Olympiad problems

2016 LMT, 5

Tags:

An isosceles triangle has angles of $50^\circ,x^\circ,$ and $y^\circ$. Find the maximum possible value of $x-y$. [i]Proposed by Nathan Ramesh

2022 Bulgarian Spring Math Competition, Problem 9.1

Let $f(x)$ be a quadratic function with integer coefficients. If we know that $f(0)$, $f(3)$ and $f(4)$ are all different and elements of the set $\{2, 20, 202, 2022\}$, determine all possible values of $f(1)$.

2018 Yasinsky Geometry Olympiad, 6

Tags: projection , geometry , angle chasing , equal angles , altitude

$AH$ is the altitude of the acute triangle $ABC$, $K$ and $L$ are the feet of the perpendiculars, from point $H$ on sides $AB$ and $AC$ respectively. Prove that the angles $BKC$ and $BLC$ are equal.

2023 Yasinsky Geometry Olympiad, 6

Tags: geometry , equilateral

An acute triangle $ABC$ is surrounded by equilateral triangles $KLM$ and $PQR$ such that its vertices lie on the sides of these equilateral triangle as shown on the picture. Lines $PK$ and $QL$ intersect at point $D$. Prove that $\angle ABC + \angle PDQ = 120^o$. (Yurii Biletskyi) [img]https://cdn.artofproblemsolving.com/attachments/4/6/32d3f74f07ca6a8edcabe4a08aa321eb3a5010.png[/img]

1986 Traian Lălescu, 2.1

Tags: combinatorics , algebra

Consider the numbers $ a_n=1-\binom{n}{3} +\binom{n}{6} -\cdots, b_n= -\binom{n}{1} +\binom{n}{4}-\binom{n}{7} +\cdots $ and $ c_n=\binom{n}{2} -\binom{n}{5} +\binom{n}{8} -\cdots , $ for a natural number $ n\ge 2. $ Prove that $$ a_n^2+b_n^2+c_n^2-a_nb_n-b_nc_n-c_na_n =3^{n-1}. $$

2005 Georgia Team Selection Test, 7

Tags: number theory

Determine all positive integers $ n$, for which $ 2^{n\minus{}1}n\plus{}1$ is a perfect square.

2020 Sharygin Geometry Olympiad, 17

Tags: geometry

Chords $A_1A_2$ and $B_1B_2$ meet at point $D$. Suppose $D'$ is the inversion image of $D$ and the line $A_1B_1$ meets the perpendicular bisector to $DD'$ at a point $C$. Prove that $CD\parallel A_2B_2$.

2020 CHMMC Winter (2020-21), 7

Tags: combinatorics

Given $10$ points on a plane such that no three are collinear, we connect each pair of points with a segment and color each segment either red or blue. Assume that there exists some point $A$ among the $10$ points such that: 1. There is an odd number of red segments connected to $A$} 2. The number of red segments connected to each of the other points are all different Find the number of red triangles (i.e, a triangle whose three sides are all red segments) on the plane.

2000 Harvard-MIT Mathematics Tournament, 10

Tags: geometry , algebra

How many times per day do at least two of the three hands on a clock coincide?

2024 Korea Junior Math Olympiad, 4

Tags: number theory

find all positive integer n such that there exists positive integers (a,b) such that (a^n + b^n)/n! is a positive integer smaller than 101

2000 Putnam, 5

Tags: induction

Let $S_0$ be a finite set of positive integers. We define finite sets $S_1, S_2, \cdots$ of positive integers as follows: the integer $a$ in $S_{n+1}$ if and only if exactly one of $a-1$ or $a$ is in $S_n$. Show that there exist infinitely many integers $N$ for which $S_N = S_0 \cup \{ N + a: a \in S_0 \}$.

2005 SNSB Admission, 3

Tags: complex analysis , function , complex number , algebra , polynomial

Let $ f:\mathbb{C}\longrightarrow\mathbb{C} $ be an holomorphic function which has the property that there exist three positive real numbers $ a,b,c $ such that $ |f(z)|\geqslant a|z|^b , $ for any complex numbers $ z $ with $ |z|\geqslant c. $ Prove that $ f $ is polynomial with degree at least $ \lceil b\rceil . $

2021 AMC 10 Fall, 14

Tags: algebra , system of equations , absolute value

How many ordered pairs $(x,y)$ of real numbers satisfy the following system of equations? \begin{align*} x^2+3y&=9\\ (|x|+|y|-4)^2&=1\\ \end{align*} $\textbf{(A)}\: 1\qquad\textbf{(B)} \: 2\qquad\textbf{(C)} \: 3\qquad\textbf{(D)} \: 5\qquad\textbf{(E)} \: 7$

2005 Purple Comet Problems, 1

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A cubic inch of the newly discovered material madelbromium weighs $5$ ounces. How many pounds will a cubic yard of madelbromium weigh?

