Olympiad problems

2019 Canadian Mathematical Olympiad Qualification, 3

Let $f(x) = x^3 + 3x^2 - 1$ have roots $a,b,c$. (a) Find the value of $a^3 + b^3 + c^3$ (b) Find all possible values of $a^2b + b^2c + c^2a$

2020 Novosibirsk Oral Olympiad in Geometry, 2

Tags: geometry , perimeter , Chessboard , combinatorics

Vitya cut the chessboard along the borders of the cells into pieces of the same perimeter. It turned out that not all of the received parts are equal. What is the largest possible number of parts that Vitya could get?

2009 Hungary-Israel Binational, 2

Tags: algebra unsolved , algebra

Denote the three real roots of the cubic $ x^3 \minus{} 3x \minus{} 1 \equal{} 0$ by $ x_1$, $ x_2$, $ x_3$ in order of increasing magnitude. (You may assume that the equation in fact has three distinct real roots.) Prove that $ x_3^2 \minus{} x_2^2 \equal{} x_3 \minus{} x_1$.

1975 Poland - Second Round, 4

Tags: algebra , inequalities

Prove that the non-negative numbers $ a_1, a_2, \ldots, a_n $ ($ n = 1, 2, \ldots $) satisfy the inequality $ x_1, x_2, \ldots, x_n $ for any real numbers $$ \left( \sum_{i=1}^n a_i x_i^2 \right)^2 \leq \sum_{i=1}^n a_i x_i^4.$$ it is necessary and sufficient that $ \sum_{i=1}^n a_i \leq 1 $.

2007 Princeton University Math Competition, 1

Tags: geometry

A pirate ship spots, $10$ nautical miles to the east, an oblivious caravel sailing $60$ south of west at a steady $12 \text{ nm/hour}$. What is the minimum speed that the pirate ship must maintain at to be able to catch the caravel?

2014 Online Math Open Problems, 1

Tags: Online Math Open , geometry , rectangle

Carl has a rectangle whose side lengths are positive integers. This rectangle has the property that when he increases the width by 1 unit and decreases the length by 1 unit, the area increases by $x$ square units. What is the smallest possible positive value of $x$? [i]Proposed by Ray Li[/i]

2023 Austrian MO Beginners' Competition, 2

Tags: geometry , collinear , hexagon

Let $ABCDEF$ be a regular hexagon with sidelength s. The points $P$ and $Q$ are on the diagonals $BD$ and $DF$, respectively, such that $BP = DQ = s$. Prove that the three points $C$, $P$ and $Q$ are on a line. [i](Walther Janous)[/i]

1996 Irish Math Olympiad, 1

Tags: number theory proposed , number theory

The Fibonacci sequence is defined by $ F_0\equal{}0, F_1\equal{}1$ and $ F_{n\plus{}2}\equal{}F_n\plus{}F_{n\plus{}1}$ for $ n \ge 0$. Prove that: $ (a)$ The statement $ "F_{n\plus{}k}\minus{}F_n$ is divisible by $ 10$ for all $ n \in \mathbb{N}"$ is true if $ k\equal{}60$ but false for any positive integer $ k<60$. $ (b)$ The statement $ "F_{n\plus{}t}\minus{}F_n$ is divisible by $ 100$ for all $ n \in \mathbb{N}"$ is true if $ t\equal{}300$ but false for any positive integer $ t<300$.

2017 Peru IMO TST, 6

Tags: combinatorics , IMO Shortlist , Coloring , Ramsey Theory

Let $n$ be a positive integer relatively prime to $6$. We paint the vertices of a regular $n$-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours.

1987 All Soviet Union Mathematical Olympiad, 444

Tags: square table , game , game strategy , combinatorics

The "Sea battle" game. a) You are trying to find the $4$-field ship -- a rectangle $1x4$, situated on the $7x7$ playing board. You are allowed to ask a question, whether it occupies the particular field or not. How many questions is it necessary to ask to find that ship surely? b) The same question, but the ship is a connected (i.e. its fields have common sides) set of $4$ fields.

2013 Saudi Arabia IMO TST, 1

Tags: geometry , concurrency , collinear , desargue , Pascal , incircle , tangent

Triangle $ABC$ is inscribed in circle $\omega$. Point $P$ lies inside triangle $ABC$.Lines $AP,BP$ and $CP$ intersect $\omega$ again at points $A_1$, $B_1$ and $C_1$ (other than $A, B, C$), respectively. The tangent lines to $\omega$ at $A_1$ and $B_1$ intersect at $C_2$.The tangent lines to $\omega$ at $B_1$ and $C_1$ intersect at $A_2$. The tangent lines to $\omega$ at $C_1$ and $A_1$ intersect at $B_2$. Prove that the lines $AA_2,BB_2$ and $CC_2$ are concurrent.

2009 Vietnam Team Selection Test, 1

Tags: inequalities , calculus , quadratics , function , logarithms , inequalities proposed

Let $ a,b,c$ be positive numbers.Find $ k$ such that: $ (k \plus{} \frac {a}{b \plus{} c})(k \plus{} \frac {b}{c \plus{} a})(k \plus{} \frac {c}{a \plus{} b}) \ge (k \plus{} \frac {1}{2})^3$

2003 Estonia Team Selection Test, 1

Tags: combinatorics

Two treasure-hunters found a treasure containing coins of value $a_1< a_2 < ... < a_{2003}$ (the quantity of coins of each value is unlimited). The first treasure-hunter forms all the possible sets of different coins containing odd number of elements, and takes the most valuable coin of each such set. The second treasure-hunter forms all the possible sets of different coins containing even number of elements, and takes the most valuable coin of each such set. Which one of them is going to have more money and how much more? (H. Nestra)

