Olympiad problems

2018 Malaysia National Olympiad, B3

There are $200$ numbers on a blackboard: $ 1! , 2! , 3! , 4! , ... ... , 199! , 200!$. Julia erases one of the numbers. When Julia multiplies the remaining $199$ numbers, the product is a perfect square. Which number was erased?

2004 Pre-Preparation Course Examination, 2

Tags: limit , inequalities , combinatorics

Let $ H(n)$ be the number of simply connected subsets with $ n$ hexagons in an infinite hexagonal network. Also let $ P(n)$ be the number of paths starting from a fixed vertex (that do not connect itself) with lentgh $ n$ in this hexagonal network. a) Prove that the limits \[ \alpha: \equal{}\lim_{n\rightarrow\infty}H(n)^{\frac1n}, \beta: \equal{}\lim_{n\rightarrow\infty}P(n)^{\frac1n}\]exist. b) Prove the following inequalities: $ \sqrt2\leq\beta\leq2$ $ \alpha\leq 12.5$ $ \alpha\geq3.5$ $ \alpha\leq\beta^4$

2016 Balkan MO, 3

Tags: polynomial , number theory

Find all monic polynomials $f$ with integer coefficients satisfying the following condition: there exists a positive integer $N$ such that $p$ divides $2(f(p)!)+1$ for every prime $p>N$ for which $f(p)$ is a positive integer. [i]Note: A monic polynomial has a leading coefficient equal to 1.[/i] [i](Greece - Panagiotis Lolas and Silouanos Brazitikos)[/i]

1994 India Regional Mathematical Olympiad, 3

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Find all 6-digit numbers $a_1a_2a_3a_4a_5a_6$ formed by using the digits $1,2,3,4,5,6$ once each such that the number $a_1a_2a_2\ldots a_k$ is divisible by $k$ for $1 \leq k \leq 6$.

2016 Bosnia And Herzegovina - Regional Olympiad, 3

Tags: geometry , circumcircle

$h_a$, $h_b$ and $h_c$ are altitudes, $t_a$, $t_b$ and $t_c$ are medians of acute triangle, $r$ radius of incircle, and $R$ radius of circumcircle of acute triangle $ABC$. Prove that $$\frac{t_a}{h_a}+\frac{t_b}{h_b}+\frac{t_c}{h_c} \leq 1+ \frac{R}{r}$$

2002 Federal Math Competition of S&M, Problem 1

Tags: algebra , floor function , inequalities

Determine all real numbers $x$ such that $$\frac{2002\lfloor x\rfloor}{\lfloor-x\rfloor+x}>\frac{\lfloor2x\rfloor}{x-\lfloor1+x\rfloor}.$$

2021 Argentina National Olympiad, 3

Tags: geometry , right triangle , angle chasing

Let $ABC$ be an isosceles right triangle at $A$ with $AB=AC$. Let $M$ and $N$ be on side $BC$, with $M$ between $B$ and $N,$ such that $$BM^2+ NC^2= MN^2.$$ Determine the measure of the angle $\angle MAN.$

1952 AMC 12/AHSME, 40

Tags: quadratic , function

In order to draw a graph of $ f(x) \equal{} ax^2 \plus{} bx \plus{} c$, a table of values was constructed. These values of the function for a set of equally spaced increasing values of $ x$ were $ 3844$, $ 3969$, $ 4096$, $ 4227$, $ 4356$, $ 4489$, $ 4624$, and $ 4761$. The one which is incorrect is: $ \textbf{(A)}\ 4096 \qquad\textbf{(B)}\ 4356 \qquad\textbf{(C)}\ 4489 \qquad\textbf{(D)}\ 4761 \qquad\textbf{(E)}\ \text{none of these}$

2020 Iran Team Selection Test, 4

Tags: incenter , circumcircle , homothety , harmonic , geometry

Let $ABC$ be an isosceles triangle ($AB=AC$) with incenter $I$. Circle $\omega$ passes through $C$ and $I$ and is tangent to $AI$. $\omega$ intersects $AC$ and circumcircle of $ABC$ at $Q$ and $D$, respectively. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of $CQ$. Prove that $AD$, $MN$ and $BC$ are concurrent. [i]Proposed by Alireza Dadgarnia[/i]

2023 Puerto Rico Team Selection Test, 6

Tags: algebra , number theory

Find all possible integer values of the sum: $$\frac{a}{b}+ \frac{2023 \times b}{4 \times a},$$ where $a$ and $b$ are positive integers with no prime factors in common.

1998 India Regional Mathematical Olympiad, 4

Tags: geometry , geometric transformation , reflection

Let $ABC$ be a triangle with $AB = AC$ and $\angle BAC = 30^{\circ}$, Let $A'$ be the reflection of $A$ in the line $BC$; $B'$ be the reflection of $B$ in the line $CA$; $C'$ be the reflection of $C$ in line $AB$, Show that $A'B'C'$ is an equilateral triangle.

