Olympiad problems

2018 Purple Comet Problems, 23

Tags: number theory

Let $a, b$, and $c$ be integers simultaneously satisfying the equations $4abc + a + b + c = 2018$ and $ab + bc + ca = -507$. Find $|a| + |b|+ |c|$.

2023 Belarusian National Olympiad, 9.7

Tags: geometry

On one of the sides of the $60$ degree angle with vertex $O$ a fixed point $F$ is marked. On the other side of the angle a point $A$ is chosen, and on the ray $OF$, but not the segment $OF$, a point $B$ such that $OA=FB$. On the segment $AB$ equilateral triangle $ABC$ and $ABD$ are built such that points $O$ and $C$ lie in the same half-plane with respect to $AB$, and $D$ in the other. a) Prove that the point $C$ does not depend on $A$. b) Prove that all points $D$ lie on a line.

2021 Saudi Arabia Training Tests, 34

Tags: polynomial , number theory , divide

Let coefficients of the polynomial$ P (x) = a_dx^d + ... + a_2x^2 + a_0$ where $d \ge 2$, are positive integers. The sequences $(b_n)$ is defined by $b_1 = a_0$ and $b_{n+1} = P (b_n)$ for $n \ge 1$. Prove that for any $n \ge 2$, there exists a prime number $p$ such that $p|b_n$ but it does not divide $b_1, b_2, ..., b_{n-1}$.

1993 Rioplatense Mathematical Olympiad, Level 3, 4

Tags: algebra , inequalities

$x$ and $y$ are real numbers such that $6 -x$, $3 + y^2$, $11 + x$, $14 - y^2$ are greater than zero. Find the maximum of the function $$f(x,y) = \sqrt{(6 -x)(3 + y^2)} + \sqrt{(11 + x)(14 - y^2)}.$$

PEN B Problems, 7

Tags: algebra , polynomial , modular arithmetic , primitive root

Suppose that $p>3$ is prime. Prove that the products of the primitive roots of $p$ between $1$ and $p-1$ is congruent to $1$ modulo $p$.

1998 IMO Shortlist, 2

Tags: floor function , number theory , algebraic identities , algebra

Determine all pairs $(a,b)$ of real numbers such that $a \lfloor bn \rfloor =b \lfloor an \rfloor $ for all positive integers $n$. (Note that $\lfloor x\rfloor $ denotes the greatest integer less than or equal to $x$.)

2024 Euler Olympiad, Round 1, 9

Tags: 3d geometry , geometry

Ants, named Anna and Bob, are located at vertices $A$ and $B$ respectively of a cube $ABCD A_1 B_1 C_1 D_1$, with a sugar cube placed at vertex $C_1$. It is known that Bob can move at a speed of $20$ meters per minute. Determine the minimum speed in integer meters per minute that Anna must be able to travel in order to reach the sugar cube at $C_1$ before Bob. [i]Proposed by Tamar Turashvili, Georgia [/i]

2023-IMOC, C2

Tags: combinatorics

A square house is partitioned into an $n \times n$ grid, where each cell is a room. All neighboring rooms have a door connecting them, and each door can either be normalor inversive. If USJL walks over an inversive door, he would become inverted-USJL,and vice versa. USJL must choose a room to begin and walk pass each room exactly once. If it is inverted-USJL showing up after finishing, then he would be trapped for all eternity. Prove that USJL could always escape.

2017 VJIMC, 1

Tags: algebra , polynomial , rational function , function

Let $(a_n)_{n=1}^{\infty}$ be a sequence with $a_n \in \{0,1\}$ for every $n$. Let $F:(-1,1) \to \mathbb{R}$ be defined by \[F(x)=\sum_{n=1}^{\infty} a_nx^n\] and assume that $F\left(\frac{1}{2}\right)$ is rational. Show that $F$ is the quotient of two polynomials with integer coefficients.

1998 National High School Mathematics League, 12

Tags: geometry , 3d geometry , pyramid

In $\triangle ABC$, $\angle C=90^{\circ},\angle B=30^{\circ}, AC=2$. $M$ is the midpoint of $AB$. Fold up $\triangle ACM$ along $CM$, satisfying that $|AB|=2\sqrt2$. The volume of triangular pyramid $A-BCM$ is________.

1955 Kurschak Competition, 1

Tags: geometry , trapezoid , geometric inequalities

Prove that if the two angles on the base of a trapezoid are different, then the diagonal starting from the smaller angle is longer than the other diagonal. [img]https://cdn.artofproblemsolving.com/attachments/7/1/77cf4958931df1c852c347158ff1e2bbcf45fd.png[/img]

2017 Romania Team Selection Test, P1

Tags: number theory

Let m be a positive interger, let $p$ be a prime, let $a_1=8p^m$, and let $a_n=(n+1)^{\frac{a_{n-1}}{n}}$, $n=2,3...$. Determine the primes $p$ for which the products $a_n(1-\frac{1}{a_1})(1-\frac{1}{a_2})...(1-\frac{1}{a_n})$, $n=1,2,3...$ are all integral.

