Olympiad problems

2017 AMC 12/AHSME, 2

Tags: AMC , AMC 10 , AMC 10 B , inequalities , 2017 AMC 10B

Real numbers $x$, $y$, and $z$ satisfy the inequalities $$0<x<1,\qquad-1<y<0,\qquad\text{and}\qquad1<z<2.$$ Which of the following numbers is nessecarily positive? $\textbf{(A) } y+x^2 \qquad \textbf{(B) } y+xz \qquad \textbf{(C) }y+y^2 \qquad \textbf{(D) }y+2y^2 \qquad\\ \textbf{(E) } y+z$

JOM 2024, 1

Tags: geometry

Consider $\triangle MAB$ with a right angle at $A$ and circumcircle $\omega$. Take any chord $CD$ perpendicular to $AB$ such that $A, C, B, D, M$ lie on $\omega$ in this order. Let $AC$ and $MD$ intersect at point $E$, and let $O$ be the circumcenter of $\triangle EMC$. Show that if $J$ is the intersection of $BC$ and $OM$, then $JB = JM$. [i](Proposed by Matthew Kung Wei Sheng and Ivan Chan Kai Chin)[/i]

2008 IMO Shortlist, 6

Tags: function , algebra , functional equation , range , IMO Shortlist

Let $ f: \mathbb{R}\to\mathbb{N}$ be a function which satisfies $ f\left(x \plus{} \dfrac{1}{f(y)}\right) \equal{} f\left(y \plus{} \dfrac{1}{f(x)}\right)$ for all $ x$, $ y\in\mathbb{R}$. Prove that there is a positive integer which is not a value of $ f$. [i]Proposed by Žymantas Darbėnas (Zymantas Darbenas), Lithuania[/i]

2020 HMNT (HMMO), 1

Tags: number theory

Chelsea goes to La Verde's at MIT and buys $100$ coconuts, each weighing $4$ pounds, and $100$ honeydews, each weighing $5$ pounds. She wants to distribute them among $n$ bags, so that each bag contains at most $13$ pounds of fruit. What is the minimum $n$ for which this is possible?

2017 ISI Entrance Examination, 8

Tags: algebra , polynomial , function

Let $k,n$ and $r$ be positive integers. (a) Let $Q(x)=x^k+a_1x^{k+1}+\cdots+a_nx^{k+n}$ be a polynomial with real coefficients. Show that the function $\frac{Q(x)}{x^k}$ is strictly positive for all real $x$ satisfying $$0<|x|<\frac1{1+\sum\limits_{i=1}^n |a_i|}$$ (b) Let $P(x)=b_0+b_1x+\cdots+b_rx^r$ be a non zero polynomial with real coefficients. Let $m$ be the smallest number such that $b_m \neq 0$. Prove that the graph of $y=P(x)$ cuts the $x$-axis at the origin (i.e., $P$ changes signs at $x=0$) if and only if $m$ is an odd integer.

1991 AIME Problems, 3

Tags: inequalities , floor function , ceiling function , ratio , AMC , AIME , probability

Expanding $(1+0.2)^{1000}$ by the binomial theorem and doing no further manipulation gives \begin{eqnarray*} &\ & \binom{1000}{0}(0.2)^0+\binom{1000}{1}(0.2)^1+\binom{1000}{2}(0.2)^2+\cdots+\binom{1000}{1000}(0.2)^{1000}\\ &\ & = A_0 + A_1 + A_2 + \cdots + A_{1000}, \end{eqnarray*} where $A_k = \binom{1000}{k}(0.2)^k$ for $k = 0,1,2,\ldots,1000$. For which $k$ is $A_k$ the largest?

2015 Indonesia MO, 5

Tags: Divisibility , number theory

Given positive integers $a,b,c,d$ such that $a\mid c^d$ and $b\mid d^c$. Prove that \[ ab\mid (cd)^{max(a,b)} \]

1967 IMO Longlists, 8

Tags: geometry , parallelogram , trigonometry , geometric inequality , Trigonometric inequality , IMO , IMO 1967

The parallelogram $ABCD$ has $AB=a,AD=1,$ $\angle BAD=A$, and the triangle $ABD$ has all angles acute. Prove that circles radius $1$ and center $A,B,C,D$ cover the parallelogram if and only \[a\le\cos A+\sqrt3\sin A.\]

1994 Vietnam Team Selection Test, 2

Tags: function , logarithms , algebra unsolved , algebra

Determine all functions $f: \mathbb{R} \mapsto \mathbb{R}$ satisfying \[f\left(\sqrt{2} \cdot x\right) + f\left(4 + 3 \cdot \sqrt{2} \cdot x \right) = 2 \cdot f\left(\left(2 + \sqrt{2}\right) \cdot x\right)\] for all $x$.

2013 SDMO (Middle School), 3

Tags: ratio , geometry

Let $ABCD$ be a square, and let $\Gamma$ be the circle that is inscribed in square $ABCD$. Let $E$ and $F$ be points on line segments $AB$ and $AD$, respectively, so that $EF$ is tangent to $\Gamma$. Find the ratio of the area of triangle $CEF$ to the area of square $ABCD$.