2023 Indonesia TST, A

Tags: algebra , inequalities

Let $a,b,c$ positive real numbers and $a+b+c = 1$. Prove that \[a^2 + b^2 + c^2 + \frac{3}{\frac{1}{a} + \frac{1}{b} + \frac{1}{c}} \ge 2(ab + bc + ac)\]

1993 Tournament Of Towns, (360) 3

Tags: number theory , perfect square , diophantine , diophantine equation

Positive integers $a$, $b$ and $c$ are positive integers with greatest common divisor equal to $1$ (i.e. they have no common divisors greater than $1$), and $$\frac{ab}{a-b}=c$$ Prove that $a -b$ is a perfect square. (SL Berlov)

2014 IMO Shortlist, N5

Tags: prime , number theory

Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$. [i]Proposed by Belgium[/i]

2001 Junior Balkan Team Selection Tests - Moldova, 5

Tags: number theory , rational

Determine if there is a non-natural natural number $n$ with the property that $\sqrt{n + 1} + \sqrt{n - 1}$ is rational.

1997 IMC, 4

Tags: linear algebra

(a) Let $f: \mathbb{R}^{n\times n}\rightarrow\mathbb{R}$ be a linear mapping. Prove that $\exists ! C\in\mathbb{R}^{n\times n}$ such that $f(A)=Tr(AC), \forall A \in \mathbb{R}^{n\times n}$. (b) Suppose in addtion that $\forall A,B \in \mathbb{R}^{n\times n}: f(AB)=f(BA)$. Prove that $\exists \lambda \in \mathbb{R}: f(A)=\lambda Tr(A)$

2012 AMC 12/AHSME, 4

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In a bag of marbles, $\frac{3}{5}$ of the marbles are blue and the rest are red. If the number of red marbles is doubled and the number of blue marbles stays the same, what fraction of the marbles will be red? $ \textbf{(A)}\ \dfrac{2}{5} \qquad\textbf{(B)}\ \dfrac{3}{7} \qquad\textbf{(C)}\ \dfrac{4}{7} \qquad\textbf{(D)}\ \dfrac{3}{5} \qquad\textbf{(E)}\ \dfrac{4}{5} $

2022 Malaysia IMONST 2, 6

Tags: number theory , algebra , combinatorics

A football league has $n$ teams. Each team plays one game with every other team. Each win is awarded $2$ points, each tie $1$ point, and each loss $0$ points. After the league is over, the following statement is true: for every subset $S$ of teams in the league, there is a team (which may or may not be in $S$) such that the total points the team obtained by playing all the teams in $S$ is odd. Prove that $n$ is even.

2015 HMMT Geometry, 7

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Let $ABCD$ be a square pyramid of height $\frac{1}{2}$ with square base $ABCD$ of side length $AB=12$ (so $E$ is the vertex of the pyramid, and the foot of the altitude from $E$ to $ABCD$ is the center of square $ABCD$). The faces $ADE$ and $CDE$ meet at an acute angle of measure $\alpha$ (so that $0^{\circ}<\alpha<90^{\circ}$). Find $\tan \alpha$.

2022 Greece JBMO TST, 2

Tags: geometry , tangent

Let $ABC$ be an acute triangle with $AB<AC < BC$, inscirbed in circle $\Gamma_1$, with center $O$. Circle $\Gamma_2$, with center point $A$ and radius $AC$ intersects $BC$ at point $D$ and the circle $\Gamma_1$ at point $E$. Line $AD$ intersects circle $\Gamma_1$ at point $F$. The circumscribed circle $\Gamma_3$ of triangle $DEF$, intersects $BC$ at point $G$. Prove that: a) Point $B$ is the center of circle $\Gamma_3$ b) Circumscribed circle of triangle $CEG$ is tangent to $AC$.

2012 Princeton University Math Competition, A4 / B6

Tags: geometry

A square is inscribed in an ellipse such that two sides of the square respectively pass through the two foci of the ellipse. The square has a side length of $4$. The square of the length of the minor axis of the ellipse can be written in the form $a + b\sqrt{c}$ where $a, b$, and $c$ are integers, and $c$ is not divisible by the square of any prime. Find the sum $a + b + c$.

2002 Tournament Of Towns, 3

Tags: combinatorics

The vertices of a $50\text{-gon}$ divide a circumference into $50$ arcs, whose lengths are $1,2,\ldots 50$ in some order. It is known that any two opposite arcs (corresponding to opposite sides) differ by $25$. Prove that the polygon has two parallel sides.

1969 Vietnam National Olympiad, 1

Tags: combinatorics , graph , subset

A graph $G$ has $n + k$ vertices. Let $A$ be a subset of $n$ vertices of the graph $G$, and $B$ be a subset of other $k$ vertices. Each vertex of $A$ is joined to at least $k - p$ vertices of $B$. Prove that if $np < k$ then there is a vertex in $B$ that can be joined to all vertices of $A$.

1999 Harvard-MIT Mathematics Tournament, 1

Tags: number theory

A combination lock has a $3$ number combination, with each number an integer between $0$ and $39$ inclusive. Call the numbers $n_1$, $n_2$, and $n_3$. If you know that $n_1$ and $n_3$ leave the same remainder when divided by $4$, and $n_2$ and $n_1 + 2$ leave the same remainder when divided by $4$, how many possible combinations are there?

2016 Czech-Polish-Slovak Junior Match, 3

Tags: combinatorial geometry , divide , geometry , 3d geometry , prism , combinatorics

Find all integers $n \ge 3$ with the following property: it is possible to assign pairwise different positive integers to the vertices of an $n$-gonal prism in such a way that vertices with labels $a$ and $b$ are connected by an edge if and only if $a | b$ or $b | a$. Poland