Olympiad problems

2011 Switzerland - Final Round, 7

For a given rational number $r$, find all integers $z$ such that \[2^z + 2 = r^2\mbox{.}\] [i](Swiss Mathematical Olympiad 2011, Final round, problem 7)[/i]

1999 Brazil Team Selection Test, Problem 2

Tags: algebra , polynomial

If $a,b,c,d$ are Distinct Real no. such that $a = \sqrt{4+\sqrt{5+a}}$ $b = \sqrt{4-\sqrt{5+b}}$ $c = \sqrt{4+\sqrt{5-c}}$ $d = \sqrt{4-\sqrt{5-d}}$ Then $abcd = $

2000 Vietnam National Olympiad, 1

Tags: trigonometry , polynomial , algebra

For every integer $ n \ge 3$ and any given angle $ \alpha$ with $ 0 < \alpha < \pi$, let $ P_n(x) \equal{} x^n \sin\alpha \minus{} x \sin n\alpha \plus{} \sin(n \minus{} 1)\alpha$. (a) Prove that there is a unique polynomial of the form $ f(x) \equal{} x^2 \plus{} ax \plus{} b$ which divides $ P_n(x)$ for every $ n \ge 3$. (b) Prove that there is no polynomial $ g(x) \equal{} x \plus{} c$ which divides $ P_n(x)$ for every $ n \ge 3$.

1996 Turkey Junior National Olympiad, 3

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Let $P$ be a point inside of equilateral $\triangle ABC$ such that $m(\widehat{APB})=150^\circ$, $|AP|=2\sqrt 3$, and $|BP|=2$. Find $|PC|$.

2023 MOAA, 9

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Let $ABCD$ be a trapezoid with $AB \parallel CD$ and $BC \perp CD$. There exists a point $P$ on $BC$ such that $\triangle{PAD}$ is equilateral. If $PB = 20$ and $PC = 23$, the area of $ABCD$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ where $b$ is square-free and $a$ and $c$ are relatively prime. Find $a+b+c$. [i]Proposed by Andy Xu[/i]

2017 Vietnam Team Selection Test, 3

Tags: combinatorics

For each integer $n>0$, a permutation $a_1,a_2,\dots ,a_{2n}$ of $1,2,\dots 2n$ is called [i]beautiful[/i] if for every $1\leq i<j \leq 2n$, $a_i+a_{n+i}=2n+1$ and $a_i-a_{i+1}\not \equiv a_j-a_{j+1}$ (mod $2n+1$) (suppose that $a_{2n+1}=a_1$). a. For $n=6$, point out a [i]beautiful [/i] permutation. b. Prove that there exists a [i]beautiful[/i] permutation for every $n$.

2005 MOP Homework, 6

Tags: algebra , quadratic , system of equations

Solve the system of equations: $x^2=\frac{1}{y}+\frac{1}{z}$, $y^2=\frac{1}{z}+\frac{1}{x}$, $z^2=\frac{1}{x}+\frac{1}{y}$. in the real numbers.

1995 Belarus Team Selection Test, 3

Tags: sequence , number theory , function , functional equation

Show that there is no infinite sequence an of natural numbers such that \[a_{a_n}=a_{n+1}a_{n-1}-a_{n}^2\] for all $n\geq 2$

2008 AMC 12/AHSME, 17

Tags: modular arithmetic

Let $ a_1,a_2,\dots$ be a sequence of integers determined by the rule $ a_n\equal{}a_{n\minus{}1}/2$ if $ a_{n\minus{}1}$ is even and $ a_n\equal{}3a_{n\minus{}1}\plus{}1$ if $ a_{n\minus{}1}$ is odd. For how many positive integers $ a_1 \le 2008$ is it true that $ a_1$ is less than each of $ a_2$, $ a_3$, and $ a_4$? $ \textbf{(A)}\ 250 \qquad \textbf{(B)}\ 251 \qquad \textbf{(C)}\ 501 \qquad \textbf{(D)}\ 502 \qquad \textbf{(E)}\ 1004$

1996 IMC, 7

Tags: function , real analysis , convergence

Prove that if $f:[0,1]\rightarrow[0,1]$ is a continuous function, then the sequence of iterates $x_{n+1}=f(x_{n})$ converges if and only if $$\lim_{n\to \infty}(x_{n+1}-x_{n})=0$$

2017 IMO Shortlist, C6

Tags: combinatorics

Let $n > 1$ be a given integer. An $n \times n \times n$ cube is composed of $n^3$ unit cubes. Each unit cube is painted with one colour. For each $n \times n \times 1$ box consisting of $n^2$ unit cubes (in any of the three possible orientations), we consider the set of colours present in that box (each colour is listed only once). This way, we get $3n$ sets of colours, split into three groups according to the orientation. It happens that for every set in any group, the same set appears in both of the other groups. Determine, in terms of $n$, the maximal possible number of colours that are present.

2024 Cono Sur Olympiad, 4

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Let $N$ be a positive integer with $2k$ digits. Its chunks are defined by the two numbers formed by the digits from $1$ to $k$ and $k+1$ to $2k$ (e.g. the chunks of 142856 are 142 and 856). We define the $N$-[i]reverse[/i] as the number formed by switching its chunks (e.g. the reverse of 142856 is 856142 and for 1401 it is 114). We call a number [i]cearense[/i] is it satisfies the following conditions: [list=i] [*] Has an even number of digits [*] Its chunks are relatively prime [*]Divides its reverse [/list] Find the two smallest cearense integer.

