Olympiad problems

2010 AMC 12/AHSME, 7

Logan is constructing a scaled model of his town. The city's water tower stands $ 40$ meters high, and the top portion is a sphere that holds $ 100,000$ liters of water. Logan's miniature water tower holds $ 0.1$ liters. How tall, in meters, should Logan make his tower? $ \textbf{(A)}\ 0.04\qquad \textbf{(B)}\ \frac{0.4}{\pi}\qquad \textbf{(C)}\ 0.4\qquad \textbf{(D)}\ \frac{4}{\pi}\qquad \textbf{(E)}\ 4$

2000 Harvard-MIT Mathematics Tournament, 7

Tags: number theory

$8712$ is an integral multiple of its reversal, $2178$, as $8712=4 * 2178$. Find another $4$-digit number which is a non-trivial integral multiple of its reversal.

2020 Harvard-MIT Mathematics Tournament, 10

Tags:

Max repeatedly throws a fair coin in a hurricane. For each throw, there is a $4\%$ chance that the coin gets blown away. He records the number of heads $H$ and the number of tails $T$ before the coin is lost. (If the coin is blown away on a toss, no result is recorded for that toss.) What is the expected value of $|H-T|$? [i]Proposed by Krit Boonsiriseth.[/i]

2011 N.N. Mihăileanu Individual, 1

Tags: quadratic , polynom , algebra

Let be a quadratic polynom that has the property that the modulus of the sum between the leading and the free coefficient is smaller than the modulus of the middle coefficient. Prove that this polynom admits two distinct real roots, one belonging to the interval $ (-1,1) , $ and the other belonging outside of the interval $ (-1,1). $

1982 Spain Mathematical Olympiad, 8

Tags: analytic geometry , geometry , locus

Given a set $C$ of points in the plane, it is called the distance of a point $P$ from the plane to the set $C$ at the smallest of the distances from $P$ to each of the points of $C$. Let the sets be $C = \{A,B\}$, with $A = (1, 0)$ and $B = (2, 0)$; and $C'= \{A',B'\}$ with $A' = (0, 1)$ and $B' = (0, 7)$, in an orthogonal reference system. Find and draw the set $M$ of points in the plane that are equidistant from $C$ and $C'$ . Study whether the function whose graph is the set $M$ previously obtained is derivable.

1985 Traian Lălescu, 1.2

Tags: algebra , polynomial

Prove that all real roots of the polynomial $$ P=X^{1985}-X^{1984}+1983\cdot X^{1983}+1994\cdot X^{992} -884064 $$ are positive.

1985 Miklós Schweitzer, 4

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[b]4.[/b] Call a subset $S$ of the set $\{1,\dots,n\}$ [i]exceptional[/i] if any pair of distinct elements of $S$ are coprime. Consider an exceptional set with a maximal sum of elements (among all exceptional sets for a fixed $n$). Prove that if $n$ is sufficiently large, then each element of $S$ has at most two distinct prime divisors. ([b]N.17[/b]) [P. Erdos]

1999 May Olympiad, 5

Tags: combinatorial geometry , combinatorics

There are $12$ points that are vertices of a regular polygon with $12$ sides. Rafael must draw segments that have their two ends at two of the points drawn. He is allowed to have each point be an endpoint of more than one segment and for the segments to intersect, but he is prohibited from drawing three segments that are the three sides of a triangle in which each vertex is one of the $12$ starting points. Find the maximum number of segments Rafael can draw and justify why he cannot draw a greater number of segments.

2010 Balkan MO Shortlist, G8

Tags: tangent , concurrency , circles , geometry

Let $c(0, R)$ be a circle with diameter $AB$ and $C$ a point, on it different than $A$ and $B$ such that $\angle AOC > 90^o$. On the radius $OC$ we consider the point $K$ and the circle $(c_1)$ with center $K$ and radius $KC = R_1$. We draw the tangents $AD$ and $AE$ from $A$ to the circle $(c_1)$. Prove that the straight lines $AC, BK$ and $DE$ are concurrent

2001 Poland - Second Round, 1

Tags: modular arithmetic , arithmetic sequence , irrational number , number theory

Find all integers $n\ge 3$ for which the following statement is true: Any arithmetic progression $a_1,\ldots ,a_n$ with $n$ terms for which $a_1+2a_2+\ldots+na_n$ is rational contains at least one rational term.

1969 Vietnam National Olympiad, 2

Tags: trigonometry , algebra , trigonometric equation

Find all real $x$ such that $0 < x < \pi $ and $\frac{8}{3 sin x - sin 3x} + 3 sin^2x \le 5$.

2013 Benelux, 4

Tags: modular arithmetic , quadratic , diophantine equation , number theory

a) Find all positive integers $g$ with the following property: for each odd prime number $p$ there exists a positive integer $n$ such that $p$ divides the two integers \[g^n - n\quad\text{ and }\quad g^{n+1} - (n + 1).\] b) Find all positive integers $g$ with the following property: for each odd prime number $p$ there exists a positive integer $n$ such that $p$ divides the two integers \[g^n - n^2\quad\text{ and }g^{n+1} - (n + 1)^2.\]

2023 Indonesia TST, C

Tags:

Let $n$ be a positive integer. Each cell on an $n \times n$ board will be filled with a positive integer less than or equal to $2n-1$ such that for each index $i$ with $1 \leq i \leq n$, the $2n-1$ cells in the $i^{\text{th}}$ row or $i^{\text{th}}$ collumn contain distinct integers. (a) Is this filling possible for $n=4$? (b) Is this filling possible for $n=5$?