1992 National High School Mathematics League, 14

Tags: 3d geometry , geometric transformation , reflection , geometry

$l,m$ are skew lines. Three points $A,B,C$ on line $l$ satisfy that $AB=BC$. Projection of $A,B,C$ on $m$ are $D,E,F$. If $|AD|=\sqrt{15},|BE|=\frac{7}{2}|CF|=\sqrt{10}$, find the distance between $l$ and $m$.

2020 USMCA, 5

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Call a positive integer $n$ an $A-B$ number if the base $A$ and base $B$ representations of $n$ are three-digit numbers that are reverses of each other. For example, $87$ is a $5-6$ number because $87 = 223_6 = 322_5$. Compute the sum of all $7-11$ numbers.

2009 Italy TST, 1

Tags: polynomial , algebra

Let $n$ be an even positive integer. An $n$-degree monic polynomial $P(x)$ has $n$ real roots (not necessarily distinct). Suppose $y$ is a positive real number such that for any real number $t<y$, we have $P(t)>0$. Prove that \[P(0)^{\frac{1}{n}}-P(y)^{\frac{1}{n}}\ge y.\]

1994 India National Olympiad, 5

Tags: rectangle , geometry

A circle passes through the vertex of a rectangle $ABCD$ and touches its sides $AB$ and $AD$ at $M$ and $N$ respectively. If the distance from $C$ to the line segment $MN$ is equal to $5$ units, find the area of rectangle $ABCD$.

2018 May Olympiad, 5

Tags: combinatorics

In each square of a $5 \times 5$ board one of the numbers $2, 3, 4$ or $5$ is written so that the the sum of all the numbers in each row, in each column and on each diagonal is always even. How many ways can we fill the board? Clarification. A $5\times 5$ board has exactly $18$ diagonals of different sizes. In particular, the corners are size $ 1$ diagonals.

2020 CCA Math Bonanza, L5.4

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Submit a positive integer less than or equal to $15$. Your goal is to submit a number that is close to the number of teams submitting it. If you submit $N$ and the total number of teams at the competition (including your own team) who submit $N$ is $T$, your score will be $\frac{2}{0.5|N-T|+1}$. [i]2020 CCA Math Bonanza Lightning Round #5.4[/i]

2001 District Olympiad, 3

Tags: function , algebra , polynomial , integration , calculus , real analysis

Consider a continuous function $f:[0,1]\rightarrow \mathbb{R}$ such that for any third degree polynomial function $P:[0,1]\to [0,1]$, we have \[\int_0^1f(P(x))dx=0\] Prove that $f(x)=0,\ (\forall)x\in [0,1]$. [i]Mihai Piticari[/i]

2010 Today's Calculation Of Integral, 532

Tags: calculus , integration , geometry , trigonometry , calculus computations

For a curve $ C: y \equal{} x\sqrt {9 \minus{} x^2}\ (x\geq 0)$, (1) Find the maximum value of the function. (2) Find the area of the figure bounded by the curve $ C$ and the $ x$-axis. (3) Find the volume of the solid by revolution of the figure in (2) around the $ y$-axis. Please find the volume without using cylindrical shells for my students. Last Edited.

1979 Bulgaria National Olympiad, Problem 2

Tags: 3d geometry , tetrahedron , geometry

Points $P,Q,R,S$ are taken on respective edges $AC$, $AB$, $BD$, and $CD$ of a tetrahedron $ABCD$ so that $PR$ and $QS$ intersect at point $N$ and $PS$ and $QR$ intersect at point $M$. The line $MN$ meets the plane $ABC$ at point $L$. Prove that the lines $AL$, $BP$, and $CQ$ are concurrent.

2019 HMNT, 7

Tags: geometry

Carl only eats food in the shape of equilateral pentagons. Unfortunately, for dinner he receives a piece of steak in the shape of an equilateral triangle. So that he can eat it, he cuts off two corners with straight cuts to form an equilateral pentagon. The set of possible perimeters of the pentagon he obtains is exactly the interval $[a, b)$, where $a$ and $b$ are positive real numbers. Compute $\frac{a}{b}$ .

2020 China Second Round Olympiad, 1

Tags: geometry , incenter

In triangle $ABC,$ $AB=BC,$ and let $I$ be the incentre of $\triangle ABC.$ $M$ is the midpoint of segment $BI.$ $P$ lies on segment $AC,$ such that $AP=3PC.$ $H$ lies on line $PI,$ such that $MH\perp PH.$ $Q$ is the midpoint of the arc $AB$ of the circumcircle of $\triangle ABC$. Prove that $BH\perp QH.$