Olympiad problems

2024/2025 TOURNAMENT OF TOWNS, P5

Tags: algebra

Given $15$ coins of the same appearance. It is known that one of them weighs $1$g, two coins weigh $2$g each, three coins weigh $3$g each, four coins weigh $4$g each, and five coins weigh $5$g each. There are inscriptions on the coins, indicating their weight. It is allowed to perform two weighings on a balance without additional weights. Find a way to check that there are no wrong inscriptions. (It is not required to check which inscriptions are wrong and which ones are correct.) (8 marks)

2020 Purple Comet Problems, 9

Tags: algebra

Let $a, b$, and $c$ be real numbers such that $3^a = 125$, $5^b = 49$,and $7^c = 8$1. Find the product $abc$.

2023 HMNT, 1

Tags: algebra

Tyler has an infinite geometric series with sum $10$. He increases the first term of his sequence by $4$ and swiftly changes the subsequent terms so that the common ratio remains the same, creating a new geometric series with sum $15$. Compute the common ratio of Tyler’s series.

2020 Junior Balkan Team Selection Tests - Moldova, 2

Tags: algebra , inequality , 3-variable inequality

The positive real numbers $a, b, c$ satisfy the equation $a+b+c=1$. Prove the identity: $\sqrt{\frac{(a+bc)(b+ca)}{c+ab}}+\sqrt{\frac{(b+ca)(c+ab)}{a+bc}}+\sqrt{\frac{(c+ab)(a+bc)}{b+ca}} = 2$

1994 China National Olympiad, 5

Tags: polynomial , algebra , integer part

For arbitrary natural number $n$, prove that $\sum^n_{k=0}C^k_n2^kC^{[(n-k)/2]}_{n-k}=C^n_{2n+1}$, where $C^0_0=1$ and $[\dfrac{n-k}{2}]$ denotes the integer part of $\dfrac{n-k}{2}$.

2008 International Zhautykov Olympiad, 2

Tags: polynomial , algebra , factorization , sum of cubes

A polynomial $ P(x)$ with integer coefficients is called good,if it can be represented as a sum of cubes of several polynomials (in variable $ x$) with integer coefficients.For example,the polynomials $ x^3 \minus{} 1$ and $ 9x^3 \minus{} 3x^2 \plus{} 3x \plus{} 7 \equal{} (x \minus{} 1)^3 \plus{} (2x)^3 \plus{} 2^3$ are good. a)Is the polynomial $ P(x) \equal{} 3x \plus{} 3x^7$ good? b)Is the polynomial $ P(x) \equal{} 3x \plus{} 3x^7 \plus{} 3x^{2008}$ good? Justify your answers.

III Soros Olympiad 1996 - 97 (Russia), 9.5

Tags: algebra , geometric progression

For what largest $n$ are there $n$ seven-digit numbers that are successive members of one geometric progression?

1991 Romania Team Selection Test, 7

Tags: algebra , inequalities , sum

Let $x_1,x_2,...,x_{2n}$ be positive real numbers with the sum $1$. Prove that $$x_1^2x_2^2...x_n^2+x_2^2x_3^2...x_{n+1}^2+...+x_{2n}^2x_1^2...x_{n-1}^2 <\frac{1}{n^{2n}}$$

2012 ELMO Shortlist, 10

Tags: algebra , projective geometry , geometry , circumcircle , conic

Let $A_1A_2A_3A_4A_5A_6A_7A_8$ be a cyclic octagon. Let $B_i$ by the intersection of $A_iA_{i+1}$ and $A_{i+3}A_{i+4}$. (Take $A_9 = A_1$, $A_{10} = A_2$, etc.) Prove that $B_1, B_2, \ldots , B_8$ lie on a conic. [i]David Yang.[/i]

2018 Singapore Senior Math Olympiad, 4

Tags: inequalities , algebra

Let $a,b,c,d$ be positive integers such that $a+c=20$ and $\frac{a}{b}+\frac{c}{d}<1$. Find the maximum possible value of $\frac{a}{b}+\frac{c}{d}$.

2015 ISI Entrance Examination, 2

Tags: algebra , geometry , coordinates , parabola , conic

Let $y = x^2 + ax + b$ be a parabola that cuts the coordinate axes at three distinct points. Show that the circle passing through these three points also passes through $(0,1)$.

2023 New Zealand MO, 5

Tags: system of equations , algebra

Let $x, y$ and $z$ be real numbers such that: $x^2 = y + 2$, and $y^2 = z + 2$, and $z^2 = x + 2$. Prove that $x + y + z$ is an integer.

