Olympiad problems

2010 ISI B.Stat Entrance Exam, 4

A real valued function $f$ is defined on the interval $(-1,2)$. A point $x_0$ is said to be a fixed point of $f$ if $f(x_0)=x_0$. Suppose that $f$ is a differentiable function such that $f(0)>0$ and $f(1)=1$. Show that if $f'(1)>1$, then $f$ has a fixed point in the interval $(0,1)$.

2011 Serbia National Math Olympiad, 1

Tags: circumcircle , parallelogram , geometric transformation , reflection , geometry

On sides $AB, AC, BC$ are points $M, X, Y$, respectively, such that $AX=MX$; $BY=MY$. $K$, $L$ are midpoints of $AY$ and $BX$. $O$ is circumcenter of $ABC$, $O_1$, $O_2$ are symmetric with $O$ with respect to $K$ and $L$. Prove that $X, Y, O_1, O_2$ are concyclic.

2004 239 Open Mathematical Olympiad, 1

Tags: polynomial , algebra , function

Given non-constant linear functions $p_1(x), p_2(x), \dots p_n(x)$. Prove that at least $n-2$ of polynomials $p_1p_2\dots p_{n-1}+p_n, p_1p_2\dots p_{n-2} p_n + p_{n-1},\dots p_2p_3\dots p_n+p_1$ have a real root.

2021 USA TSTST, 5

Tags: tree , graph theory , combinatorics

Let $T$ be a tree on $n$ vertices with exactly $k$ leaves. Suppose that there exists a subset of at least $\frac{n+k-1}{2}$ vertices of $T$, no two of which are adjacent. Show that the longest path in $T$ contains an even number of edges. [hide=*]A tree is a connected graph with no cycles. A leaf is a vertex of degree 1[/hide] [i]Vincent Huang[/i]

1990 Putnam, B2

Tags: induction

Prove that for $ |x| < 1 $, $ |z| > 1 $, \[ 1 + \displaystyle\sum_{j=1}^{\infty} \left( 1 + x^j \right) P_j = 0, \]where $P_j$ is \[ \dfrac {(1-z)(1-zx)(1-zx^2) \cdots (1-zx^{j-1})}{(z-x)(z-x^2)(z-x^3)\cdots(z-x^j)}. \]

Swiss NMO - geometry, 2015.8

Tags: geometry , trapezoid , concurrency

Let $ABCD$ be a trapezoid, where $AB$ and $CD$ are parallel. Let $P$ be a point on the side $BC$. Show that the parallels to $AP$ and $PD$ intersect through $C$ and $B$ to $DA$, respectively.

2006 Putnam, B2

Tags: inequalities , pigeonhole principle

Prove that, for every set $X=\{x_{1},x_{2},\dots,x_{n}\}$ of $n$ real numbers, there exists a non-empty subset $S$ of $X$ and an integer $m$ such that \[\left|m+\sum_{s\in S}s\right|\le\frac1{n+1}\]

2018 HMNT, 9

Tags: probability , expected value

$20$ players are playing in a Super Mario Smash Bros. Melee tournament. They are ranked $1-20$, and player $n$ will always beat player $m$ if $n<m$. Out of all possible tournaments where each player plays $18$ distinct other players exactly once, one is chosen uniformly at random. Find the expected number of pairs of players that win the same number of games.

2023 Azerbaijan IZhO TST, 4

Tags: number theory

A positive integer $t$ is called a Jane's integer if $t = x^3+y^2$ for some positive integers $x$ and $y$. Prove that for every integer $n \ge 2$ there exist infinitely many positive integers $m$ such that the set of $n^2$ consecutive integers $\{m+1,m+2,\dotsc,m+n^2\}$ contains exactly $n + 1$ Jane's integers.

May Olympiad L1 - geometry, 2005.4

Tags: rectangle , equilateral triangle , paper , geometry

There are two paper figures: an equilateral triangle and a rectangle. The height of rectangle is equal to the height of the triangle and the base of the rectangle is equal to the base of the triangle. Divide the triangle into three parts and the rectangle into two, using straight cuts, so that with the five pieces can be assembled, without gaps or overlays, a equilateral triangle. To assemble the figure, each part can be rotated and / or turned around.

2025 Kosovo National Mathematical Olympiad`, P2

Tags: inequality , algebra , inequalities

Let $x$ and $y$ be real numbers where at least one of them is bigger than $2$ and $xy+4 > 2(x+y)$ holds. Show that $xy>x+y$.

2024 UMD Math Competition Part I, #25

Tags: geometry

An equilateral triangle $T$ and a circle $C$ are on the same plane. Suppose each side length of $T$ is $6\sqrt3$ and the radius of $C$ is $2.$ The distance between the centers of $T$ and $C$ is $15.$ For every two points $X$ on $T$ and $Y$ on $C,$ let $M(X, Y)$ be the midpoint of segment $\overline{XY}.$ The points $M(X, Y)$ as $X$ varies on $T$ and $Y$ varies on $C$ create a region whose area is $A.$ Find $A.$ \[\mathrm a. ~\pi + 14\sqrt3 \qquad \mathrm b. ~3\pi + 10\sqrt3 \qquad \mathrm c. ~4\pi+9\sqrt3 \qquad\mathrm d. ~\pi + 15\sqrt3 \qquad\mathrm e. ~4\pi+6\sqrt3\]

2014 France Team Selection Test, 3

Tags: number theory , prime divisor , polynomial

Prove that there exist infinitely many positive integers $n$ such that the largest prime divisor of $n^4 + n^2 + 1$ is equal to the largest prime divisor of $(n+1)^4 + (n+1)^2 +1$.

