Olympiad problems

2012 Today's Calculation Of Integral, 790

Define a parabola $C$ by $y=x^2+1$ on the coordinate plane. Let $s,\ t$ be real numbers with $t<0$. Denote by $l_1,\ l_2$ the tangent lines drawn from the point $(s,\ t)$ to the parabola $C$. (1) Find the equations of the tangents $l_1,\ l_2$. (2) Let $a$ be positive real number. Find the pairs of $(s,\ t)$ such that the area of the region enclosed by $C,\ l_1,\ l_2$ is $a$.

2022 AMC 12/AHSME, 4

Tags: number theory

The least common multiple of a positive integer $n$ and 18 is 180, and the greatest common divisor of $n$ and 45 is 15. What is the sum of the digits of $n$? $\textbf{(A) }3\qquad\textbf{(B) }6\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }12$

2004 Germany Team Selection Test, 1

Tags: geometry , circles , intersection

Let $D_1$, $D_2$, ..., $D_n$ be closed discs in the plane. (A closed disc is the region limited by a circle, taken jointly with this circle.) Suppose that every point in the plane is contained in at most $2003$ discs $D_i$. Prove that there exists a disc $D_k$ which intersects at most $7\cdot 2003 - 1 = 14020$ other discs $D_i$.

1973 Canada National Olympiad, 7

Tags: induction

Observe that \[\frac{1}{1}= \frac{1}{2}+\frac{1}{2};\quad \frac{1}{2}=\frac{1}{3}+\frac{1}{6};\quad \frac{1}{3}=\frac{1}{4}+\frac{1}{12};\quad \frac{1}{4}= \frac{1}{5}+\frac{1}{20}. \] State a general law suggested by these examples, and prove it. Prove that for any integer $n$ greater than 1 there exist positive integers $i$ and $j$ such that \[\frac{1}{n}= \frac{1}{i(i+1)}+\frac{1}{(i+1)(i+2)}+\frac{1}{(i+2)(i+3)}+\cdots+\frac{1}{j(j+1)}. \] [hide="Remark."] It seems that this is a two-part problem. [/hide]

2023 Miklós Schweitzer, 4

Tags: polynomial , analysis

Determine the pairs of sets $X,Y\subset\mathbb{R}$ for which the following is true: if $f(x, y)$ is a function on $X\times Y{}$ such that for every $x\in X$ it is equal to a polynomial in $y$ on $Y$ and for every $y\in Y$ it is equal to a polynomial in $x$ on $X$ then $f$ is a bivariate polynomial on $X\times Y.$

2010 Today's Calculation Of Integral, 617

Tags: calculus , integration , function , calculus computations

Let $y=f(x)$ be a function of the graph of broken line connected by points $(-1,\ 0),\ (0,\ 1),\ (1,\ 4)$ in the $x$ -$y$ plane. Find the minimum value of $\int_{-1}^1 \{f(x)-(a|x|+b)\}^2dx.$ [i]2010 Tohoku University entrance exam/Economics, 2nd exam[/i]

1993 Miklós Schweitzer, 2

Tags: number theory

Let A be a subset of natural numbers and let k , r be positive integers. Suppose that for any r different elements selected from A , their greatest common divisor has at most k different prime factors. Prove that A can be partitioned into B and C , where any element of B has at most k + 1 different prime divisors and $$\sum_{n\in C} \frac{1}{n} <\infty$$

2005 Sharygin Geometry Olympiad, 15

Tags: geometry , lattice points , minimum , radius , circles

Given a circle centered at the origin. Prove that there is a circle of smaller radius that has no less points with integer coordinates.

2011 ELMO Shortlist, 6

Tags: algebra , polynomial , counting , derangement , induction , geometry , geometric transformation

Let $Q(x)$ be a polynomial with integer coefficients. Prove that there exists a polynomial $P(x)$ with integer coefficients such that for every integer $n\ge\deg{Q}$, \[\sum_{i=0}^{n}\frac{!i P(i)}{i!(n-i)!} = Q(n),\]where $!i$ denotes the number of derangements (permutations with no fixed points) of $1,2,\ldots,i$. [i]Calvin Deng.[/i]

2024 AMC 10, 1

Tags:

In a long line of people, the 1013th person from the left is also the 1010th person from the right. How many people are in the line? $ \textbf{(A) }2021 \qquad \textbf{(B) }2022 \qquad \textbf{(C) }2023 \qquad \textbf{(D) }2024 \qquad \textbf{(E) }2025 \qquad $

2001 Vietnam National Olympiad, 1

Tags: ratio , power of a point , radical axis , perpendicular bisector , geometry

A circle center $O$ meets a circle center $O'$ at $A$ and $B.$ The line $TT'$ touches the first circle at $T$ and the second at $T'$. The perpendiculars from $T$ and $T'$ meet the line $OO'$ at $S$ and $S'$. The ray $AS$ meets the first circle again at $R$, and the ray $AS'$ meets the second circle again at $R'$. Show that $R, B$ and $R'$ are collinear.