2012 NIMO Problems, 9

Tags:

In how many ways can the following figure be tiled with $2 \times 1$ dominos? [asy] defaultpen(linewidth(.8)); size(5.5cm); int i; for(i = 1; i<6; i = i+1) { draw((.5 + i,6-i)--(.5 + i,i-6)--(-(.5 + i),i-6)--(-(.5 + i),6-i)--cycle);} draw((.5,5)--(.5,-5)^^(-.5,5)--(-.5,-5)^^(5.5,0)--(-5.5,0)); [/asy] [i]Proposed by Lewis Chen[/i]

2002 Romania Team Selection Test, 1

Tags: symmetry , trigonometry , geometry proposed , geometry

Let $ABCDE$ be a cyclic pentagon inscribed in a circle of centre $O$ which has angles $\angle B=120^{\circ},\angle C=120^{\circ},$ $\angle D=130^{\circ},\angle E=100^{\circ}$. Show that the diagonals $BD$ and $CE$ meet at a point belonging to the diameter $AO$. [i]Dinu Șerbănescu[/i]

2004 Tournament Of Towns, 6

Tags: geometry unsolved , geometry

Let n be a fixed prime number >3. A triangle is said to be admissible if the measure of each of its angles is of the form $\frac{m\cdot 180^{\circ}}{n}$ for some positive integer m. We are given one admissible triangle. Every minute we cut one of the triangles we already have into two admissible triangles so that no two of the triangles we have after cutting are similar. After some time, it turns out that no more cuttings are possible. Prove that at this moment, the triangles we have contain all possible admissible triangles (we do not distinguish between triangles which have same sets of angles, i.e. similar triangles).

2025 Poland - Second Round, 1

Tags: algebra

Determine all integers $n\ge 2$ with the following property: there exist nonzero real numbers $x_1, x_2, \ldots, x_n,y$ such that \[(x_1+x_2+\ldots+x_k)(x_{k+1}+x_{k+2}+\ldots+x_n)=y\] for all $k\in\{1,2,\ldots,n-1\}$.

2007 Today's Calculation Of Integral, 256

Tags: calculus , integration , trigonometry , calculus computations

Find the value of $ a$ for which $ \int_0^{\pi} \{ax(\pi ^ 2 \minus{} x^2) \minus{} \sin x\}^2dx$ is minimized.

1999 Chile National Olympiad, 4

Tags: combinatorics

Given a $ n \times n$ grid board . How many ways can an $X$ and an $O$ be placed in such a way that they are not in adjacent squares?

2009 CIIM, Problem 4

Tags: CIIM 2009 , CIIM , undergraduate

Let $m$ be a line in the plane and $M$ a point not in $m$. Find the locus of the focus of the parabolas with vertex $M$ that are tangent to $m$.

2004 Tournament Of Towns, 5

Tags: combinatorics unsolved , combinatorics

Two 10-digit integers are called neighbours if they diﬀer in exactly one digit (for example, integers $1234567890$ and $1234507890$ are neighbours). Find the maximal number of elements in the set of 10-digit integers with no two integers being neighbours.

2014-2015 SDML (High School), 13

Tags: 2014-15 SDML Round 1 , SDML High School , SDML Middle School

How many triangles formed by three vertices of a regular $17$-gon are obtuse? $\text{(A) }156\qquad\text{(B) }204\qquad\text{(C) }357\qquad\text{(D) }476\qquad\text{(E) }524$

2014 Mexico National Olympiad, 2

Tags: number theory proposed , number theory

A positive integer $a$ is said to [i]reduce[/i] to a positive integer $b$ if when dividing $a$ by its units digits the result is $b$. For example, 2015 reduces to $\frac{2015}{5} = 403$. Find all the positive integers that become 1 after some amount of reductions. For example, 12 is one such number because 12 reduces to 6 and 6 reduces to 1.

2003 Canada National Olympiad, 4

Tags: ratio , geometry unsolved , geometry

Prove that when three circles share the same chord $AB$, every line through $A$ different from $AB$ determines the same ratio $X Y : Y Z$, where $X$ is an arbitrary point different from $B$ on the ﬁrst circle while $Y$ and $Z$ are the points where AX intersects the other two circles (labeled so that $Y$ is between $X$ and $Z$).

2018 HMIC, 4

Tags: Harvard , college , MIT , combinatorics , HMIC

Find all functions $f: \mathbb{R}^+\to\mathbb{R}^+$ such that \[f(x+f(y+xy))=(y+1)f(x+1)-1\]for all $x,y\in\mathbb{R}^+$. ($\mathbb{R}^+$ denotes the set of positive real numbers.)

2019 AMC 12/AHSME, 11

Tags: AMC 10 , AMC 12 , AMC , 2019 AMC 12A , 2019 AMC 10A , 2019 AMC

For some positive integer $k$, the repeating base-$k$ representation of the (base-ten) fraction $\frac{7}{51}$ is $0.\overline{23}_k = 0.232323..._k$. What is $k$? $\textbf{(A) } 13 \qquad\textbf{(B) } 14 \qquad\textbf{(C) } 15 \qquad\textbf{(D) } 16 \qquad\textbf{(E) } 17$

2001 May Olympiad, 4

Tags: number theory , prime numbers

Using only prime numbers, a set is formed with the following conditions: Any one-digit prime number can be in the set. For a prime number with more than one digit to be in the set, the number that results from deleting only the first digit and also the number that results from deleting only the last digit must be in the set. Write, of the sets that meet these conditions, the one with the greatest number of elements. Justify why there cannot be one with more elements. Remember that the number $1$ is not prime.

2007 iTest Tournament of Champions, 3

Tags: iTest Tournament of Champions

Find the sum of all integers $n$ such that \[n^4+n^3+n^2+n+1\] is a perfect square.