1986 Tournament Of Towns, (113) 7

Tags: combinatorics

Thirty pupils from the same class decided to exchange visits. Any pupil may make several visits during one evening, but must stay home if he is receiving guests that evening. Prove that in order that each pupil visit each of his classmates (a) four evenings are not enough (b) five evenings are not enough (c) ten evenings are enough (d) even seven evenings are enough

2020 USMCA, 6

Tags:

Let $P$ be a non-constant polynomial with integer coefficients such that if $n$ is a perfect power, so is $P(n)$. Prove that $P(x) = x$ or $P$ is a perfect power of a polynomial with integer coefficients. A perfect power is an integer $n^k$, where $n \in \mathbb Z$ and $k \ge 2$. A perfect power of a polynomial is a polynomial $P(x)^k$, where $P$ has integer coefficients and $k \ge 2$.

Kyiv City MO Seniors 2003+ geometry, 2014.10.4.1

Tags: geometry , angle bisector , angle

In the triangle $ABC$ the side $AC = \tfrac {1} {2} (AB + BC) $, $BL$ is the bisector $\angle ABC$, $K, \, \, M $ - the midpoints of the sides $AB$ and $BC$, respectively. Find the value $\angle KLM$ if $\angle ABC = \beta$

1998 Harvard-MIT Mathematics Tournament, 3

Tags: calculus , geometry , asymptote , integration

Find the area of the region bounded by the graphs $y=x^2$, $y=x$, and $x=2$.

1997 IMO Shortlist, 3

Tags: vector , combinatorics , combinatorial geometry , extremal combinatorics , geometry

For each finite set $ U$ of nonzero vectors in the plane we define $ l(U)$ to be the length of the vector that is the sum of all vectors in $ U.$ Given a finite set $ V$ of nonzero vectors in the plane, a subset $ B$ of $ V$ is said to be maximal if $ l(B)$ is greater than or equal to $ l(A)$ for each nonempty subset $ A$ of $ V.$ (a) Construct sets of 4 and 5 vectors that have 8 and 10 maximal subsets respectively. (b) Show that, for any set $ V$ consisting of $ n \geq 1$ vectors the number of maximal subsets is less than or equal to $ 2n.$

2022 District Olympiad, P4

Tags:

Find all functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy the following relationship for all $x,y\in\mathbb{R}:$\[f(f(y-x)-xf(y))+f(x)=y\cdot(1-f(x)).\]

2012 Online Math Open Problems, 17

Tags:

Find the number of integers $a$ with $1\le a\le 2012$ for which there exist nonnegative integers $x,y,z$ satisfying the equation \[x^2(x^2+2z) - y^2(y^2+2z)=a.\] [i]Ray Li.[/i] [hide="Clarifications"][list=1][*]$x,y,z$ are not necessarily distinct.[/list][/hide]

2023 MMATHS, 12

Tags:

Let $ABC$ be a triangle with incenter $I,$ circumcenter $O,$ and $A$-excenter $J_A.$ The incircle of $\triangle{ABC}$ touches side $BC$ at a point $D.$ Lines $OI$ and $J_AD$ meet at a point $K.$ Line $AK$ meets the circumcircle of $\triangle{ABC}$ again at a point $L \neq A.$ If $BD=11, CD=5,$ and $AO=10,$ the length of $DL$ can be expressed as $\tfrac{m\sqrt{p}}{n},$ where $m,n,p$ are positive integers, $m$ and $n$ are relatively prime, and $p$ is not divisible by the square of any prime. Find $m+n+p.$

2022 China Team Selection Test, 6

Tags: number theory , divisibility , divisor

Given a positive integer $n$, let $D$ be the set of all positive divisors of $n$. The subsets $A,B$ of $D$ satisfies that for any $a \in A$ and $b \in B$, it holds that $a \nmid b$ and $b \nmid a$. Show that \[ \sqrt{|A|}+\sqrt{|B|} \le \sqrt{|D|}. \]

2001 Romania National Olympiad, 4

Tags: 3d geometry , geometry

In the cube $ABCDA'B'C'D'$, with side $a$, the plane $(AB'D')$ intersects the planes $(A'BC),(A'CD),(A'DB)$ after the lines $d_1,d_2$ and $d_3$ respectively. a) Show that the lines $d_1,d_2,d_3$ intersect pairwise. b) Determine the area of the triangle formed by these three lines.

2008 Mathcenter Contest, 10

Tags: combinatorics , this problem is a nightmare

One test is a multiple choice test with $5$ questions, each with $4$ options, $2000$ candidates, each choosing only one answer for each item.Find the smallest possible integer $n$ that gives a student's answer sheet the following properties: In the student's answer sheet $n$, there are four sheets in it. Any two of the four tiles have exactly the same three answers. [i](tatari/nightmare)[/i]

2014 Iran MO (3rd Round), 2

Tags: geometric transformation , reflection , circumcircle , angle bisector , geometry

$\triangle{ABC}$ is isosceles$(AB=AC)$. Points $P$ and $Q$ exist inside the triangle such that $Q$ lies inside $\widehat{PAC}$ and $\widehat{PAQ} = \frac{\widehat{BAC}}{2}$. We also have $BP=PQ=CQ$.Let $X$ and $Y$ be the intersection points of $(AP,BQ)$ and $(AQ,CP)$ respectively. Prove that quadrilateral $PQYX$ is cyclic. [i](20 Points)[/i]