Olympiad problems

2021 Science ON all problems, 4

Tags: number theory

Find the least positive integer which is a multiple of $13$ and all its digits are the same. [i](Adapted from Gazeta Matematică 1/1982, Florin Nicolăită)[/i]

2018 AMC 12/AHSME, 20

Tags:

Let $ABCDEF$ be a regular hexagon with side length $1$. Denote by $X$, $Y$, and $Z$ the midpoints of sides $\overline{AB},\overline{CD},\overline{EF}$, respectively. What is the area of the convex hexagon whose interior is the intersection of the interiors of $\triangle{ACE}$ and $\triangle{XYZ}$? $\textbf{(A) }\dfrac{3}{8}\sqrt{3}\qquad\textbf{(B) }\dfrac{7}{16}\sqrt{3}\qquad\textbf{(C) }\dfrac{15}{32}\sqrt{3}\qquad\textbf{(D) }\dfrac{1}{2}\sqrt{3}\qquad\textbf{(E) }\dfrac{9}{16}\sqrt{3}$

1967 IMO, 5

Tags: algebra , sequence , algebraic identities

Let $a_1,\ldots,a_8$ be reals, not all equal to zero. Let \[ c_n = \sum^8_{k=1} a^n_k\] for $n=1,2,3,\ldots$. Given that among the numbers of the sequence $(c_n)$, there are infinitely many equal to zero, determine all the values of $n$ for which $c_n = 0.$

2012 Today's Calculation Of Integral, 779

Tags: calculus , analytic geometry , conic , integration , parabola , calculus computations , geometry

Consider parabolas $C_a: y=-2x^2+4ax-2a^2+a+1$ and $C: y=x^2-2x$ in the coordinate plane. When $C_a$ and $C$ have two intersection points, find the maximum area enclosed by these parabolas.

1998 Spain Mathematical Olympiad, 2

Tags: number theory , 3d geometry , modular arithmetic , geometry

Find all four-digit numbers which are equal to the cube of the sum of their digits.

2017 District Olympiad, 4

Tags: algebra , inequalities , inequality

If $ a,b,c>0 $ and $ ab+bc+ca+abc=4, $ then $ \sqrt{ab} +\sqrt{bc} +\sqrt{ca} \le 3\le a+b+c. $

2015 Rioplatense Mathematical Olympiad, Level 3, 4

Tags: combinatorics , board , coloring

You have a $9 \times 9$ board with white squares. A tile can be moved from one square to another neighbor (tiles that share one side). If we paint some squares of black, we say that such coloration is [i]good [/i] if there is a white square where we can place a chip that moving through white squares can return to the initial square having passed through at least $3$ boxes, without passing the same square twice. Find the highest possible value of $k$ such that any form of painting $k$ squares of black are a [i]good [/i] coloring.

Kvant 2025, M2833

Tags: combinatorics , weighing

There are a) $26$; b) $30$ identical-looking coins in a circle. It is known that exactly two of them are fake. Real coins weigh the same, fake ones too, but they are lighter than the real ones. How can you determine in three weighings on a cup scale without weights whether there are fake coins lying nearby or not?? [i]Proposed by A. Gribalko[/i]

2002 Singapore Team Selection Test, 3

Tags: functional equation , algebra , functional

Find all functions $f : [0,\infty) \to [0,\infty)$ such that $f(f(x)) +f(x) = 12x$, for all $x \ge 0$.

2005 Slovenia National Olympiad, Problem 1

Tags: function , floor function

Evaluate the sum $\left\lfloor\log_21\right\rfloor+\left\lfloor\log_22\right\rfloor+\left\lfloor\log_23\right\rfloor+\ldots+\left\lfloor\log_2256\right\rfloor$.

2020 Taiwan TST Round 1, 1

Tags: combinatorics , binomial coefficient , trivial

The infinite sequence $a_0,a _1, a_2, \dots$ of (not necessarily distinct) integers has the following properties: $0\le a_i \le i$ for all integers $i\ge 0$, and \[\binom{k}{a_0} + \binom{k}{a_1} + \dots + \binom{k}{a_k} = 2^k\] for all integers $k\ge 0$. Prove that all integers $N\ge 0$ occur in the sequence (that is, for all $N\ge 0$, there exists $i\ge 0$ with $a_i=N$).

2018 Thailand TST, 3

Tags: number theory , arithmetic sequence

Does there exist an arithmetic progression with $2017$ terms such that each term is not a perfect power, but the product of all $2017$ terms is?