2023 Princeton University Math Competition, B1

Tags: algebra

Consider the equations $x^2+y^2=16$ and $xy=\tfrac{9}{2}.$ Find the sum, over all ordered pairs $(x,y)$ satisfying these equations, of $|x+y|.$

2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, 2

Tags: linear algebra , matrix , algebra , polynomial

Justify your answer whether $A=\left( \begin{array}{ccc} -4 & -1& -1 \\ 1 & -2& 1 \\ 0 & 0& -3 \end{array} \right)$ is similar to $B=\left( \begin{array}{ccc} -2 & 1& 0 \\ -1 & -4& 1 \\ 0 & 0& -3 \end{array} \right),\ A,\ B\in{M(\mathbb{C})}$ or not.

1966 Swedish Mathematical Competition, 4

Tags: algebra , composition , sequence

Let $f(x) = 1 + \frac{2}{x}$. Put $f_1(x) = f(x)$, $f_2(x) = f(f_1(x))$, $f_3(x) = f(f_2(x))$, $... $. Find the solutions to $x = f_n(x)$ for $n > 0$.

2015 Belarus Team Selection Test, 3

Tags: algebra , polynomial , functional inequality

Consider all polynomials $P(x)$ with real coefficients that have the following property: for any two real numbers $x$ and $y$ one has \[|y^2-P(x)|\le 2|x|\quad\text{if and only if}\quad |x^2-P(y)|\le 2|y|.\] Determine all possible values of $P(0)$. [i]Proposed by Belgium[/i]

1989 IMO Longlists, 4

Tags: algebra , rectangle , area , equation , calculus

Ali Barber, the carpet merchant, has a rectangular piece of carpet whose dimensions are unknown. Unfortunately, his tape measure is broken and he has no other measuring instruments. However, he finds that if he lays it flat on the floor of either of his storerooms, then each corner of the carpet touches a different wall of that room. He knows that the sides of the carpet are integral numbers of feet and that his two storerooms have the same (unknown) length, but widths of 38 feet and 50 feet respectively. What are the carpet dimensions?

1992 Romania Team Selection Test, 4

Tags: inequalities , algebra , sum

Let $x_1,x_2,...,x_n$ be real numbers with $1 \ge x_1 \ge x_2\ge ... \ge x_n \ge 0$ and $x_1^2 +x_2^2+...+x_n^2= 1$. If $[x_1 +x_2 +...+x_n] = m$, prove that $x_1 +x_2 +...+x_m \ge 1$.

2022 Bolivia IMO TST, P1

Tags: algebra , gauss identity

Find all possible values of $\frac{1}{x}+\frac{1}{y}$, if $x,y$ are real numbers not equal to $0$ that satisfy $$x^3+y^3+3x^2y^2=x^3y^3$$

2023 Indonesia Regional, 3

Tags: algebra , combinatorics , maximization

Find the maximum value of an integer $B$ such that for every 9 distinct natural number with the sum of $2023$, there must exist a sum of 4 of the number that is greater than or equal to $B$

2002 HKIMO Preliminary Selection Contest, 8

Tags: algebra

Given that $0.3010<\log 2<0.3011$ and $0.4771<\log 3<0.4772$. Find the leftmost digit of $12^{37}$

1984 Vietnam National Olympiad, 2

Tags: polynomial , algebra

Given two real numbers $a, b$ with $a \neq 0$, find all polynomials $P(x)$ which satisfy \[xP(x - a) = (x - b)P(x).\]

2020 HK IMO Preliminary Selection Contest, 7

Tags: algebra

Solve the equation $\sqrt{7-x}=7-x^2$, where $x>0$.

2020-IMOC, A1

Tags: functional equation , algebra , inequalities , am-gm

$\definecolor{A}{RGB}{190,0,60}\color{A}\fbox{A1.}$ Find all $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $$\definecolor{A}{RGB}{80,0,200}\color{A} x^4+y^4+z^4\ge f(xy)+f(yz)+f(zx)\ge xyz(x+y+z)$$holds for all $a,b,c\in\mathbb{R}$. [i]Proposed by [/i][b][color=#FFFF00]usjl[/color][/b]. [color=#B6D7A8]#1733[/color]

2018 Turkey MO (2nd Round), 1

Tags: algebra , inequalities , system of equations

Find all pairs $(x,y)$ of real numbers that satisfy, \begin{align*} x^2+y^2+x+y &= xy(x+y)-\frac{10}{27}\\ |xy| & \leq \frac{25}{9}. \end{align*}