Olympiad problems

1973 Czech and Slovak Olympiad III A, 4

For any integer $n\ge2$ evaluate the sum \[\sum_{k=1}^{n^2-1}\bigl\lfloor\sqrt k\bigr\rfloor.\]

2013 National Chemistry Olympiad, 56

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All of the following are condensation polymers except: $ \textbf{(A) }\text{Nylon} \qquad\textbf{(B) }\text{Polyethylene}\qquad\textbf{(C) }\text{Protein} \qquad\textbf{(D) }\text{Starch}\qquad $

2014 CHMMC (Fall), 10

Tags: combinatorics , analytic geometry

Consider a grid of all lattice points $(m, n)$ with $m, n$ between $1$ and $125$. There exists a “path” between two lattice points $(m_1, n_1)$ and $(m_2, n_2)$ on the grid if $m_1n_1 = m_2n_2$ or if $m_1/n_1 = m_2/n_2$. For how many lattice points $(m, n)$ on the grid is there a sequence of paths that goes from $(1, 1)$ to $(m, n$)?

CIME II 2018, 14

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Positive rational numbers $x<y<z$ sum to $1$ and satisfy the equation $$(x^2+y^2+z^2-1)^3+8xyz=0.$$ Given that $\sqrt{z}$ is also rational, it can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. If $m+n < 1000$, find the maximum value of $m+n$. [I]Proposed by [b] Th3Numb3rThr33 [/b][/I]

2000 Singapore Team Selection Test, 1

Tags: geometry , circumcircle , angle bisector

In a triangle $ABC$, $AB > AC$, the external bisector of angle $A$ meets the circumcircle of triangle $ABC$ at $E$, and $F$ is the foot of the perpendicular from $E$ onto $AB$. Prove that $2AF = AB - AC$

2005 Junior Balkan Team Selection Tests - Romania, 12

Tags: modular arithmetic , number theory

Find all positive integers $n$ and $p$ if $p$ is prime and \[ n^8 - p^5 = n^2+p^2 . \] [i]Adrian Stoica[/i]

2007 Baltic Way, 6

Tags: algorithm , induction , combinatorics

Freddy writes down numbers $1, 2,\ldots ,n$ in some order. Then he makes a list of all pairs $(i, j)$ such that $1\le i<j\le n$ and the $i$-th number is bigger than the $j$-th number in his permutation. After that, Freddy repeats the following action while possible: choose a pair $(i, j)$ from the current list, interchange the $i$-th and the $j$-th number in the current permutation, and delete $(i, j)$ from the list. Prove that Freddy can choose pairs in such an order that, after the process finishes, the numbers in the permutation are in ascending order.

2013 Today's Calculation Of Integral, 874

Tags: calculus , integration , parabola , geometry , calculus computations , conic

Given a parabola $C : y=1-x^2$ in $xy$-palne with the origin $O$. Take two points $P(p,\ 1-p^2),\ Q(q,\ 1-q^2)\ (p<q)$ on $C$. (1) Express the area $S$ of the part enclosed by two segments $OP,\ OQ$ and the parabalola $C$ in terms of $p,\ q$. (2) If $q=p+1$, then find the minimum value of $S$. (3) If $pq=-1$, then find the minimum value of $S$.

2016 PUMaC Number Theory B, 3

Tags: number theory

For positive integers $i$ and $j$, define $d(i,j)$ as follows: $d(1,j) = 1, d(i,1) = 1$ for all $i$ and $j$, and for $i, j > 1$, $d(i,j) = d(i-1,j) + d(i,j-1) + d(i-1,j-1)$. Compute the remainder when $d(3,2016)$ is divided by $1000$.

1995 India National Olympiad, 3

Tags: symmetry , combinatorics

Show that the number of $3-$element subsets $\{ a , b, c \}$ of $\{ 1 , 2, 3, \ldots, 63 \}$ with $a+b +c < 95$ is less than the number of those with $a + b +c \geq 95.$

2007 Peru IMO TST, 1

Tags: incenter , geometry

Let $P$ be an interior point of the semicircle whose diameter is $AB$ ($\angle APB$ is obtuse). The incircle of $\triangle ABP$ touches $AP$ and $BP$ at $M$ and $N$ respectively. The line $MN$ intersects the semicircle in $X$ and $Y$. Prove that $\widehat{XY}= \angle APB$.

2012 Singapore MO Open, 1

Tags: trigonometry , geometry

The incircle with centre $I$ of the triangle $ABC$ touches the sides $BC, CA$ and $AB$ at $D, E, F$ respectively. The line $ID$ intersects the segment $EF$ at $K$. Proof that $A, K, M$ collinear, where $M$ is the midpoint of $BC$.

