bananas and $2018$ oranges.Two monkeys Adu and Bakar play the following game: alternatively each of them takes and eats one fruit of any kind except for the one that he took in previous turn (in the first turn, each of them can take a fruit of any kind). Who can not take a fruit is the loser. Which monkey has the winning strategy if Adu plays first?

2010 AMC 8, 9
Tags: percent
Ryan got $80\%$ of the problems on a $25$-problem test, $90\%$ on a $40$-problem test, and $70\%$ on a $10$-problem test. What percent of all problems did Ryan answer correctly? $ \textbf{(A)}\ 64 \qquad\textbf{(B)}\ 75\qquad\textbf{(C)}\ 80\qquad\textbf{(D)}\ 84\qquad\textbf{(E)}\ 86 $

2006 Taiwan TST Round 1, 2
Tags: floor function , geometry , rectangle , polynomial , quadratics , number theory proposed , number theory
Let $p,q$ be two distinct odd primes. Calculate $\displaystyle \sum_{j=1}^{\frac{p-1}{2}}\left \lfloor \frac{qj}{p}\right \rfloor +\sum_{j=1}^{\frac{q-1}{2}}\left \lfloor \frac{pj}{q}\right\rfloor$.

2010 Bosnia And Herzegovina - Regional Olympiad, 3
Tags: number theory , Divisibility
If $a$ and $b$ are positive integers such that $ab \mid a^2+b^2$ prove that $a=b$

2012-2013 SDML (High School), 5
Tags: number theory , prime numbers , SDML High School , 2012-13 SDML Round 2
Palmer correctly computes the product of the first $1,001$ prime numbers. Which of the following is NOT a factor of Palmer's product? $\text{(A) }2,002\qquad\text{(B) }3,003\qquad\text{(C) }5,005\qquad\text{(D) }6,006\qquad\text{(E) }7,007$

1983 Federal Competition For Advanced Students, P2, 5
Tags: number theory unsolved , number theory
Given positive integers $ a,b,$ find all positive integers $ x,y$ satisfying the equation: $ x^{a\plus{}b}\plus{}y\equal{}x^a y^b$.

2015 Romania Team Selection Test, 2
Tags: circles , triangle inequality , inradius , inequalities , geometry
Let $ABC$ be a triangle, and let $r$ denote its inradius. Let $R_A$ denote the radius of the circle internally tangent at $A$ to the circle $ABC$ and tangent to the line $BC$; the radii $R_B$ and $R_C$ are defined similarly. Show that $\frac{1}{R_A} + \frac{1}{R_B} + \frac{1}{R_C}\leq\frac{2}{r}$.

2003 Croatia National Olympiad, Problem 4
Tags: geometry , 3D geometry
Given $8$ unit cubes, $24$ of their faces are painted in blue and the remaining $24$ faces in red. Show that it is always possible to assemble these cubes into a cube of edge $2$ on whose surface there are equally many blue and red unit squares.

2019 Azerbaijan Junior NMO, 3
Tags: number theory , difference of squares
A positive number $a$ is given, such that $a$ could be expressed as difference of two inverses of perfect squares ($a=\frac1{n^2}-\frac1{m^2}$). Is it possible for $2a$ to be expressed as difference of two perfect squares?

1979 Polish MO Finals, 3
Tags: probability , combinatorics
An experiment consists of performing $n$ independent tests. The $i$-th test is successful with the probability equal to $p_i$. Let $r_k$ be the probability that exactly $k$ tests succeed. Prove that $$\sum_{i=1}^n p_i =\sum_{k=0}^n kr_k.$$

1985 IMO Longlists, 2
Tags: geometry , rectangle , perimeter , LaTeX , geometry proposed
We are given a triangle $ABC$ and three rectangles $R_1,R_2,R_3$ with sides parallel to two fixed perpendicular directions and such that their union covers the sides $AB,BC$, and $CA$; i.e., each point on the perimeter of $ABC$ is contained in or on at least one of the rectangles. Prove that all points inside the triangle are also covered by the union of $R_1,R_2,R_3.$

Novosibirsk Oral Geo Oly VII, 2020.6
Tags: geometry , angle bisector , angles
Angle bisectors $AA', BB'$and $CC'$ are drawn in triangle $ABC$ with angle $\angle B= 120^o$. Find $\angle A'B'C'$.

2010 LMT, 19
Tags:
Two integers are called [i]relatively prime[/i] if they share no common factors other than $1.$ Determine the sum of all positive integers less than $162$ that are relatively prime to $162.$

2011 Iran MO (3rd Round), 4
Tags: combinatorics proposed , combinatorics
We say the point $i$ in the permutation $\sigma$ [b]ongoing[/b] if for every $j<i$ we have $\sigma (j)<\sigma (i)$. [b]a)[/b] prove that the number of permutations of the set $\{1,....,n\}$ with exactly $r$ ongoing points is $s(n,r)$. [b]b)[/b] prove that the number of $n$-letter words with letters $\{a_1,....,a_k\},a_1<.....<a_k$. with exactly $r$ ongoing points is $\sum_{m}\dbinom{k}{m} S(n,m) s(m,r)$.

2009 Stars Of Mathematics, 5
Tags: combinatorics , Coloring , Color problem , linear algebra , matrix
The cells of a $(n^2-n+1)\times(n^2-n+1)$ matrix are coloured using $n$ colours. A colour is called [i]dominant[/i] on a row (or a column) if there are at least $n$ cells of this colour on that row (or column). A cell is called [i]extremal[/i] if its colour is [i]dominant [/i] both on its row, and its column. Find all $n \ge 2$ for which there is a colouring with no [i]extremal [/i] cells. Iurie Boreico (Moldova)

2021 239 Open Mathematical Olympiad, 5
Tags: geometry , triangle inequality , geometric inequality
The median $AD$ is drawn in triangle $ABC$. Point $E$ is selected on segment $AC$, and on the ray $DE$ there is a point $F$, and $\angle ABC = \angle AED$ and $AF // BC$. Prove that from segments $BD, DF$ and $AF$, you can make a triangle, the area of ​​which is not less half the area of ​​triangle $ABC$.

1998 National Olympiad First Round, 7
Tags: geometry , geometric transformation
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 Belarusian National Olympiad, 3
Tags: combinatorics
Given a $2n \times 2m$ table $(m,n \in \mathbb{N})$ with one of two signs ”+” or ”-” in each of its cells. A union of all the cells of some row and some column is called a cross. The cell on the intersectin of this row and this column is called the center of the cross. The following procedure we call a transformation of the table: we mark all cells which contain ”−” and then, in turn, we replace the signs in all cells of the crosses which centers are marked by the opposite signs. (It is easy to see that the order of the choice of the crosses doesn’t matter.) We call a table attainable if it can be obtained from some table applying such transformations one time. Find the number of all attainable tables.

2022 Malaysia IMONST 2, 2
Tags: combinatorics
The following list shows every number for which more than half of its digits are digits $2$, in increasing order: $$2, 22, 122, 202, 212, 220, 221, 222, 223, 224, \dots$$ If the $n$th term in the list is $2022$, what is $n$?