Olympiad problems

2020 Serbian Mathematical Olympiad, Problem 5

For a natural number $n$, with $v_2(n)$ we denote the largest integer $k\geq0$ such that $2^k|n$. Let us assume that the function $f\colon\mathbb{N}\to\mathbb{N}$ meets the conditions: $(i)$ $f(x)\leq3x$ for all natural numbers $x\in\mathbb{N}$. $(ii)$ $v_2(f(x)+f(y))=v_2(x+y)$ for all natural numbers $x,y\in\mathbb{N}$. Prove that for every natural number $a$ there exists exactly one natural number $x$ such that $f(x)=3a$.

2019 Serbia National Math Olympiad, 6

Tags: algebra , sequence

Sequences $(a_n)_{n=0}^{\infty}$ and $(b_n)_{n=0}^{\infty}$ are defined with recurrent relations : $$a_0=0 , \;\;\; a_1=1, \;\;\;\; a_{n+1}=\frac{2018}{n} a_n+ a_{n-1}\;\;\; \text {for }\;\;\; n\geq 1$$ and $$b_0=0 , \;\;\; b_1=1, \;\;\;\; b_{n+1}=\frac{2020}{n} b_n+ b_{n-1}\;\;\; \text {for }\;\;\; n\geq 1$$ Prove that :$$\frac{a_{1010}}{1010}=\frac{b_{1009}}{1009}$$

2023 MMATHS, 8

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Find the number of ordered pairs of integers $(m,n)$ such that $0 \le m,n \le 2023$ and $$m^2 \equiv \sum_{d \mid 2023} n^d \pmod{2024}.$$

2019 Tuymaada Olympiad, 5

Tags: graph theory , graph , combinatorics , combinatorial geometry

Is it possible to draw in the plane the graph presented in the figure so that all the vertices are different points and all the edges are unit segments? (The segments can intersect at points different from vertices.)

2018 Taiwan APMO Preliminary, 2

Tags: number theory

Let $k,x,y$ be postive integers. The quotients of $k$ divided by $x^2, y^2$ are $n,n+148$ respectively.($k$ is divisible by $x^2$ and $y^2$) (a) If $\gcd(x,y)=1$, then find $k$. (b) If $\gcd(x,y)=4$, then find $k$.

2017 Romania Team Selection Test, P2

Tags: irrational number , polynomial , number theory

Determine all intergers $n\geq 2$ such that $a+\sqrt{2}$ and $a^n+\sqrt{2}$ are both rational for some real number $a$ depending on $n$

1967 IMO Shortlist, 6

Tags: linear algebra , matrix , algebra , system of equations

Solve the system of equations: $ \begin{matrix} |x+y| + |1-x| = 6 \\ |x+y+1| + |1-y| = 4. \end{matrix} $

2020 Malaysia IMONST 2, 3

Tags: number theory , quadratic

Find all possible integer values of $n$ such that $12n^2 + 12n + 11$ is a $4$-digit number with equal digits.

Kvant 2023, M2770

Tags: combinatorial geometry

2. A unit square paper has a triangle-shaped hole (vertices of the hole are not on the border of the paper). Prove that a triangle with area of $1 / 6$ can be cut from the remaining paper. Alexandr Yuran

2024 Chile National Olympiad., 2

Tags: combinatorics

On a table, there are many coins and a container with two coins. Vale and Diego play the following game, where Vale starts and then Diego plays, alternating turns. If at the beginning of a turn the container contains $ n $ coins, the player can add a number $ d $ of coins, where $ d $ divides exactly into $ n $ and $ d < n $. The first player to complete at least 2024 coins in the container wins. Prove that there exists a strategy for Vale to win, no matter the decisions made by Diego.

2017 Putnam, B4

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Evaluate the sum \[\sum_{k=0}^{\infty}\left(3\cdot\frac{\ln(4k+2)}{4k+2}-\frac{\ln(4k+3)}{4k+3}-\frac{\ln(4k+4)}{4k+4}-\frac{\ln(4k+5)}{4k+5}\right)\] \[=3\cdot\frac{\ln 2}2-\frac{\ln 3}3-\frac{\ln 4}4-\frac{\ln 5}5+3\cdot\frac{\ln 6}6-\frac{\ln 7}7-\frac{\ln 8}8-\frac{\ln 9}9+3\cdot\frac{\ln 10}{10}-\cdots.\] (As usual, $\ln x$ denotes the natural logarithm of $x.$)

2018 Mediterranean Mathematics OIympiad, 2

Tags: geometry , circumcircle , ugly

Let $ABC$ be acute triangle. Let $E$ and $F$ be points on $BC$, such that angles $BAE$ and $FAC$ are equal. Lines $AE$ and $AF$ intersect cirumcircle of $ABC$ at points $M$ and $N$. On rays $AB$ and $AC$ we have points $P$ and $R$, such that angle $PEA$ is equal to angle $B$ and angle $AER$ is equal to angle $C$. Let $L$ be intersection of $AE$ and $PR$ and $D$ be intersection of $BC$ and $LN$. Prove that $$\frac{1}{|MN|}+\frac{1}{|EF|}=\frac{1}{|ED|}.$$

2010 Indonesia TST, 2

Tags: combinatorics

A government’s land with dimensions $n \times n$ are going to be sold in phases. The land is divided into $n^2$ squares with dimension $1 \times 1$. In the first phase, $n$ farmers bought a square, and for each rows and columns there is only one square that is bought by a farmer. After one season, each farmer could buy one more square, with the conditions that the newly-bought square has a common side with the farmer’s land and it hasn’t been bought by other farmers. Determine all values of n such that the government’s land could not be entirely sold within $n$ seasons.

