Olympiad problems

2023 Brazil National Olympiad, 5

Let $m$ be a positive integer with $m \leq 2024$. Ana and Banana play a game alternately on a $1\times2024$ board, with squares initially painted white. Ana starts the game. Each move by Ana consists of choosing any $k \leq m$ white squares on the board and painting them all green. Each Banana play consists of choosing any sequence of consecutive green squares and painting them all white. What is the smallest value of $m$ for which Ana can guarantee that, after one of her moves, the entire board will be painted green?

2020 Kazakhstan National Olympiad, 3

Tags: order of an element , number theory , prime number

Let $p$ be a prime number and $k,r$ are positive integers such that $p>r$. If $pk+r$ divides $p^p+1$ then prove that $r$ divides $k$.

2023 MMATHS, 1

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Cat and Claire are having a conversation about Cat’s favorite number. Cat says, “My favorite number is a two-digit multiple of $7$.” Claire asks, “If you just told me the tens digit of the number, would I know your number?” Cat says, “No. However, without knowing that, if I told you the tens digit of $100$ minus my number, you could determine my favorite number.” Claire says, “Now I know your favorite number!" What is Cat’s favorite number?

1999 USAMO, 6

Tags: geometry , trapezoid , geometric transformation , reflection , incenter , symmetry , isosceles triangle

Let $ABCD$ be an isosceles trapezoid with $AB \parallel CD$. The inscribed circle $\omega$ of triangle $BCD$ meets $CD$ at $E$. Let $F$ be a point on the (internal) angle bisector of $\angle DAC$ such that $EF \perp CD$. Let the circumscribed circle of triangle $ACF$ meet line $CD$ at $C$ and $G$. Prove that the triangle $AFG$ is isosceles.

2022 Indonesia Regional, 4

Tags: geometry , circumcircle , geometric transformation , reflection

Suppose $ABC$ is a triangle with circumcenter $O$. Point $D$ is the reflection of $A$ with respect to $BC$. Suppose $\ell$ is the line which is parallel to $BC$ and passes through $O$. The line through $B$ and parallel to $CD$ meets $\ell$ at $B_1$. Lines $CB_1$ and $BD$ intersect at point $B_2$. The line through $C$ parallel to $BD$ and $\ell$ meet at $C_1$. Finally, $BC_1$ and $CD$ intersects at point $C_2$. Prove that points $A, B_2, C_2, D$ lie on a circle.

2023 Silk Road, 4

Tags: algebra , polynomial , multivariable

Let $\mathcal{M}=\mathbb{Q}[x,y,z]$ be the set of three-variable polynomials with rational coefficients. Prove that for any non-zero polynomial $P\in \mathcal{M}$ there exists non-zero polynomials $Q,R\in \mathcal{M}$ such that \[ R(x^2y,y^2z,z^2x) = P(x,y,z)Q(x,y,z). \]

1916 Eotvos Mathematical Competition, 3

Tags: number theory , combinatorics

Divide the numbers $$1, 2,3, 4,5$$ into two arbitrarily chosen sets. Prove that one of the sets contains two numbers and their difference.

2017 ASDAN Math Tournament, 3

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Alex and Zev are two members of a group of $2017$ friends who all know each other. Alex is trying to send a package to Zev. The delivery process goes as follows: Alex sends the package randomly to one of the people in the group. If this person is Zev, the delivery is done. Otherwise, the person who received the package also randomly sends it to someone in the group who hasn't held the package before and this process repeats until Zev gets the package. What is the expected number of deliveries made?

2008 Tournament Of Towns, 5

Tags: table , combinatorics

We may permute the rows and the columns of the table below. How may different tables can we generate? 1 2 3 4 5 6 7 7 1 2 3 4 5 6 6 7 1 2 3 4 5 5 6 7 1 2 3 4 4 5 6 7 1 2 3 3 4 5 6 7 1 2 2 3 4 5 6 7 1

1998 Romania National Olympiad, 4

Tags: geometry , 3d geometry , tetrahedron

Let $ABCD$ be an arbitrary tetrahedron. The bisectors of the angles $\angle BDC$, $\angle CDA$ and $\angle ADB$ intersect $BC$, $CA$ and $AB$, in the points $M$, $N$, $P$, respectively. a) Show that the planes $(ADM)$, $(BDN)$ and $(CDP)$ have a common line $d$. b) Let the points $A' \in (AD)$, $B' \in (BD)$ and $C' \in (CD)$ be such that $(AA') = (BB') = (CC')$ ; show that if $G$ and $G'$ are the centroids of $ABC$ and $A'B'C'$ then the lines $GG'$ and $d$ are either parallel or identical.

2016 Auckland Mathematical Olympiad, 1

Tags: combinatorics

How many $3 \times 5$ rectangular pieces of cardboard can be cut from a $17 \times 22$ rectangular piece of cardboard, when the amount of waste is minimised?