2017 NZMOC Camp Selection Problems, 4

Tags: combinatorics

Ross wants to play solitaire with his deck of $n$ playing cards, but he’s discovered that the deck is “boxed”: some cards are face up, and others are face down. He wants to turn them all face down again, by repeatedly choosing a block of consecutive cards, removing the block from the deck, turning it over, and replacing it back in the deck at the same point. What is the smallest number of such steps Ross needs in order to guarantee that he can turn all the cards face down again, regardless of how they start out?

2018 South East Mathematical Olympiad, 3

Tags: parallelogram , geometry

Let $O$ be the circumcenter of $\triangle ABC$, where $\angle ABC> 90^{\circ}$ and $M$ is the midpoint of $BC$. Point $P$ lies inside $\triangle ABC$ such that $PB\perp PC$. $D,E$ distinct from $P$ lies on the perpendicular to $AP$ through $P$ such that $BD=BP, CE=CP$. If quadrilateral $ADOE$ is a parallelogram, prove that $$\angle OPE = \angle AMB.$$

2007 Pre-Preparation Course Examination, 11

Tags: number theory

Let $p \geq 3$ be a prime and $a_1,a_2,\cdots , a_{p-2}$ be a sequence of positive integers such that for every $k \in \{1,2,\cdots,p-2\}$ neither $a_k$ nor $a_k^k-1$ is divisible by $p$. Prove that product of some of members of this sequence is equivalent to $2$ modulo $p$.

2016 Regional Olympiad of Mexico Southeast, 2

Tags: geometry , trapezoid , circumcircle , cyclic quadrilateral

Let $ABCD$ a trapezium with $AB$ parallel to $CD, \Omega$ the circumcircle of $ABCD$ and $A_1,B_1$ points on segments $AC$ and $BC$ respectively, such that $DA_1B_1C$ is a cyclic cuadrilateral. Let $A_2$ and $B_2$ the symmetric points of $A_1$ and $B_1$ with respect of the midpoint of $AC$ and $BC$, respectively. Prove that points $A, B, A_2, B_2$ are concyclic.

2013 USAJMO, 1

Tags: number theory

Are there integers $a$ and $b$ such that $a^5b+3$ and $ab^5+3$ are both perfect cubes of integers?

2009 Peru MO (ONEM), 4

Tags: domino , combinatorics , tiles , tiling

Let $ n$ be a positive integer. A $4\times n$ rectangular grid is divided in$ 2\times 1$ or $1\times 2$ rectangles (as if it were completely covered with tiles of domino, no overlaps or gaps). Then all the grid points which are vertices of one of the $2\times 1$ or $1\times 2$ rectangles, are painted red. What is the least amount of red points you can get?

1964 IMO Shortlist, 4

Tags: combinatorics , ramsey theory , coloring , graph theory , extremal combinatorics

Seventeen people correspond by mail with one another-each one with all the rest. In their letters only three different topics are discussed. each pair of correspondents deals with only one of these topics. Prove that there are at least three people who write to each other about the same topic.

1995 IMO Shortlist, 5

Tags: function , algebra , functional equation

Let $ \mathbb{R}$ be the set of real numbers. Does there exist a function $ f: \mathbb{R} \mapsto \mathbb{R}$ which simultaneously satisfies the following three conditions? [b](a)[/b] There is a positive number $ M$ such that $ \forall x:$ $ \minus{} M \leq f(x) \leq M.$ [b](b)[/b] The value of $f(1)$ is $1$. [b](c)[/b] If $ x \neq 0,$ then \[ f \left(x \plus{} \frac {1}{x^2} \right) \equal{} f(x) \plus{} \left[ f \left(\frac {1}{x} \right) \right]^2 \]

2009 Brazil Team Selection Test, 1

Tags: combinatorics , permutation , divisibility

Let $n \in \mathbb N$ and $A_n$ set of all permutations $(a_1, \ldots, a_n)$ of the set $\{1, 2, \ldots , n\}$ for which \[k|2(a_1 + \cdots+ a_k), \text{ for all } 1 \leq k \leq n.\] Find the number of elements of the set $A_n$. [i]Proposed by Vidan Govedarica, Serbia[/i]

2004 Vietnam National Olympiad, 1

Tags: trigonometry , limit , algebra

The sequence $ (x_n)^{\infty}_{n\equal{}1}$ is defined by $ x_1 \equal{} 1$ and $ x_{n\plus{}1} \equal{}\frac{(2 \plus{} \cos 2\alpha)x_n \minus{} \cos^2\alpha}{(2 \minus{} 2 \cos 2\alpha)x_n \plus{} 2 \minus{} \cos 2\alpha}$, for all $ n \in\mathbb{N}$, where $ \alpha$ is a given real parameter. Find all values of $ \alpha$ for which the sequence $ (y_n)$ given by $ y_n \equal{} \sum_{k\equal{}1}^{n}\frac{1}{2x_k\plus{}1}$ has a finite limit when $ n \to \plus{}\infty$ and find that limit.

2016 Kosovo Team Selection Test, 2

Tags: number theory

Show that for all positive integers $n\geq 2$ the last digit of the number $2^{2^n}+1$ is $7$ .

2011 Princeton University Math Competition, A7 / B8

Tags: algebra

Let $\alpha_1,\alpha_2,\dots,\alpha_6$ be a fixed labeling of the complex roots of $x^6-1$. Find the number of permutations $\{\alpha_{i_1},\alpha_{i_2},\dots,\alpha_{i_6}\}$ of these roots such that if $P(\alpha_1, \dots, \alpha_6) = 0$, then $P(\alpha_{i_1},\dots,\alpha_{i_6}) = 0$, where $P$ is any polynomial with rational coefficients.