Olympiad problems

2017 VJIMC, 1

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

2022 Yasinsky Geometry Olympiad, 6

Tags: geometry , incenter , fixed point , fixed , circumcircle

Let $s$ be an arbitrary straight line passing through the incenter $I$ of the triangle $ABC$ . Line $s$ intersects lines $AB$ and $BC$ at points $D$ and $E$, respectively. Points $P$ and $Q$ are the centers of the circumscribed circles of triangles $DAI$ and $CEI$, respectively, and point $F$ is the second intersection point of these circles. Prove that the circumcircle of the triangle $PQF$ is always passes through a fixed point on the plane regardless of the position of the straight line $s$. (Matvii Kurskyi)

2017 F = ma, 11

Tags: angular momentum , momentum , collision

A small hard solid sphere of mass m and negligible radius is connected to a thin rod of length L and mass 2m. A second small hard solid sphere, of mass M and negligible radius, is fired perpendicularly at the rod at a distance h above the sphere attached to the rod, and sticks to it. A: h = 0 B: h = L/3 C: h = L/2 D: h = L E: Any L will work

2007 AIME Problems, 12

Tags: rotation , geometry

In isosceles triangle $ABC$, $A$ is located at the origin and $B$ is located at $(20, 0)$. Point $C$ is in the first quadrant with $AC = BC$ and $\angle BAC = 75^\circ$. If $\triangle ABC$ is rotated counterclockwise about point $A$ until the image of $C$ lies on the positive y-axis, the area of the region common to the original and the rotated triangle is in the form $p\sqrt{2}+q\sqrt{3}+r\sqrt{6}+s$ where $p$, $q$, $r$, $s$ are integers. Find $(p-q+r-s)/2$.

2015 IFYM, Sozopol, 2

Tags: geometry

Let $ABCD$ be an inscribed quadrilateral and $P$ be an inner point for it so that $\angle PAB=\angle PBC=\angle PCD=\angle PDA$. The lines $AD$ and $BC$ intersect in point $Q$ and lines $AB$ and $CD$ – in point $R$. Prove that $\angle (PQ,PR)=\angle (AC,BD)$.

1991 China Team Selection Test, 1

Tags: geometry

We choose 5 points $A_1, A_2, \ldots, A_5$ on a circumference of radius 1 and centre $O.$ $P$ is a point inside the circle. Denote $Q_i$ as the intersection of $A_iA_{i+2}$ and $A_{i+1}P$, where $A_7 = A_2$ and $A_6 = A_1.$ Let $OQ_i = d_i, i = 1,2, \ldots, 5.$ Find the product $\prod^5_{i=1} A_iQ_i$ in terms of $d_i.$

1991 French Mathematical Olympiad, Problem 1

Tags: number theory

(a) Suppose that $x_n~(n\ge0)$ is a sequence of real numbers with the property that $x_0^3+x_1^3+\ldots+x_n^3=(x_0+x_1+\ldots+x_n)^2$ for each $n\in\mathbb N$. Prove that for each $n\in\mathbb N_0$ there exists $m\in\mathbb N_0$ such that $x_0+x_1+\ldots+x_n=\frac{m(m+1)}2$. (b) For natural numbers $n$ and $p$, we define $S_{n,p}=1^p+2^p+\ldots+n^p$. Find all natural numbers $p$ such that $S_{n,p}$ is a perfect square for each $n\in\mathbb N$.

2017 China National Olympiad, 6

Tags: prc , inequalities , convex-concave inequalities , sqing orz , inequality

Given an integer $n \geq2$ and real numbers $a,b$ such that $0<a<b$. Let $x_1,x_2,\ldots, x_n\in [a,b]$ be real numbers. Find the maximum value of $$\frac{\frac{x^2_1}{x_2}+\frac{x^2_2}{x_3}+\cdots+\frac{x^2_{n-1}}{x_n}+\frac{x^2_n}{x_1}}{x_1+x_2+\cdots +x_{n-1}+x_n}.$$

1984 Czech And Slovak Olympiad IIIA, 3

Tags: algebra , recurrence relation , floor function

Let the sequence $\{a_n\}$ , $n \ge 0$ satisfy the recurrence relation $$a_{n + 2} =4a_{n + 1}-3a_n, \ \ (1) $$ Let us define the sequence $\{b_n\}$ , $n \ge 1$ by the relation $$b_n= \left[ \frac{a_{n+1}}{a_{n-1}} \right]$$ where we put $b_n =1$ for $a_{n-1}=0$. Prove that, starting from a certain term, the sequence also satisfies the recurrence relation (1). Note: $[x]$ indicates the whole part of the number $x$.