2021 Centroamerican and Caribbean Math Olympiad, 2

Tags: circumcircle , cyclic quadrilateral , geometry

Let $ABC$ be a triangle and let $\Gamma$ be its circumcircle. Let $D$ be a point on $AB$ such that $CD$ is parallel to the line tangent to $\Gamma$ at $A$. Let $E$ be the intersection of $CD$ with $\Gamma$ distinct from $C$, and $F$ the intersection of $BC$ with the circumcircle of $\bigtriangleup ADC$ distinct from $C$. Finally, let $G$ be the intersection of the line $AB$ and the internal bisector of $\angle DCF$. Show that $E,\ G,\ F$ and $C$ lie on the same circle.

2012 Uzbekistan National Olympiad, 2

Tags: modular arithmetic , calculus , integration , number theory

For any positive integers $n$ and $m$ satisfying the equation $n^3+(n+1)^3+(n+2)^3=m^3$, prove that $4\mid n+1$.

2012 Baltic Way, 20

Tags: number theory

Find all integer solutions of the equation $2x^6 + y^7 = 11$.

1962 AMC 12/AHSME, 11

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The difference between the larger root and the smaller root of $ x^2 \minus{} px \plus{} (p^2 \minus{} 1)/4 \equal{} 0$ is: $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ p \qquad \textbf{(E)}\ p\plus{}1$

Taiwan TST 2015 Round 1, 1

Tags: number theory , equation

Determine all pairs $(x, y)$ of positive integers such that \[\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.\] [i]Proposed by Titu Andreescu, USA[/i]

2006 Czech and Slovak Olympiad III A, 1

Tags: number theory

Define a sequence of positive integers $\{a_n\}$ through the recursive formula: $a_{n+1}=a_n+b_n(n\ge 1)$,where $b_n$ is obtained by rearranging the digits of $a_n$ (in decimal representation) in reverse order (for example,if $a_1=250$,then $b_1=52,a_2=302$,and so on). Can $a_7$ be a prime?

1992 Poland - First Round, 1

Tags: number theory , diophantine equation

Solve the following equation in real numbers: $\frac{(x^2-1)(|x|+1)}{x+sgnx}=[x+1].$

2016 NIMO Problems, 3

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How many positive integers divide at least two of the numbers $120$, $144$, and $180$? [i]Proposed by Evan Chen[/i]

1964 Czech and Slovak Olympiad III A, 1

Tags: number theory , divisible , last digit

Show that the number $11^{100}-1$ is both divisible by $6000$ and its last four decimal digits are $6000$.

2006 May Olympiad, 4

Tags: geometry , trapezoid , area

Let $ABCD$ be a trapezoid of bases $AB$ and $CD$ . Let $O$ be the intersection point of the diagonals $AC$ and $BD$. If the area of the triangle $ABC$ is $150$ and the area of the triangle $ACD$ is $120$, calculate the area of the triangle $BCO$.

1969 IMO Longlists, 15

Tags: inequalities , factorial , algebra , combinatorics , combinatorial inequality

$(CZS 4)$ Let $K_1,\cdots , K_n$ be nonnegative integers. Prove that $K_1!K_2!\cdots K_n! \ge \left[\frac{K}{n}\right]!^n$, where $K = K_1 + \cdots + K_n$

2019 Jozsef Wildt International Math Competition, W. 25

Tags: gamma function , summation , inequalities , function

Let $x_i$, $y_i$, $z_i$, $w_i \in \mathbb{R}^+, i = 1, 2,\cdots n$, such that$$\sum \limits_{i=1}^nx_i=nx,\ \sum \limits_{i=1}^ny_i=ny,\ \sum \limits_{i=1}^nw_i=nw $$ $$\Gamma \left(z_i\right)\geq \Gamma \left(w_i\right),\ \sum \limits_{i=1}^n\Gamma \left(z_i\right)=n\Gamma^* (z)$$Then$$\sum \limits_{i=1}^n \frac{\left(\Gamma \left(x_i\right)+\Gamma \left(y_i\right)\right)^2}{\Gamma \left(z_i\right)-\Gamma \left(w_i\right)}\geq n\frac{\left(\Gamma \left(x\right)+\Gamma \left(y\right)\right)^2}{\Gamma^* \left(z\right)-\Gamma \left(w\right)}$$

2022 Czech-Austrian-Polish-Slovak Match, 2

Tags: algebra , functional equation

Find all functions $f: \mathbb{R^{+}} \rightarrow \mathbb {R^{+}}$ such that $f(f(x)+\frac{y+1}{f(y)})=\frac{1}{f(y)}+x+1$ for all $x, y>0$. [i]Proposed by Dominik Burek, Poland[/i]

India EGMO 2022 TST, 3

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Let $I$ be incentre of scalene $\triangle ABC$ and let $L$ be midpoint of arc $BAC$. Let $M$ be midpoint of $BC$ and let the line through $M$ parallel to $AI$ intersect $LI$ at point $P$. Let $Q$ lie on $BC$ such that $PQ\perp LI$. Let $S$ be midpoint of $AM$ and $T$ be midpoint of $LI$. Prove that $IS\perp BC$ if and only if $AQ\perp ST$. [i]~Mahavir Gandhi[/i]