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Stage 7 · Algebra

10 problems · National olympiad second round; IMO P1/P4 · mathsolympiadprep.com

Print or save as PDF The answer key prints on its own page at the end.

1. $P(x)=ax^2+bx+c$ has exactly $1$ different real root where $a,b,c$ are real numbers. If $P(P(P(x)))$ has exactly $3$ different real roots, what is the minimum possible value of $abc$?

   1. A-3
   2. B-2
   3. C$2\sqrt 3$
   4. D$3\sqrt 3$
   5. E$\text{None of above}$

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2. Solve the following system of equations for real $x,y$ and $z$:
   $\text{}$
   x &=& $\sqrt{2y+3}\\$
   y &=& $\sqrt{2z+3}\\$
   z &=& $\text{2x+3 .}$

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3. Determine all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ satisfying $$f(f(m)+n)+f(m)=f(n)+f(3 m)+2014$$ for all integers $m$ and $n$. (Netherlands) Answer. There is only one such function, namely $n \longmapsto 2 n+1007$.

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4. Find the smallest real number $C$, such that for any positive integers $x \neq y$ holds the following:

   $$\min(\{\sqrt{x^2 + 2y}\}, \{\sqrt{y^2 + 2x}\})<C$$

   Here $\{x\}$ denotes the fractional part of $x$. For example, $\{3.14\} = 0.14$.

   Proposed by Anton Trygub

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5. A polynomial $p(x)$ with real coefficients is said to be almeriense if it is of the form:

   $$p(x) = x^3+ax^2+bx+a$$

   And its three roots are positive real numbers in arithmetic progression. Find all almeriense polynomials such that $p\left(\frac{7}{4}\right) = 0$

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6. $a, b, c$ are positive real numbers such that $$(\sqrt {ab}-1)(\sqrt {bc}-1)(\sqrt {ca}-1)=1$$
   At most, how many of the numbers: $$a-\frac {b}{c}, a-\frac {c}{b}, b-\frac {a}{c}, b-\frac {c}{a}, c-\frac {a}{b}, c-\frac {b}{a}$$ can be bigger than $1$?

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7. p1. Find all real numbers that satisfy the equation $$(1 + x^2 + x^4 + .... + x^{2014})(x^{2016} + 1) = 2016x^{2015}$$

   p2. Let $A$ be an integer and $A = 2 + 20 + 201 + 2016 + 20162 + ... + \underbrace{20162016...2016}_{40\,\, digits}$
   Find the last seven digits of $A$, in order from millions to units.

   p3. In triangle $ABC$, points $P$ and $Q$ are on sides of $BC$ so that the length of $BP$ is equal to $CQ$, $\angle BAP = \angle CAQ$ and $\angle APB$ is acute. Is triangle $ABC$ isosceles? Write down your reasons.

   p4. Ayu is about to open the suitcase but she forgets the key. The suitcase code consists of nine digits, namely four $0$s (zero) and five $1$s. Ayu remembers that no four consecutive numbers are the same. How many codes might have to try to make sure the suitcase is open?

   p5. Fulan keeps $100$ turkeys with the weight of the $i$-th turkey, being $x_i$ for $i\in\{1, 2, 3, ... , 100\}$. The weight of the $i$-th turkey in grams is assumed to follow the function $x_i(t) = S_it + 200 - i$ where $t$ represents the time in days and $S_i$ is the $i$-th term of an arithmetic sequence where the first term is a positive number $a$ with a difference of $b =\frac15$. It is known that the average data on the weight of the hundred turkeys at $t = a$ is $150.5$ grams. Calculate the median weight of the turkey at time $t = 20$ days.

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8. 165 Find the integer solution of the equation $\underbrace{\sqrt{x+\sqrt{x+\sqrt{x+\cdots+\sqrt{x}}}}}_{1964 \text { terms }}=y$.

   untranslated part:
   将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。

   translated part:
   Find the integer solution of the equation $\underbrace{\sqrt{x+\sqrt{x+\sqrt{x+\cdots+\sqrt{x}}}}}_{1964 \text { terms }}=y$.

   4. 165

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9. Let $m$ and $n$ two given integers. Ana thinks of a pair of real numbers $x$, $y$ and then she tells Beto the values of $x^m+y^m$ and $x^n+y^n$, in this order. Beto's goal is to determine the value of $xy$ using that information. Find all values of $m$ and $n$ for which it is possible for Beto to fulfill his wish, whatever numbers that Ana had chosen.

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10. The function $f(n)$ is defined on the positive integers and takes non-negative integer values. $f(2)=0,f(3)>0,f(9999)=3333$ and for all $m,n:$ $$f(m+n)-f(m)-f(n)=0 \text{ or } 1.$$ Determine $f(1982)$.

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Answer key — Stage 7 · Algebra

Worked solutions for every problem are on the site, one page per problem.

1. $-2$ open
2. $x = y = z = 3$ open
3. $f(n) = 2n + 1007$ open
4. $\varphi - 1$ open
5. $p(x) = x^3 - \frac{21}{4}x^2 + \frac{73}{8}x - \frac{21}{4}$ open
6. $4$ open
7. $200.5$ open
8. $x=0, y=0$ open
9. $(m, n) = (2k+1, 2t(2k+1))$ open
10. $660$ open

Problems belong to the competitions that set them and are reproduced from open datasets under their licences; every problem page names its source. Free to copy for classroom use.