2024 SG Originals, Q1

Tags: permutation , number theory , polynomial

Find all permutations $(a_1, a_2, \cdots, a_{2024})$ of $(1, 2, \cdots, 2024)$ such that there exists a polynomial $P$ with integer coefficients satisfying $P(i) = a_i$ for each $i = 1, 2, \cdots, 2024$.

2008 Hanoi Open Mathematics Competitions, 8

Tags: geometry

Consider a convex quadrilateral $ABCD$. Let $O$ be the intersection of $AC$ and $BD$; $M, N$ be the centroid of $\Delta AOB$ and $\Delta COD$ and $P, Q$ be orthocenter of $\Delta BOC$ and $\Delta DOA$, respectively. Prove that $MN\bot PQ$.

1997 Tournament Of Towns, (524) 1

Tags: number theory , sum of digits , divisible

How many integers from $1$ to $1997$ have the sum of their digits divisible by $5$? (AI Galochkin)

2004 Regional Olympiad - Republic of Srpska, 3

Tags: quadratic , algebra

Determine all pairs of positive integers $(a,b)$, such that the roots of the equations \[x^2-ax+a+b-3=0,\] \[x^2-bx+a+b-3=0,\] are also positive integers.

1981 AMC 12/AHSME, 5

Tags: geometry , trapezoid

In trapezoid $ABCD$, sides $AB$ and $CD$ are parallel, and diagonal $BD$ and side $AD$ have equal length. If $m\angle DBC=110^\circ$ and $m\angle CBD =30^\circ$, then $m \angle ADB=$ $\text{(A)}\ 80^\circ \qquad \text{(B)}\ 90^\circ \qquad \text{(C)}\ 100^\circ \qquad \text{(D)}\ 110^\circ \qquad \text{(E)}\ 120^\circ$

2001 Bulgaria National Olympiad, 2

Tags: algebra

Find all real values $t$ for which there exist real numbers $x$, $y$, $z$ satisfying : $3x^2 + 3xz + z^2 = 1$ , $3y^2 + 3yz + z^2 = 4$, $x^2 - xy + y^2 = t$.

2001 CentroAmerican, 3

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Find all the real numbers $ N$ that satisfy these requirements: 1. Only two of the digits of $ N$ are distinct from $ 0$, and one of them is $ 3$. 2. $ N$ is a perfect square.

2016 Japan Mathematical Olympiad Preliminary, 1

Tags: square root , number theory

Calculate the value of $\sqrt{\dfrac{11^4+100^4+111^4}{2}}$ and answer in the form of an integer.

2012 IFYM, Sozopol, 7

Tags: geometry , rotation , quadrilateral , parallelogram

A quadrilateral $ABCD$ is inscribed in a circle with center $O$. Let $A_1 B_1 C_1 D_1$ be the image of $ABCD$ after rotation with center $O$ and angle $\alpha \in (0,90^\circ)$. The points $P,Q,R$ and $S$ are intersections of $AB$ and $A_1 B_1$, $BC$ and $B_1 C_1$, $CD$ and $C_1 D_1$, and $DA$ and $D_1 A_1$. Prove that $PQRS$ is a parallelogram.

2023 AMC 12/AHSME, 3

Tags: geometry

A $3-4-5$ right triangle is inscribed in circle $A$, and a $5-12-13$ right triangle is inscribed in circle $B$. What is the ratio of the area of circle $A$ to the area of circle $B$? $\textbf{(A)}~\frac{9}{25}\qquad\textbf{(B)}~\frac{1}{9}\qquad\textbf{(C)}~\frac{1}{5}\qquad\textbf{(D)}~\frac{25}{169}\qquad\textbf{(E)}~\frac{4}{25}$

2014 Contests, 2

Tags: geometry , 3d geometry , prism , algorithm , geometric transformation , reflection , induction

The $100$ vertices of a prism, whose base is a $50$-gon, are labeled with numbers $1, 2, 3, \ldots, 100$ in any order. Prove that there are two vertices, which are connected by an edge of the prism, with labels differing by not more than $48$. Note: In all the triangles the three vertices do not lie on a straight line.

2012 Vietnam National Olympiad, 3

Tags: function , limit , algebra

Find all $f:\mathbb{R} \to \mathbb{R}$ such that: (a) For every real number $a$ there exist real number $b$:$f(b)=a$ (b) If $x>y$ then $f(x)>f(y)$ (c) $f(f(x))=f(x)+12x.$