Olympiad problems

2018 Junior Balkan Team Selection Tests - Moldova, 2

Tags: algebra

Let $x$,$y$ be positive real numbers such that $\frac{1}{1+x+x^2}+\frac{1}{1+y+y^2}+\frac{1}{1+x+y}=1$.Prove that $xy=1.$

2014 AMC 10, 17

Tags: modular arithmetic

What is the greatest power of 2 that is a factor of $10^{1002}-4^{501}$? $ \textbf{(A) }2^{1002}\qquad\textbf{(B) }2^{1003}\qquad\textbf{(C) }2^{1004}\qquad\textbf{(D) }2^{1005} \qquad\textbf{(E) }2^{1006} \qquad $

2020 USMCA, 17

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An \textit{island} is a contiguous set of at least two equal digits. Let $b(n)$ be the number of islands in the binary representation of $n$. For example, $2020_{10} = 11111100100_2$, so $b(2020) = 3$. Compute \[b(1) + b(2) + \cdots + b(2^{2020}).\]

2009 Princeton University Math Competition, 1

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The sequence of positive real numbers $(x_n)$ is defined recursively as follows: $x_1 = 1$, and for $n \geq 2$, \[ x_{n} = \begin{cases} nx_{n-1} & \quad \text{when }nx_{n-1}\leq 1, \\ \frac{x_{n-1}}{n} &\quad \text{otherwise}. \end{cases} \] Show that there is an integer $N > 1$ such that $|x_N - 1| < 0.00001$. Thus the elements of the sequence can get very close to 1 for large $N$; however, it is easy to see that they can never be 1 unless $N = 1$.

2002 Spain Mathematical Olympiad, Problem 1

Tags: polynomial , algebra , number theory

Find all the polynomials $P(t)$ of one variable that fullfill the following for all real numbers $x$ and $y$: $P(x^2-y^2) = P(x+y)P(x-y)$.

2013 National Olympiad First Round, 4

Tags: number theory , prime number , combinatorics

The numbers $1,2,\dots, 49$ are written on unit squares of a $7\times 7$ chessboard such that consequtive numbers are on unit squares sharing a common edge. At most how many prime numbers can a row have? $ \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 3 $

1984 AIME Problems, 10

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Mary told John her score on the American High School Mathematics Examination (AHSME), which was over 80. From this, John was able to determine the number of problems Mary solved correctly. If Mary's score had been any lower, but still over 80, John could not have determined this. What was Mary's score? (Recall that the AHSME consists of 30 multiple-choice problems and that one's score, $s$, is computed by the formula $s = 30 + 4c - w$, where $c$ is the number of correct and $w$ is the number of wrong answers; students are not penalized for problems left unanswered.)

2006 South East Mathematical Olympiad, 1

Tags: trigonometry , symmetry , geometry

[size=130]In $\triangle ABC$, $\angle A=60^\circ$. $\odot I$ is the incircle of $\triangle ABC$. $\odot I$ is tangent to sides $AB$, $AC$ at $D$, $E$, respectively. Line $DE$ intersects line $BI$ and $CI$ at $F$, $G$ respectively. Prove that [/size]$FG=\frac{BC}{2}$.

1985 Tournament Of Towns, (104) 1

Tags: convex , diagonal , sum , perimeter , geometric inequality

We are given a convex quadrilateral and point $M$ inside it . The perimeter of the quadrilateral has length $L$ while the lengths of the diagonals are $D_1$ and $D_2$. Prove that the sum of the distances from $M$ to the vertices of the quadrilateral are not greater than $L + D_1 + D_2$ . (V. Prasolov)

2011 Baltic Way, 9

Tags: geometry , rectangle , combinatorics

Given a rectangular grid, split into $m\times n$ squares, a colouring of the squares in two colours (black and white) is called valid if it satisfies the following conditions: [list] [*]All squares touching the border of the grid are coloured black. [*]No four squares forming a $2\times 2$ square are coloured in the same colour. [*]No four squares forming a $2\times 2$ square are coloured in such a way that only diagonally touching squares have the same colour.[/list] Which grid sizes $m\times n$ (with $m,n\ge 3$) have a valid colouring?