2003 National Olympiad First Round, 2

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How many prime divisors does the number $1\cdot 2003 + 2\cdot 2002 + 3\cdot 2001 + \cdots + 2001 \cdot 3 + 2002 \cdot 2 + 2003 \cdot 1$ have? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 7 $

1982 AMC 12/AHSME, 23

Tags: trigonometry , geometry , trig identities , law of sines , law of cosines

The lengths of the sides of a triangle are consescutive integers, and the largest angle is twice the smallest angle. The cosine of the smallest angle is $\textbf {(A) } \frac 34 \qquad \textbf {(B) } \frac{7}{10} \qquad \textbf {(C) } \frac 23 \qquad \textbf {(D) } \frac{9}{14} \qquad \textbf {(E) } \text{none of these}$

2014 Cuba MO, 2

Tags: algebra , inequalities

Let $a$ and $b$ be real numbers with $0 \le a, b \le 1$. (a) Prove that $ \frac{a}{b+1} +\frac{b}{a+1} \le 1.$ (b) Find the case of equality.

1996 Bundeswettbewerb Mathematik, 1

Tags: combinatorial geometry , geometry , combinatorics , cube

For a given set of points in space it is allowed to mirror a point from the set with respect to another point from the set, and to include the image in the set. Starting with a set of seven vertices of a cube, is it possible to include the eight vertex in the set after finitely many such steps?

2021 Indonesia TST, C

Tags: combinatorics

Several square-shaped papers are situated on a table such that every side of the paper is positioned parallel to the sides of the table. Each paper has a colour, and there are $n$ different coloured papers. It is known that for every $n$ papers with distinct colors, we can always find an overlapping pair of papers. Prove that, using $2n- 2$ nails, it is possible to hammer all the squares of a certain colour to the table.

1967 AMC 12/AHSME, 7

Tags: inequalities

If $\frac{a}{b}<-\frac{c}{d}$ where $a$, $b$, $c$, and $d$ are real numbers and $bd \not= 0$, then: $ \text{(A)}\ a \; \text{must be negative} \qquad \text{(B)}\ a \; \text{must be positive} \qquad$ $\text{(C)}\ a \; \text{must not be zero} \qquad \text{(D)}\ a \; \text{can be negative or zero, but not positive } \\ \text{(E)}\ a \; \text{can be positive, negative, or zero}$

2009 Sharygin Geometry Olympiad, 3

Tags: altitude , geometry , projection

Let $AH_a $ and $BH_b$ be the altitudes of triangle $ABC$. Points $P$ and $Q$ are the projections of $H_a$ to $AB$ and $AC$. Prove that line $PQ $ bisects segment $H_aH_b$. (A.Akopjan, K.Savenkov)

Russian TST 2018, P2

Tags: homothety , power of a point , excircle , geometry , tangent circle

In triangle $ABC$, let $\omega$ be the excircle opposite to $A$. Let $D, E$ and $F$ be the points where $\omega$ is tangent to $BC, CA$, and $AB$, respectively. The circle $AEF$ intersects line $BC$ at $P$ and $Q$. Let $M$ be the midpoint of $AD$. Prove that the circle $MPQ$ is tangent to $\omega$.

2023 All-Russian Olympiad, 7

Tags: algebra , polynomial

We call a polynomial $P(x)$ good if the numbers $P(k)$ and $P'(k)$ are integers for all integers $k$. Let $P(x)$ be a good polynomial of degree $d$, and let $N_d$ be the product of all composite numbers not exceeding $d$. Prove that the leading coefficient of the polynomial $N_d \cdot P(x)$ is integer.

2012 Putnam, 2

Tags: geometry , inequalities , 3d geometry , sphere , prism

Let $P$ be a given (non-degenerate) polyhedron. Prove that there is a constant $c(P)>0$ with the following property: If a collection of $n$ balls whose volumes sum to $V$ contains the entire surface of $P,$ then $n>c(P)/V^2.$

2016 Portugal MO, 4

Tags: tangent , geometry , parallelogram , equal angles , angle

Let $[ABCD]$ be a parallelogram with $AB <BC$ and let $E, F$ be points on the circle that passes through $A, B$ and $C$ such that $DE$ and $DF$ are tangents to this circle. Knowing that $\angle ADE = \angle CDF$ , determine $\angle ABC$. [img]https://cdn.artofproblemsolving.com/attachments/5/e/4140b92730e9d382df49ac05ca4e8ba48332dc.png[/img]

1995 National High School Mathematics League, 10

Tags: calculus , integration

The number of integral points satisfy $\begin{cases} y\leq 3x\\ y\geq \frac{x}{3}\\ x+y\geq100 \end{cases}$ on the coordinate plane is________.

2006 Korea Junior Math Olympiad, 2

Tags: integer , positive integer , number theory , coprime

Find all positive integers that can be written in the following way $\frac{b}{a}+\frac{c}{a}+\frac{c}{b}+\frac{a}{b}+\frac{a}{c}+\frac{b}{c}$ . Also, $a,b, c$ are positive integers that are pairwise relatively prime.