Olympiad problems

2024 Malaysia IMONST 2, 3

Tags: number theory

Ivan claims that for all positive integers $n$, $$\left\lfloor\sqrt[2]{\frac{n}{1^3}}\right\rfloor + \left\lfloor\sqrt[2]{\frac{n}{2^3}}\right\rfloor + \left\lfloor\sqrt[2]{\frac{n}{3^3}}\right\rfloor + \cdots = \left\lfloor\sqrt[3]{\frac{n}{1^2}}\right\rfloor + \left\lfloor\sqrt[3]{\frac{n}{2^2}}\right\rfloor + \left\lfloor\sqrt[3]{\frac{n}{3^2}}\right\rfloor + \cdots$$ Why is he correct? (Note: $\lfloor x \rfloor$ denotes the floor function.)

2021 Alibaba Global Math Competition, 3

Tags: matrices , binary matrix

Given positive integers $k \ge 2$ and $m$ sufficiently large. Let $\mathcal{F}_m$ be the infinite family of all the (not necessarily square) binary matrices which contain exactly $m$ 1's. Denote by $f(m)$ the maximum integer $L$ such that for every matrix $A \in \mathcal{F}_m$, there always exists a binary matrix $B$ of the same dimension such that (1) $B$ has at least $L$ 1-entries; (2) every entry of $B$ is less or equal to the corresponding entry of $A$; (3) $B$ does not contain any $k \times k$ all-1 submatrix. Show the equality \[\lim_{m \to \infty} \frac{\ln f(m)}{\ln m}=\frac{k}{k+1}.\]

1991 Flanders Math Olympiad, 2

Tags: limit , algebra , polynomial

(a) Show that for every $n\in\mathbb{N}$ there is exactly one $x\in\mathbb{R}^+$ so that $x^n+x^{n+1}=1$. Call this $x_n$. (b) Find $\lim\limits_{n\rightarrow+\infty}x_n$.

2013 Pan African, 1

Tags: trapezoid , parallelogram , ratio , geometry

Let $ABCD$ be a convex quadrilateral with $AB$ parallel to $CD$. Let $P$ and $Q$ be the midpoints of $AC$ and $BD$, respectively. Prove that if $\angle ABP=\angle CBD$, then $\angle BCQ=\angle ACD$.

2014 Saudi Arabia IMO TST, 1

Tags: number theory

Tarik and Sultan are playing the following game. Tarik thinks of a number that is greater than $100$. Then Sultan is telling a number greater than $1$. If Tarik’s number is divisible by Sultan’s number, Sultan wins, otherwise Tarik subtracts Sultan’s number from his number and Sultan tells his next number. Sultan is forbidden to repeat his numbers. If Tarik’s number becomes negative, Sultan loses. Does Sultan have a winning strategy?

2020 BMT Fall, 18

Tags: combinatorics

Let $T$ be the answer to question $17$, and let $N =\frac{24}{T}$. Leanne flips a fair coin $N$ times. Let $X$ be the number of times that within a series of three consecutive flips, there were exactly two heads or two tails. What is the expected value of $X$?

2015 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: inequalities , algebra

Let $a$, $b$ and $c$ be positive real numbers such that $abc=1$. Prove the inequality: $$\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a} \leq \frac{a^2+b^2+c^2}{2}$$

1990 National High School Mathematics League, 10

Tags: calculus , integration

Define $f(n):$ the number of integral points of line segment $OA_n$ ($O$ and $A_n$ not included), where $A_n(n,n+3)$. Then, $f(1)+f(2)+\cdots+f(1990)=$________.

2024/2025 TOURNAMENT OF TOWNS, P1

Tags: combinatorics

Peter writes a positive integer on the whiteboard. Each minute Basil multiplies the last written number by 2 or by 3 and writes the product on the whiteboard too. Can Peter choose the starting integer such that, irrespective of Basil's strategy, at any given moment the number of integers on the whiteboard starting with 1 or 2 would exceed the number of the ones starting with 7, 8 or 9 ? Maxim Didin

2007 iTest Tournament of Champions, 5

Tags:

Convex quadrilateral $ABCD$ has the property that the circles with diameters $AB$ and $CD$ are tangent at point $X$ inside the quadrilateral, and likewise, the circles with diameters $BC$ and $DA$ are tangent at a point $Y$ inside the quadrilateral. Given that the perimeter of $ABCD$ is $96$, and the maximum possible length of $XY$ is $m$, find $\lfloor 2007m\rfloor$.

2011 SEEMOUS, Problem 1

Tags: function , real analysis , integral inequality , inequalities

Let $f:[0,1]\rightarrow R$ be a continuous function and n be an integer number,n>0.Prove that $\int_0^1f(x)dx \le (n+1)*\int_0^1 x^n*f(x)dx $

2019 Thailand TSTST, 3

Tags: coefficient , algebra , polynomial , combinatorics

Let $n\geq 2$ be an integer. Determine the number of terms in the polynomial $$\prod_{1\leq i< j\leq n}(x_i+x_j)$$ whose coefficients are odd integers.

