Olympiad problems

2022 SAFEST Olympiad, 3

Tags: game , combinatorics

A hunter and an invisible rabbit play a game on an infinite square grid. First the hunter fixes a colouring of the cells with finitely many colours. The rabbit then secretly chooses a cell to start in. Every minute, the rabbit reports the colour of its current cell to the hunter, and then secretly moves to an adjacent cell that it has not visited before (two cells are adjacent if they share an edge). The hunter wins if after some finite time either:[list][*]the rabbit cannot move; or [*]the hunter can determine the cell in which the rabbit started.[/list]Decide whether there exists a winning strategy for the hunter. [i]Proposed by Aron Thomas[/i]

1989 India National Olympiad, 1

Tags: polynomial , calculus , integration , quadratic , modular arithmetic , algebra

Prove that the Polynomial $ f(x) \equal{} x^{4} \plus{} 26x^{3} \plus{} 56x^{2} \plus{} 78x \plus{} 1989$ can't be expressed as a product $ f(x) \equal{} p(x)q(x)$ , where $ p(x)$ and $ q(x)$ are both polynomial with integral coefficients and with degree at least $ 1$.

2018 Swedish Mathematical Competition, 1

Tags: angle bisector , parallel , cyclic quadrilateral , geometry

Let the $ABCD$ be a quadrilateral without parallel sides, inscribed in a circle. Let $P$ and $Q$ be the intersection points between the lines containing the quadrilateral opposite sides. Show that the bisectors to the angles at $P$ and $Q$ are parallel to the bisectors of the angles at the intersection point of the diagonals of the quadrilateral.

1996 Iran MO (3rd Round), 4

Tags: function , algebra

Determine all functions $f : \mathbb N_0 \rightarrow \mathbb N_0 - \{1\}$ such that \[f(n + 1) + f(n + 3) = f(n + 5)f(n + 7) - 1375, \qquad \forall n \in \mathbb N.\]

2023 Czech and Slovak Olympiad III A., 3

Tags: geometry

In acute triangle $ABC$ let $H$ be its orthocenter and $I$ be its incenter. Let $D$ be the projection of point $I$ onto the line $BC$ and $E$ be the reflection of point $A$ in point $I$. Further, let $F$ be the projection of point $H$ onto the line $ED$. Prove that points $B, H, F$ and $C$ lie on circle.

1999 Mongolian Mathematical Olympiad, Problem 3

Tags: geometry

Three squares $ABB_1B_2,BCC_1C_2,CAA_1A_2$ are constructed in the exterior of a triangle $ABC$. In the exterior of these squares, another three squares $A_1B_2B_3B_4,B_1C_2C_3C_4,C_1A_2A_3A_4$ are constructed. Prove that the area of a triangle with sides $C_3A_4,A_3B_4,B_3C_4$ is $16$ times the area of $\triangle ABC$.

2020 CCA Math Bonanza, L5.1

Tags:

Professor Shian Bray is buying CCA Math Bananas$^{\text{TM}}$. He starts with $\$500$. The first CCA Math Bananas$^{\text{TM}}$ he buys costs $\$1$. Each time after he buys a CCA Math Banana$^{\text{TM}}$, the cost of a CCA Math Bananas$^{\text{TM}}$ doubles with probability $\frac{1}{2}$ (otherwise staying the same). Professor Bray buys CCA Math Bananas$^{\text{TM}}$ until he cannot afford any more, ending with $n$ CCA Math Bananas$^{\text{TM}}$. Estimate the expected value of $n$. An estimate of $E$ earns $2^{1-0.25|E-A|}$ points, where $A$ is the actual answer. [i]2020 CCA Math Bonanza Lightning Round #5.1[/i]

2021 MOAA, 6

Tags: team

Find the sum of all two-digit prime numbers whose digits are also both prime numbers. [i]Proposed by Nathan Xiong[/i]

2007 Hanoi Open Mathematics Competitions, 15

Tags: quadratic , polynomial , algebra

Let $p = \overline{abcd}$ be a $4$-digit prime number. Prove that the equation $ax^3+bx^2+cx+d=0$ has no rational roots.

2018 CCA Math Bonanza, T8

Tags: 3d geometry , prism , geometry

A rectangular prism with positive integer side lengths formed by stacking unit cubes is called [i]bipartisan[/i] if the same number of unit cubes can be seen on the surface as those which cannot be seen on the surface. How many non-congruent bipartisan rectangular prisms are there? [i]2018 CCA Math Bonanza Team Round #8[/i]

2010 Iran MO (3rd Round), 2

Tags: abstract algebra , calculus , integration , algebra , function , domain , number theory

$R$ is a ring such that $xy=yx$ for every $x,y\in R$ and if $ab=0$ then $a=0$ or $b=0$. if for every Ideal $I\subset R$ there exist $x_1,x_2,..,x_n$ in $R$ ($n$ is not constant) such that $I=(x_1,x_2,...,x_n)$, prove that every element in $R$ that is not $0$ and it's not a unit, is the product of finite irreducible elements.($\frac{100}{6}$ points)

2011 China Team Selection Test, 1

Tags: circumcircle , geometric transformation , reflection , perpendicular bisector , geometry

Let $AA',BB',CC'$ be three diameters of the circumcircle of an acute triangle $ABC$. Let $P$ be an arbitrary point in the interior of $\triangle ABC$, and let $D,E,F$ be the orthogonal projection of $P$ on $BC,CA,AB$, respectively. Let $X$ be the point such that $D$ is the midpoint of $A'X$, let $Y$ be the point such that $E$ is the midpoint of $B'Y$, and similarly let $Z$ be the point such that $F$ is the midpoint of $C'Z$. Prove that triangle $XYZ$ is similar to triangle $ABC$.