2021 Novosibirsk Oral Olympiad in Geometry, 3

Tags: geometry , angle

Find the angle $BCA$ in the quadrilateral of the figure. [img]https://cdn.artofproblemsolving.com/attachments/0/2/974e23be54125cde8610a78254b59685833b5b.png[/img]

2006 Bulgaria National Olympiad, 2

Tags: algebra , function , inequalities , induction , limit

Let $f:\mathbb{R}^+\to\mathbb{R}^+$ be a function that satisfies for all $x>y>0$ \[f(x+y)-f(x-y)=4\sqrt{f(x)f(y)}\] a) Prove that $f(2x)=4f(x)$ for all $x>0$; b) Find all such functions. [i]Nikolai Nikolov, Oleg Mushkarov [/i]

2024-IMOC, A2

Tags: algebra , inequalities , minimum value , absolute value

Given integer $n \geq 3$ and $x_1$, $x_2$, …, $x_n$ be $n$ real numbers satisfying $|x_1|+|x_2|+…+|x_n|=1$. Find the minimum of \[|x_1+x_2|+|x_2+x_3|+…+|x_{n-1}+x_n|+|x_n+x_1|.\] [i]Proposed by snap7822[/i]

1978 All Soviet Union Mathematical Olympiad, 259

Tags: combinatorics , combinatorial geometry , square

Prove that there exists such a number $A$ that you can inscribe $1978$ different size squares in the plot of the function $y = A sin(x)$. (The square is inscribed if all its vertices belong to the plot.)

1991 Tournament Of Towns, (301) 2

Tags: combinatorics

The “flying rook” moves as the usual chess rook but can’t move to a neighbouring square in one move. Is it possible for the flying rook on a $4 \times 4$ chess-board to visit every square once and return to the initial square in $16$ moves? (A. Tolpygo, Kiev)

2012 Miklós Schweitzer, 2

Tags: linear algebra

Call a subset $A$ of the cyclic group $(\mathbb{Z}_n,+)$ [i]rich[/i] if for all $x,y \in \mathbb{Z}_n$ there exists $r \in \mathbb{Z}_n$ such that $x-r,x+r,y-r,y+r$ are all in $A$. For what $\alpha$ is there a constant $C_\alpha>0$ such that for each odd positive integer $n$, every rich subset $A \subset \mathbb{Z}_n$ has at least $C_\alpha n^\alpha$ elements?

2019 Israel National Olympiad, 6

Tags: number theory

A set of integers is called [b]legendary[/b] if you can reach any integer from it by using the following action multiple times: If the numbers $x,y$ are in the set, we may add the number $xy-y^2-y+x$ to the set. Prove that any legendary set contains at least 8 numbers.

2006 All-Russian Olympiad Regional Round, 10.1

Tags: geometry , combinatorics , combinatorial geometry , geometric inequality

Natural numbers from $1$ to $200$ were divided into $50$ sets. Prove that one of them contains three numbers that are the lengths of the sides some triangle.

2017 Princeton University Math Competition, A3/B5

Tags: geometry , 3d geometry , prism

A right regular hexagonal prism has bases $ABCDEF$, $A'B'C'D'E'F'$ and edges $AA'$, $BB'$, $CC'$, $DD'$, $EE'$, $FF'$, each of which is perpendicular to both hexagons. The height of the prism is $5$ and the side length of the hexagons is $6$. The plane $P$ passes through points $A$, $C'$, and $E$. The area of the portion of $P$ contained in the prism can be expressed as $m\sqrt{n}$, where $n$ is not divisible by the square of any prime. Find $m+n$.

2008 Bulgaria Team Selection Test, 1

Tags: number theory

For each positive integer $n$, denote by $a_{n}$ the first digit of $2^{n}$ (base ten). Is the number $0.a_{1}a_{2}a_{3}\cdots$ rational?

1990 Bundeswettbewerb Mathematik, 2

Tags: sequence , perfect square , recurrence relation , algebra

The sequence $a_0,a_1,a_2,...$ is defined by $a_0 = 0, a_1 = a_2 = 1$ and $a_{n+2} +a_{n-1} = 2(a_{n+1} +a_n)$ for all $n \in N$. Show that all $a_n$ are perfect squares .

2015 Saudi Arabia IMO TST, 1

Tags: geometry , incenter

Let $ABC$ be an acute-angled triangle inscribed in the circle $(O)$, $H$ the foot of the altitude of $ABC$ at $A$ and $P$ a point inside $ABC$ lying on the bisector of $\angle BAC$. The circle of diameter $AP$ cuts $(O)$ again at $G$. Let $L$ be the projection of $P$ on $AH$. Prove that if $GL$ bisects $HP$ then $P$ is the incenter of the triangle $ABC$. Lê Phúc Lữ

2018 Purple Comet Problems, 9

Tags: system of equations , algebra

For some $k > 0$ the lines $50x + ky = 1240$ and $ky = 8x + 544$ intersect at right angles at the point $(m,n)$. Find $m + n$.