2002 Bulgaria National Olympiad, 4

Tags: incenter , circumcircle , euler , geometry

Let $I$ be the incenter of a non-equilateral triangle $ABC$ and $T_1$, $T_2$, and $T_3$ be the tangency points of the incircle with the sides $BC$, $CA$ and $AB$, respectively. Prove that the orthocenter of triangle $T_1T_2T_3$ lies on the line $OI$, where $O$ is the circumcenter of triangle $ABC$. [i]Proposed by Georgi Ganchev[/i]

2021 Sharygin Geometry Olympiad, 10-11.3

Tags: geometry , concyclic , cyclic

The bisector of angle $A$ of triangle $ABC$ ($AB > AC$) meets its circumcircle at point $P$. The perpendicular to $AC$ from $C$ meets the bisector of angle $A$ at point $K$. A cừcle with center $P$ and radius $PK$ meets the minor arc $PA$ of the circumcircle at point $D$. Prove that the quadrilateral $ABDC$ is circumscribed.

1959 AMC 12/AHSME, 22

Tags: geometry , trapezoid

The line joining the midpoints of the diagonals of a trapezoid has length $3$. If the longer base is $97$, then the shorter base is: $ \textbf{(A)}\ 94 \qquad\textbf{(B)}\ 92\qquad\textbf{(C)}\ 91\qquad\textbf{(D)}\ 90\qquad\textbf{(E)}\ 89 $

2012 Gulf Math Olympiad, 2

Tags: inequalities , parameterization , algebra

Prove that if $a, b, c$ are positive real numbers, then the least possible value of \[6a^3 + 9b^3 + 32c^3 + \frac{1}{4abc}\] is $6$. For which values of $a, b$ and $c$ is equality attained?

2004 Postal Coaching, 4

Tags: combinatorics , function

In how many ways can a $2\times n$ grid be covered by (a) 2 monominoes and $n-1$ dominoes (b) 4 monominoes and $n-2$ dominoes.

2008 National Chemistry Olympiad, 11

Tags: percent

For the reaction: $2X + 3Y \rightarrow 3Z$, the combination of $2.00$ moles of $X$ with $2.00$ moles of $Y$ produces $1.75 $ moles of $Z$. What is the percent yield of this reaction? $\textbf{(A)}\hspace{.05in}43.8\%\qquad\textbf{(B)}\hspace{.05in}58.3\%\qquad\textbf{(C)}\hspace{.05in}66.7\%\qquad\textbf{(D)}\hspace{.05in}87.5\%\qquad $

2001 IMO Shortlist, 5

Tags: composite number , number theory

Let $a > b > c > d$ be positive integers and suppose that \[ ac + bd = (b+d+a-c)(b+d-a+c). \] Prove that $ab + cd$ is not prime.

1950 Moscow Mathematical Olympiad, 184

Tags: point , combinatorics , combinatorial geometry

* On a circle, $20$ points are chosen. Ten non-intersecting chords without mutual endpoints connect some of the points chosen. How many distinct such arrangements are there?

2010 Dutch IMO TST, 3

Tags: number theory , perfect square

Let $n\ge 2$ be a positive integer and $p $ a prime such that $n|p-1$ and $p | n^3-1$. Show $ 4p-3$ is a square.

2022 Tuymaada Olympiad, 3

Tags: number theory , combinatorics

Is there a colouring of all positive integers in three colours so that for each positive integer the numbers of its divisors of any two colours differ at most by $2?$

2011 Czech-Polish-Slovak Match, 1

Tags: polynomial , algebra

A polynomial $P(x)$ with integer coefficients satisfies the following: if $F(x)$, $G(x)$, and $Q(x)$ are polynomials with integer coefficients satisfying $P\Big(Q(x)\Big)=F(x)\cdot G(x)$, then $F(x)$ or $G(x)$ is a constant polynomial. Prove that $P(x)$ is a constant polynomial.

Champions Tournament Seniors - geometry, 2008.4

Tags: geometry , 3d geometry , sphere , pyramid

Given a quadrangular pyramid $SABCD$, the basis of which is a convex quadrilateral $ABCD$. It is known that the pyramid can be tangent to a sphere. Let $P$ be the point of contact of this sphere with the base $ABCD$. Prove that $\angle APB + \angle CPD = 180^o$.

2001 Swedish Mathematical Competition, 1

Tags: algebra , product

Show that if we take any six numbers from the following array, one from each row and column, then the product is always the same: 4 6 10 14 22 26 6 9 15 21 33 39 10 15 25 35 55 65 16 24 40 56 88 104 18 27 45 63 99 117 20 30 50 70 110 130