Olympiad problems

2008 Balkan MO Shortlist, N5

Tags:

Let $(a_n)$ be a sequence with $a_1=0$ and $a_{n+1}=2+a_n$ for odd $n$ and $a_{n+1}=2a_n$ for even $n$. Prove that for each prime $p>3$, the number \begin{align*} b=\frac{2^{2p}-1}{3} \mid a_n \end{align*} for infinitely many values of $n$

2006 Polish MO Finals, 1

Tags: algebra

Solve in reals: \begin{eqnarray*}a^2=b^3+c^3 \\ b^2=c^3+d^3 \\ c^2=d^3+e^3 \\ d^2=e^3+a^3 \\ e^2=a^3+b^3 \end{eqnarray*}

1992 Poland - First Round, 10

Tags: geometry , 3d geometry

Let $C$ be a cube and let $f: C \longrightarrow C$ be a surjection with $|PQ| \geq |f(P)f(Q)|$ for all $P,Q \in C$. Prove that $f$ is an isometry.

2019 Estonia Team Selection Test, 3

Tags: algebra , functional equation , functional

Find all functions $f : R \to R$ which for all $x, y \in R$ satisfy $f(x^2)f(y^2) + |x|f(-xy^2) = 3|y|f(x^2y)$.

2004 Gheorghe Vranceanu, 3

Tags: function , equation , inversability , algebra

Consider the function $ f:(-\infty,1]\longrightarrow\mathbb{R} $ defined as $$ f(x)=\left\{ \begin{matrix} \frac{5}{2} +2^x-\frac{1}{2^x} ,& \quad x<-1 \\ 3^{\sqrt{1-x^2}} ,& \quad x\in [-1,1] \end{matrix} \right. . $$ [b]a)[/b] For a fixed parameter, find the roots of $ f-m. $ [b]b)[/b] Study the inversability of the restrictions of $ f $ to $ (-\infty,-1] $ and $ [-1,1] $ and find the inverses of these that admit them. [i]D. Zaharia[/i]

2010 Math Prize For Girls Problems, 8

Tags: number theory , prime factorization

When Meena turned 16 years old, her parents gave her a cake with $n$ candles, where $n$ has exactly 16 different positive integer divisors. What is the smallest possible value of $n$?

Swiss NMO - geometry, 2015.4

Tags: geometry , construction , circles

Given a circle $k$ and two points $A$ and $B$ outside the circle. Specify how to can construct a circle with a compass and ruler, so that $A$ and $B$ lie on that circle and that circle is tangent to $k$.

2007 ITest, 20

Tags: modular arithmetic

Find the largest integer $n$ such that $2007^{1024}-1$ is divisible by $2^n$. $\textbf{(A) }1\hspace{14em}\textbf{(B) }2\hspace{14em}\textbf{(C) }3$ $\textbf{(D) }4\hspace{14em}\textbf{(E) }5\hspace{14em}\textbf{(F) }6$ $\textbf{(G) }7\hspace{14em}\textbf{(H) }8\hspace{14em}\textbf{(I) }9$ $\textbf{(J) }10\hspace{13.7em}\textbf{(K) }11\hspace{13.5em}\textbf{(L) }12$ $\textbf{(M) }13\hspace{13.3em}\textbf{(N) }14\hspace{13.4em}\textbf{(O) }15$ $\textbf{(P) }16\hspace{13.6em}\textbf{(Q) }55\hspace{13.4em}\textbf{(R) }63$ $\textbf{(S) }64\hspace{13.7em}\textbf{(T) }2007$

2020 Romania EGMO TST, P2

Tags: function , algebra

Suppose a function $f:\mathbb{R}\to\mathbb{R}$ satisfies $|f(x+y)|\geqslant|f(x)+f(y)|$ for all real numbers $x$ and $y$. Prove that equality always holds. Is the conclusion valid if the sign of the inequality is reversed?

1958 AMC 12/AHSME, 16

Tags: geometry

The area of a circle inscribed in a regular hexagon is $ 100\pi$. The area of hexagon is: $ \textbf{(A)}\ 600\qquad \textbf{(B)}\ 300\qquad \textbf{(C)}\ 200\sqrt{2}\qquad \textbf{(D)}\ 200\sqrt{3}\qquad \textbf{(E)}\ 120\sqrt{5}$

1975 AMC 12/AHSME, 11

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Let $ P$ be an interior point of circle $ K$ other than the center of $ K$. Form all chords of $ K$ which pass through $ P$, and determine their midpoints. The locus of these midpoints is $ \textbf{(A)}\ \text{a circle with one point deleted} \qquad$ $ \textbf{(B)}\ \text{a circle if the distance from } P \text{ to the center of } K \text{ is less than}$ $ \text{one half the radius of } K \text{; otherwise a circular arc of less than}$ $ 360^{\circ}\qquad$ $ \textbf{(C)}\ \text{a semicircle with one point deleted} \qquad$ $ \textbf{(D)}\ \text{a semicircle} \qquad$ $ \textbf{(E)}\ \text{a circle}$

2000 Moldova National Olympiad, Problem 8

Tags: geometry , 3d geometry

A rectangular parallelepiped has dimensions $a,b,c$ that satisfy the relation $3a+4b+10c=500$, and the length of the main diagonal $20\sqrt5$. Find the volume and the total area of the surface of the parallelepiped.