2014 MMATHS, 3

Tags: algebra , functional equation

Let $f : R^+ \to R^+$ be a function satisfying $$f(\sqrt{x_1x_2}) =\sqrt{f(x_1)f(x_2)}$$ for all positive real numbers $x_1, x_2$. Show that $$f( \sqrt[n]{x_1x_2... x_n}) = \sqrt[n]{f(x_1)f(x_2) ... f(x_n)}$$ for all positive integers $n$ and positive real numbers $x_1, x_2,..., x_n$.

1908 Eotvos Mathematical Competition, 2

Tags: inequalities , geometric inequality , number theory , geometry

Let $n$ be an integer greater than $2$. Prove that the $n$th power of the length of the hypotenuse of a right triangle is greater than the sum of the $n$th powers of the lengths of the legs.

2024 AMC 8 -, 10

Tags:

In January 1980 the Moana Loa Observation recorded carbon dioxide levels of 338 ppm (parts per million). Over the years the average carbon dioxide reading has increased by about 1.515 ppm each year. What is the expected carbon dioxide level in ppm in January 2030? Round your answer to the nearest integer. $\textbf{(A) } 399\qquad\textbf{(B) } 414\qquad\textbf{(C) } 420\qquad\textbf{(D) } 444\qquad\textbf{(E) } 459$

2002 SNSB Admission, 3

Tags: topology , algebra , polynomial , function , quadratic

Classify up to homeomorphism the topological spaces of the support of functions that are real quadratic polynoms of three variables and and irreducible over the set of real numbers.

2012 Princeton University Math Competition, B4

Tags: combinatorics

For a set $S$ of integers, define $\max (S)$ to be the maximal element of $S$. How many non-empty subsets $S \subseteq \{1, 2, 3, ... , 10\}$ satisfy $\max (S) \le |S| + 2$?

2019 Belarus Team Selection Test, 2.1

Tags: quadratic , algebra , polynomial

Given a quadratic trinomial $p(x)$ with integer coefficients such that $p(x)$ is not divisible by $3$ for all integers $x$. Prove that there exist polynomials $f(x)$ and $h(x)$ with integer coefficients such that $$ p(x)\cdot f(x)+3h(x)=x^6+x^4+x^2+1. $$ [i](I. Gorodnin)[/i]

2007 AMC 8, 24

Tags: probability

A bag contains four pieces of paper, each labeled with one of the digits $1$, $2$, $3$ or $4$, with no repeats. Three of these pieces are drawn, one at a time without replacement, to construct a three-digit number. What is the probability that the three-digit number is a multiple of $3$? $\textbf{(A)}\ \frac{1}{4} \qquad \textbf{(B)}\ \frac{1}{3} \qquad \textbf{(C)}\ \frac{1}{2} \qquad \textbf{(D)}\ \frac{2}{3} \qquad \textbf{(E)}\ \frac{3}{4}$

Russian TST 2014, P2

Tags: polygon , homothety , geometry

The polygon $M{}$ is bicentric. The polygon $P{}$ has vertices at the points of contact of the sides of $M{}$ with the inscribed circle. The polygon $Q{}$ is formed by the external bisectors of the angles of $M{}.$ Prove that $P{}$ and $Q{}$ are homothetic.

2016 Romania Team Selection Tests, 3

Tags: number theory , algebra

Prove that: [b](a)[/b] If $(a_n)_{n\geq 1}$ is a strictly increasing sequence of positive integers such that $\frac{a_{2n-1}+a_{2n}}{a_n}$ is a constant as $n$ runs through all positive integers, then this constant is an integer greater than or equal to $4$; and [b](b)[/b] Given an integer $N\geq 4$, there exists a strictly increasing sequene $(a_n)_{n\geq 1}$ of positive integers such that $\frac{a_{2n-1}+a_{2n}}{a_n}=N$ for all indices $n$.

2020 ASDAN Math Tournament, 2

Tags: team test

Sam's cup has a $400$ mL mixture of coffee and milk tea. He pours $200$ mL into Ben's empty cup. Ben then adds 100mL of coee to his cup and stirs well. Finally, Ben pours $200$ mL out of his cup back into Sam's cup. If the mixture in Sam's cup is now $50\%$ milk tea, then how many milliliters of milk tea were in it originally?

2019 India IMO Training Camp, P2

Tags: rectangle , geometry

Determine all positive integers $m$ satisfying the condition that there exists a unique positive integer $n$ such that there exists a rectangle which can be decomposed into $n$ congruent squares and can also be decomposed into $m+n$ congruent squares.

1961 IMO Shortlist, 3

Tags: trigonometry , algebra , equation , trigonometric equation

Solve the equation $\cos^n{x}-\sin^n{x}=1$ where $n$ is a natural number.

2011 Postal Coaching, 6

Tags: polynomial , algebra

Prove that there exist integers $a, b, c$ all greater than $2011$ such that \[(a+\sqrt{b})^c=\ldots 2010 \cdot 2011\ldots\] [Decimal point separates an integer ending in $2010$ and a decimal part beginning with $2011$.]