1988 IMO Longlists, 53

Tags: geometry

Given $n$ points $A_1, A_2, \ldots, A_n,$ no three collinear, show that the $n$- gon $A_1 A_2 \ldots A_n,$ is inscribed in a circle if and only if $A_1 A_2 \cdot A_3 A_n \cdot \ldots \cdot A_{n-1} A_n + A_2 A_3 \cdot A_4 A_n \cdot \ldots A_{n-1} A_n \cdot A_1 A_n + \ldots$ $+ A_{n-1} A_{n-2} \cdot A_1 A_n \cdot \ldots \cdot A_{n-3} A_n$ $= A_1 A_{n-1} \cdot A_2 A_n \cdot \ldots \cdot A_{n-2} A_n$, where $XY$ denotes the length of the segment $XY.$

1989 AIME Problems, 13

Tags:

Let $S$ be a subset of $\{1,2,3,\ldots,1989\}$ such that no two members of $S$ differ by $4$ or $7$. What is the largest number of elements $S$ can have?

2017 Balkan MO Shortlist, C2

Tags: lattice points , combinatorics , number theory , congruent triangles , coloring

Let $n,a,b,c$ be natural numbers. Every point on the coordinate plane with integer coordinates is colored in one of $n$ colors. Prove there exists $c$ triangles whose vertices are colored in the same color, which are pairwise congruent, and which have a side whose lenght is divisible by $a$ and a side whose lenght is divisible by $b$.

2011 South africa National Olympiad, 1

Tags:

Consider the sequence $2, 3, 5, 6, 7, 8, 10, ...$ of all positive integers that are not perfect squares. Determine the $2011^{th}$ term of the sequence.

2019 Malaysia National Olympiad, 4

Tags: number theory

Let $A=\{1,2,...,100\}$ and $f(k), k\in N$ be the size of the largest subset of $A$ such that no two elements differ by $k$. How many solutions are there to $f(k)=50$?

2013 BMT Spring, P2

Tags: number theory

Let $p$ be an odd prime, and let $(p^p)!=mp^k$ for some positive integers $m$ and $k$. Find in terms of $p$ the number of ordered pairs $(m,k)$ satisfying $m+k\equiv0\pmod p$.

2001 Slovenia National Olympiad, Problem 2

Tags: inequality , algebra

Tina wrote a positive number on each of five pieces of paper. She did not say which numbers she wrote, but revealed their pairwise sums instead: $17,20,28,14,42,36,28,39,25,31$. Which numbers did she write?

1982 Swedish Mathematical Competition, 4

Tags: geometry , equal segments

$ABC$ is a triangle with $AB = 33$, $AC = 21$ and $BC = m$, an integer. There are points $D$, $E$ on the sides $AB$, $AC$ respectively such that $AD = DE = EC = n$, an integer. Find $m$.

1997 Bosnia and Herzegovina Team Selection Test, 2

Tags: geometry , circumcircle , incenter , angle chasing , cyclic quadrilateral

In isosceles triangle $ABC$ with base side $AB$, on side $BC$ it is given point $M$. Let $O$ be a circumcenter and $S$ incenter of triangle $ABC$. Prove that $$ SM \mid \mid AC \Leftrightarrow OM \perp BS$$

2021 AMC 12/AHSME Fall, 20

Tags: function

For each positive integer $n$, let $f_1(n)$ be twice the number of positive integer divisors of $n$, and for $j \ge 2$, let $f_j(n) = f_1(f_{j-1}(n))$. For how many values of $n \le 50$ is $f_{50}(n) = 12?$ $\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }9\qquad\textbf{(D) }10\qquad\textbf{(E) }11$

2022 AMC 12/AHSME, 15

Tags: algebra

The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism. A new rectangular box is formed by lengthening each edge of the original box by 2 units. What is the volume of the new box? $\textbf{(A) }\frac{24}{5}\qquad\textbf{(B) }\frac{42}{5}\qquad\textbf{(C) }\frac{81}{5}\qquad\textbf{(D) }30\qquad\textbf{(E) }48$

2024-IMOC, N3

Tags: number theory

Find all positive integers $n$ such that $$n(2^n-1)$$ is a perfect square

2010 N.N. Mihăileanu Individual, 1

Tags: conic , quadratic , algebra , polynomial

Let be two real reducible quadratic polynomials $ P,Q $ in one variable. Prove that if $ P-Q $ is irreducible, then $ P+Q $ is reducible.

2009 China Team Selection Test, 4

Tags: combinatorics

Let positive real numbers $ a,b$ satisfy $ b \minus{} a > 2.$ Prove that for any two distinct integers $ m,n$ belonging to $ [a,b),$ there always exists non-empty set $ S$ consisting of certain integers belonging to $ [ab,(a \plus{} 1)(b \plus{} 1))$ such that $ \frac {\displaystyle\prod_{x\in S}}{mn}$ is square of a rational number.