2024 Serbia JBMO TST, 1

Tags: number theory

Find all non-negative integers $x, y$ and primes $p$ such that $$3^x+p^2=7 \cdot 2^y.$$

2009 Czech-Polish-Slovak Match, 3

Tags: geometry

Let $\omega$ denote the excircle tangent to side $BC$ of triangle $ABC$. A line $\ell$ parallel to $BC$ meets sides $AB$ and $AC$ at points $D$ and $E$, respectively. Let $\omega'$ denote the incircle of triangle $ADE$. The tangent from $D$ to $\omega$ (different from line $AB$) and the tangent from $E$ to $\omega$ (different from line $AC$) meet at point $P$. The tangent from $B$ to $\omega'$ (different from line $AB$) and the tangent from $C$ to $\omega'$ (different from line $AC$) meet at point $Q$. Prove that, independent of the choice of $\ell$, there is a fixed point that line $PQ$ always passes through.

1980 Spain Mathematical Olympiad, 2

Tags: combinatorics , probability

A ballot box contains the votes for the election of two candidates $A$ and $B$. It is known that candidate $A$ has $6$ votes and candidate $B$ has $9$. Find the probability that, when carrying out the scrutiny, candidate $B$ always goes first.

2015 Caucasus Mathematical Olympiad, 5

Tags: number theory , perfect square , sides of a triangle

Are there natural $a, b >1000$ , such that for any $c$ that is a perfect square, the three numbers $a, b$ and $c$ are not the lengths of the sides of a triangle?

2006 Miklós Schweitzer, 8

Tags: real analysis

let $f(x) = \sum_{n=0}^{\infty} 2^{-n} ||2^n x||$ , where ||x|| is the distance between x and the closest integer to x. Are the level sets $\{ x \in [0,1] : f(x)=y \}$ Lebesgue measurable for almost all $y \in f(R)$?

2016 Cono Sur Olympiad, 4

Tags: number theory

Let $S(n)$ be the sum of the digits of the positive integer $n$. Find all $n$ such that $S(n)(S(n)-1)=n-1$.

1997 Singapore Senior Math Olympiad, 2

Tags: semicircle , geometry , angle chasing , angle

Figure shows a semicircle with diameter $AD$. The chords $AC$ and $BD$ meet at $P$. $Q$ is the foot of the perpendicular from $P$ to $AD$. find $\angle BCQ$ in terms of $\theta$ and $\phi$ . [img]https://cdn.artofproblemsolving.com/attachments/a/2/2781050e842b2dd01b72d246187f4ed434ff69.png[/img]

2023 USA TSTST, 1

Tags: geometry

Let $ABC$ be a triangle with centroid $G$. Points $R$ and $S$ are chosen on rays $GB$ and $GC$, respectively, such that \[ \angle ABS=\angle ACR=180^\circ-\angle BGC.\] Prove that $\angle RAS+\angle BAC=\angle BGC$. [i]Merlijn Staps[/i]

2008 ITest, 91

Tags: geometry , 3d geometry

Find the sum of all positive integers $n$ such that \[x^3+y^3+z^3=nx^2y^2z^2\] is satisfied by at least one ordered triplet of positive integers $(x,y,z)$.

2007 Stanford Mathematics Tournament, 3

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A clock currently shows the time $10:10$. The obtuse angle between the hands measures $x$ degrees. What is the next time that the angle between the hands will be $x$ degrees? Round your answer to the nearest minute.

1986 Miklós Schweitzer, 4

Tags: complex number

Determine all real numbers $x$ for which the following statement is true: the field $\mathbb C$ of complex numbers contains a proper subfield $F$ such that adjoining $x$ to $F$ we get $\mathbb C$. [M. Laczkovich]

2010 Iran MO (2nd Round), 6

Tags: combinatorics , double counting

A school has $n$ students and some super classes are provided for them. Each student can participate in any number of classes that he/she wants. Every class has at least two students participating in it. We know that if two different classes have at least two common students, then the number of the students in the first of these two classes is different from the number of the students in the second one. Prove that the number of classes is not greater that $\left(n-1\right)^2$.

1987 All Soviet Union Mathematical Olympiad, 450

Tags: geometry , pentagon , equal angles

Given a convex pentagon $ABCDE$ with $\angle ABC= \angle ADE$ and $\angle AEC= \angle ADB$ . Prove that $\angle BAC = \angle DAE$ .

2014 Contests, 2

Tags: combinatorics

There are cities in country, and some cities are connected by roads. Not more than $100$ roads go from every city. Set of roads is called as ideal if all roads in set have not common ends, and we can not add one more road in set without breaking this rule. Every day minister destroy one ideal set of roads. Prove, that he need not more than $199$ days to destroy all roads in country.

1992 IMTS, 2

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In how many ways can 1992 be expressed as the sum of one or more consecutive integers?