1985 IMO Shortlist, 15

Tags: square , geometry , dissection

Let $K$ and $K'$ be two squares in the same plane, their sides of equal length. Is it possible to decompose $K$ into a finite number of triangles $T_1, T_2, \ldots, T_p$ with mutually disjoint interiors and find translations $t_1, t_2, \ldots, t_p$ such that \[K'=\bigcup_{i=1}^{p} t_i(T_i) \ ? \]

2021 Canada National Olympiad, 3

Tags: combinatorics

At a dinner party there are $N$ hosts and $N$ guests, seated around a circular table, where $N\geq 4$. A pair of two guests will chat with one another if either there is at most one person seated between them or if there are exactly two people between them, at least one of whom is a host. Prove that no matter how the $2N$ people are seated at the dinner party, at least $N$ pairs of guests will chat with one another.

2017 AIME Problems, 14

Tags:

Let $a > 1$ and $x > 1$ satisfy $\log_a(\log_a(\log_a 2) + \log_a 24 - 128) = 128$ and $\log_a(\log_a x) = 256$. Find the remainder when $x$ is divided by $1000$.

2024 Serbia Team Selection Test, 1

Tags: combinatorics

Three coins are placed at the origin of a Cartesian coordinate system. On one move one removes a coin placed at some position $(x, y)$ and places three new coins at $(x+1, y)$, $(x, y+1)$ and $(x+1, y+1)$. Prove that after finitely many moves, there will exist two coins placed at the same point.

2011 Chile National Olympiad, 3

Tags: combinatorics , coloring

Consider the following figure formed by $10$ nodes and $15$ edges: [asy] unitsize(1.5 cm); pair A, B, C, D, E, F, G, H, I, J; A = dir(90); B = dir(90 + 360/5); C = dir(90 + 2*360/5); D = dir(90 + 3*360/5); E = dir(90 + 4*360/5); F = 0.6*A; G = 0.6*B; H = 0.6*C; I = 0.6*D; J = 0.6*E; draw(A--B--C--D--E--cycle); draw(F--H--J--G--I--cycle); draw(A--F); draw(B--G); draw(C--H); draw(D--I); draw(E--J); dot(A); dot(B); dot(C); dot(D); dot(E); dot(F); dot(G); dot(H); dot(I); dot(J); [/asy] Prove that the edges of the figure cannot be colored by using $3$ different colors so that the edges that reach each node have different colors from each other.

2013 Bangladesh Mathematical Olympiad, 1

Tags: geometry

Higher Secondary P1 A polygon is called degenerate if one of its vertices falls on a line that joins its neighboring two vertices. In a pentagon $ABCDE$, $AB=AE$, $BC=DE$, $P$ and $Q$ are midpoints of $AE$ and $AB$ respectively. $PQ||CD$, $BD$ is perpendicular to both $AB$ and $DE$. Prove that $ABCDE$ is a degenerate pentagon.

2011 Canadian Open Math Challenge, 2

Tags:

Carmen selects four different numbers from the set $\{1, 2, 3, 4, 5, 6, 7\}$ whose sum is 11. If $l$ is the largest of these four numbers, what is the value of $l$?

2024 CMIMC Team, 4

Tags: team

Eric and Christina are playing a game with $n$ stones. They alternate taking some number of stones from the pile, with Eric going first. The number of stones Eric takes from the pile must be a power of $3$ (e.g. 1, 3, 9, 27, ...), while the number of stones Christina takes must be a power of $2$ (e.g. 1, 2, 4, 8, ...). Whoever takes the last stone wins. Find the sum of all $1\leq n \leq 100$ for which Eric has a winning strategy. [i]Proposed by Connor Gordon[/i]

2021 Thailand TST, 2

Tags: algebra , inequality , 4-variable inequality , inequalities

Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of $$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$ [i]Israel[/i]

2019 China Team Selection Test, 3

Tags: coloring , bijection , combinatorics

Does there exist a bijection $f:\mathbb{N}^{+} \rightarrow \mathbb{N}^{+}$, such that there exist a positive integer $k$, and it's possible to have each positive integer colored by one of $k$ chosen colors, such that for any $x \neq y$ , $f(x)+y$ and $f(y)+x$ are not the same color?

2024 Saint Petersburg Mathematical Olympiad, 4

Tags: number theory

Given a $101$-digit number $a$ and an arbitrary positive integer $b$. Prove that there is at most a $102$-digit positive integer $c$ such that any number of the form $\overline{caaa \dots ab}$ is composite.

2006 Federal Math Competition of S&M, Problem 1

Tags: quadratic , inequality , algebra , inequalities

MathLinks Contest 7th, 1.1

Tags: geometry , circumcircle , euler , symmetry , trigonometry , parameterization , ratio

Given is an acute triangle $ ABC$ and the points $ A_1,B_1,C_1$, that are the feet of its altitudes from $ A,B,C$ respectively. A circle passes through $ A_1$ and $ B_1$ and touches the smaller arc $ AB$ of the circumcircle of $ ABC$ in point $ C_2$. Points $ A_2$ and $ B_2$ are defined analogously. Prove that the lines $ A_1A_2$, $ B_1B_2$, $ C_1C_2$ have a common point, which lies on the Euler line of $ ABC$.