2012 AMC 12/AHSME, 23

Tags: algebra , polynomial , inequalities , triangle inequality , AMC

Consider all polynomials of a complex variable, $P(z)=4z^4+az^3+bz^2+cz+d$, where $a, b, c$ and $d$ are integers, $0 \le d \le c \le b \le a \le 4$, and the polynomial has a zero $z_0$ with $|z_0|=1$. What is the sum of all values $P(1)$ over all the polynomials with these properties? $ \textbf{(A)}\ 84\qquad\textbf{(B)}\ 92\qquad\textbf{(C)}\ 100\qquad\textbf{(D)}\ 108 \qquad\textbf{(E)}\ 120 $

2023 Romanian Master of Mathematics, 5

Tags: Polynomials , RMM 2023 , number theory , algebra

Let $P,Q,R,S$ be non constant polynomials with real coefficients, such that $P(Q(x))=R(S(x)) $ and the degree of $P$ is multiple of the degree of $R. $ Prove that there exists a polynomial $T$ with real coefficients such that $$\displaystyle P(x)=R(T(x))$$

1970 Miklós Schweitzer, 4

Tags: modular arithmetic , quadratics , function , absolute value , number theory proposed , number theory

If $ c$ is a positive integer and $ p$ is an odd prime, what is the smallest residue (in absolute value) of \[ \sum_{n=0}^{\frac{p-1}{2}} \binom{2n}{n}c^n \;(\textrm{mod}\;p\ ) \ ?\] J. Suranyi

2020 Thailand TST, 3

Tags: ceiling function , number theory , IMO Shortlist , IMO Shortlist 2019 , Perfect Square

Let $a$ and $b$ be two positive integers. Prove that the integer \[a^2+\left\lceil\frac{4a^2}b\right\rceil\] is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.) [i]Russia[/i]

2024 AMC 10, 2

Tags: AMC , AMC 12 , AMC 10 , 2024 AMC 10b , 2024 AMC 12B , factorial , 2024 AMC

What is $10! - 7! \cdot 6!$? $ \textbf{(A) }-120 \qquad \textbf{(B) }0 \qquad \textbf{(C) }120 \qquad \textbf{(D) }600 \qquad \textbf{(E) }720 \qquad $

2022 JBMO Shortlist, N4

Tags: number theory , JBMO Shortlist

Consider the sequence $u_0, u_1, u_2, ...$ defined by $u_0 = 0, u_1 = 1,$ and $u_n = 6u_{n - 1} + 7u_{n - 2}$ for $n \ge 2$. Show that there are no non-negative integers $a, b, c, n$ such that $$ab(a + b)(a^2 + ab + b^2) = c^{2022} + 42 = u_n.$$

2014 Iran Team Selection Test, 2

Tags: geometry , incenter , circumcircle , geometric transformation

Point $D$ is an arbitary point on side $BC$ of triangle $ABC$. $I$,$I_1$ and$I_2$ are the incenters of triangles $ABC$,$ABD$ and $ACD$ respectively. $M\not=A$ and $N\not=A$ are the intersections of circumcircle of triangle $ABC$ and circumcircles of triangles $IAI_1$ and $IAI_2$ respectively. Prove that regardless of point $D$, line $MN$ goes through a fixed point.

2025 All-Russian Olympiad Regional Round, 11.7

Tags: combinatorics

There are several bears living on the $2025$ islands of the Arctic Ocean. Every bear sometimes swims from one island to another. It turned out that every bear made at least one swim in a year, but no two bears made equal swams. At the same time, exactly one swim was made between each two islands $A$ and $B$: either from $A$ to $B$ or from $B$ to $A$. Prove that there were no bears on some island at the beginning and at the end of the year. [i]A. Kuznetsov[/i]

2013 Tournament of Towns, 4

Tags: Sum , perfect cube , number theory , Integer

Is it true that every integer is a sum of finite number of cubes of distinct integers?

2019 Ramnicean Hope, 2

Tags: geometry , circumcircle , Medians , vectorial geometry

Let $ P,Q,R $ be the intersections of the medians $ AD,BE,CF $ of a triangle $ ABC $ with its circumcircle, respectively. Show that $ ABC $ is equilateral if $ \overrightarrow{DP} +\overrightarrow{EQ} +\overrightarrow{FR} =0. $ [i]Dragoș Lăzărescu[/i]

2014 Miklós Schweitzer, 5

Tags: superior algebra , superior algebra unsolved

Let $ \alpha $ be a non-real algebraic integer of degree two, and let $ \mathbb{P} $ be the set of irreducible elements of the ring $ \mathbb{Z}[ \alpha] $. Prove that \[ \sum_{p\in \mathbb{P}}^{{}}\frac{1}{|p|^{2}}=\infty \]

2009 Today's Calculation Of Integral, 481

Tags: calculus , integration , trigonometry , calculus computations

For real numbers $ a,\ b$ such that $ |a|\neq |b|$, let $ I_n \equal{} \int \frac {1}{(a \plus{} b\cos \theta)^n}\ (n\geq 2)$. Prove that : $ \boxed{\boxed{I_n \equal{} \frac {a}{a^2 \minus{} b^2}\cdot \frac {2n \minus{} 3}{n \minus{} 1}I_{n \minus{} 1} \minus{} \frac {1}{a^2 \minus{} b^2}\cdot\frac {n \minus{} 2}{n \minus{} 1}I_{n \minus{} 2} \minus{} \frac {b}{a^2 \minus{} b^2}\cdot\frac {1}{n \minus{} 1}\cdot \frac {\sin \theta}{(a \plus{} b\cos \theta)^{n \minus{} 1}}}}$