2019 CCA Math Bonanza, I9

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Isosceles triangle $\triangle{ABC}$ has $\angle{BAC}=\angle{ABC}=30^\circ$ and $AC=BC=2$. If the midpoints of $BC$ and $AC$ are $M$ and $N$, respectively, and the circumcircle of $\triangle{CMN}$ meets $AB$ at $D$ and $E$ with $D$ closer to $A$ than $E$ is, what is the area of $MNDE$? [i]2019 CCA Math Bonanza Individual Round #9[/i]

1996 Moldova Team Selection Test, 7

Tags: geometry , 3d geometry

Let $ABCDA_1B_1C_1D_1$ be a cube. On the sides $AB{}$ and $AD{}$ there are the points $M{}$ and $N{}$, respectively, such that $AM+AN=AB$. Show that the measure of the dihedral angle between the planes $(MA_1C)$ and $(NA_1C)$ doe not depend on the positions of $M{}$ and $N{}$. Find this measure.

2010 Belarus Team Selection Test, 2.1

Tags: geometry , product , ratio , angle

Point $D$ is marked inside a triangle $ABC$ so that $\angle ADC = \angle ABC + 60^o$, $\angle CDB =\angle CAB + 60^o$, $\angle BDA = \angle BCA + 60^o$. Prove that $AB \cdot CD = BC \cdot AD = CA \cdot BD$. (A. Levin)

2020 CIIM, 3

Tags: combinatorics

Let $(m,r,s,t)$ be positive integers such that $m\geq s+1$ and $r\geq t$. Consider $m$ sets $A_1, A_2, \dots, A_m$ with $r$ elements each one. Suppose that, for each $1\leq i\leq m$, there exist at least $t$ elements of $A_i$, such that each one(element) belongs to (at least) $s$ sets $A_j$ where $j\neq i$. Determine the greatest quantity of elements in the following set $A_1 \cup A_2 \cup A_3 \dots \cup A_m$.

1985 Brazil National Olympiad, 5

Tags: algebra , equation

$A, B$ are reals. Find a necessary and sufficient condition for $Ax + B[x] = Ay + B[y]$ to have no solutions except $x = y$.

2010 China Team Selection Test, 2

Tags: induction , number theory , greatest common divisor , algebra

Given integer $a_1\geq 2$. For integer $n\geq 2$, define $a_n$ to be the smallest positive integer which is not coprime to $a_{n-1}$ and not equal to $a_1,a_2,\cdots, a_{n-1}$. Prove that every positive integer except 1 appears in this sequence $\{a_n\}$.

2023 Indonesia Regional, 4

Tags: irrational number , algebra

Find all irrational real numbers $\alpha$ such that \[ \alpha^3 - 15 \alpha \text{ and } \alpha^4 - 56 \alpha \] are both rational numbers.

2006 Taiwan TST Round 1, 2

Tags: circumcircle , geometry

$ABCD$ is a cyclic quadrilateral. Lines $AB,CD$ intersect at $E$, lines $AD,BC$ intersect at $F$, and $EM$ and $FN$ are tangents to the circumcircle of $ABCD$. Two circles are constructed with $E,F$ their centers and $EM, FN$ their radii, respectively. $K$ is one of their intersections. Prove that $EK$ is perpendicular to $FK$.

2014 Indonesia MO Shortlist, A2

Tags: number theory

A sequence of positive integers $a_1, a_2, \ldots$ satisfies $a_k + a_l = a_m + a_n$ for all positive integers $k,l,m,n$ satisfying $kl = mn$. Prove that if $p$ divides $q$ then $a_p \le a_q$.

1997 Junior Balkan MO, 5

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Let $n_1$, $n_2$, $\ldots$, $n_{1998}$ be positive integers such that \[ n_1^2 + n_2^2 + \cdots + n_{1997}^2 = n_{1998}^2. \] Show that at least two of the numbers are even.

2013 Online Math Open Problems, 24

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The real numbers $a_0, a_1, \dots, a_{2013}$ and $b_0, b_1, \dots, b_{2013}$ satisfy $a_{n} = \frac{1}{63} \sqrt{2n+2} + a_{n-1}$ and $b_{n} = \frac{1}{96} \sqrt{2n+2} - b_{n-1}$ for every integer $n = 1, 2, \dots, 2013$. If $a_0 = b_{2013}$ and $b_0 = a_{2013}$, compute \[ \sum_{k=1}^{2013} \left( a_kb_{k-1} - a_{k-1}b_k \right). \][i]Proposed by Evan Chen[/i]

1998 Austrian-Polish Competition, 3

Tags: function , algebra

Find all pairs of real numbers $(x, y)$ satisfying the following system of equations $2-x^{3}=y, 2-y^{3}=x$.

2004 National High School Mathematics League, 8

Tags: algebra , functional equation , function

Function $f:\mathbb{R}\to\mathbb{R}$, satisfies that $f(0)=1$, and $f(xy+1)=f(x)f(y)-f(y)-x+2$, then $f(x)=$________.