2018 Harvard-MIT Mathematics Tournament, 9

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Evan has a simple graph with $v$ vertices and $e$ edges. Show that he can delete at least $\frac{e-v+1}{2}$ edges so that each vertex still has at least half of its original degree.

2002 IMO Shortlist, 2

Tags: combinatorics , tiling , dissection

For $n$ an odd positive integer, the unit squares of an $n\times n$ chessboard are coloured alternately black and white, with the four corners coloured black. A it tromino is an $L$-shape formed by three connected unit squares. For which values of $n$ is it possible to cover all the black squares with non-overlapping trominos? When it is possible, what is the minimum number of trominos needed?

2010 Germany Team Selection Test, 3

Tags: algebra , polynomial , number theory , permutation

Let $P(x)$ be a non-constant polynomial with integer coefficients. Prove that there is no function $T$ from the set of integers into the set of integers such that the number of integers $x$ with $T^n(x)=x$ is equal to $P(n)$ for every $n\geq 1$, where $T^n$ denotes the $n$-fold application of $T$. [i]Proposed by Jozsef Pelikan, Hungary[/i]

2015 ASDAN Math Tournament, 3

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For a math tournament, each person is assigned an ID which consists of two uppercase letters followed by two digits. All IDs have the property that either the letters are the same, the digits are the same, or both the letters are the same and the digits are the same. Compute the number of possible IDs that the tournament can generate.

2014 Brazil Team Selection Test, 2

Tags: number theory , prime divisor , polynomial

Prove that there exist infinitely many positive integers $n$ such that the largest prime divisor of $n^4 + n^2 + 1$ is equal to the largest prime divisor of $(n+1)^4 + (n+1)^2 +1$.

1996 Iran MO (3rd Round), 2

Tags: geometry

Consider a semicircle of center $O$ and diameter $AB$. A line intersects $AB$ at $M$ and the semicircle at $C$ and $D$ s.t. $MC>MD$ and $MB<MA$. The circumcircles od the $AOC$ and $BOD$ intersect again at $K$. Prove that $MK\perp KO$.

2023 CMIMC Team, 14

Tags: team

Let $ABC$ be points such that $AB=7, BC=5, AC=10$, and $M$ be the midpoint of $AC$. Let $\omega$, $\omega_1$ be the circumcircles of $ABC$ and $BMC$. $\Omega$, $\Omega_1$ are circles through $A$ and $M$ such that $\Omega$ is tangent to $\omega_1$ and $\Omega_1$ is tangent to the line through the centers of $\omega_1$ and $\Omega$. $D, E$ be the intersection of $\Omega$ with $\omega$ and $\Omega_1$ with $\omega_1$. If $F$ is the intersection of the circumcircle of $DME$ with $BM$, find $FB$. [i]Proposed by David Tang[/i]

2006 Denmark MO - Mohr Contest, 4

Tags: combinatorics

Of the numbers $1, 2,3,..,2006$, ten different ones must be selected. Show that you can pick ten different numbers with a sum greater than $10039$ in more ways than you can select ten different numbers with a sum less than $10030$.

1998 Balkan MO, 1

Tags: floor function , number theory

Consider the finite sequence $\left\lfloor \frac{k^2}{1998} \right\rfloor$, for $k=1,2,\ldots, 1997$. How many distinct terms are there in this sequence? [i]Greece[/i]

2020 CCA Math Bonanza, I5

Tags: quadratic

Let $f(x)=x^2-kx+(k-1)^2$ for some constant $k$. What is the largest possible real value of $k$ such that $f$ has at least one real root? [i]2020 CCA Math Bonanza Individual Round #5[/i]

2017 Iran Team Selection Test, 2

Tags: geometry

Let $P$ be a point in the interior of quadrilateral $ABCD$ such that: $$\angle BPC=2\angle BAC \ \ ,\ \ \angle PCA = \angle PAD \ \ ,\ \ \angle PDA=\angle PAC$$ Prove that: $$\angle PBD= \left | \angle BCA - \angle PCA \right |$$ [i]Proposed by Ali Zamani[/i]

1991 Arnold's Trivium, 13

Tags: integration

Calculate with $5\%$ relative error \[\int_1^{10}x^xdx\]

2010 Malaysia National Olympiad, 4

Tags: geometry , angle

In the diagram, $\angle AOB = \angle BOC$ and$\angle COD = \angle DOE = \angle EOF$. Given that $\angle AOD = 82^o$ and $\angle BOE = 68^o$. Find $\angle AOF$. [img]https://cdn.artofproblemsolving.com/attachments/b/2/deba6cd740adbf033ad884fff8e13cd21d9c5a.png[/img]