Olympiad problems

2024 German National Olympiad, 2

Tags: 3d geometry , geometry

Six quadratic mirrors are put together to form a cube $ABCDEFGH$ with a mirrored interior. At each of the eight vertices, there is a tiny hole through which a laser beam can enter and leave the cube. A laser beam enters the cube at vertex $A$ in a direction not parallel to any of the cube's sides. If the beam hits a side, it is reflected; if it hits an edge, the light is absorbed, and if it hits a vertex, it leaves the cube. For each positive integer $n$, determine the set of vertices where the laser beam can leave the cube after exactly $n$ reflections.

2018 CCA Math Bonanza, L5.3

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Choose an integer $n$ from $1$ to $10$ inclusive as your answer to this problem. Let $m$ be the number of distinct values in $\left\{1,2,\ldots,10\right\}$ chosen by all teams at the Math Bonanza for this problem which are greater than or equal to $n$. Your score on this problem will be $\frac{mn}{15}$. For example, if $5$ teams choose $1$, $2$ teams choose $2$, and $6$ teams choose $3$ with these being the only values chosen, and you choose $2$, you will receive $\frac{4}{15}$ points. [i]2018 CCA Math Bonanza Lightning Round #5.3[/i]

2024 India IMOTC, 2

Tags: inequalities

Let $x_1, x_2 \dots, x_{2024}$ be non-negative real numbers such that $x_1 \le x_2\cdots \le x_{2024}$, and $x_1^3 + x_2^3 + \dots + x_{2024}^3 = 2024$. Prove that \[\sum_{1 \le i < j \le 2024} (-1)^{i+j} x_i^2 x_j \ge -1012.\] [i]Proposed by Shantanu Nene[/i]

2017 IMAR Test, 3

Tags: combinatorics

We consider $S$ a set of odd positive interger numbers with $n\geq 3$ elements such that no element divides another element. We say that a set $S$ is $beautiful$ if for any 3 elements from $S$, there is one the divides the sum of the other 2. We call a beautiful set $S$ $maximal$ if we can't add another number to the set such that $S$ will still be beautiful. Find the values of $n$ for which there exists a $maximal$ set.

2018 District Olympiad, 3

Tags: limit , convergence , real analysis , sequence

Let $(a_n)_{n\ge 1}$ be a sequence such that $a_n > 1$ and $a_{n+1}^2 \ge a_n a_{n + 2}$, for any $n\ge 1$. Show that the sequence $(x_n)_{n\ge 1}$ given by $x_n = \log_{a_n} a_{n + 1}$ for $n\ge 1$ is convergent and compute its limit.

2018 ISI Entrance Examination, 5

Tags: calculus

Let $f:\mathbb{R}\to\mathbb{R}$ be a differentiable function such that its derivative $f'$ is a continuous function. Moreover, assume that for all $x\in\mathbb{R}$, $$0\leqslant \vert f'(x)\vert\leqslant \frac{1}{2}$$ Define a sequence of real numbers $\{a_n\}_{n\in\mathbb{N}}$ by :$$a_1=1~~\text{and}~~a_{n+1}=f(a_n)~\text{for all}~n\in\mathbb{N}$$ Prove that there exists a positive real number $M$ such that for all $n\in\mathbb{N}$, $$\vert a_n\vert \leqslant M$$

1959 AMC 12/AHSME, 47

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Assume that the following three statements are true: $I$. All freshmen are human. $II$. All students are human. $III$. Some students think. Given the following four statements: $ \textbf{(1)}\ \text{All freshmen are students.}\qquad$ $\textbf{(2)}\ \text{Some humans think.}\qquad$ $\textbf{(3)}\ \text{No freshmen think.}\qquad$ $\textbf{(4)}\ \text{Some humans who think are not students.}$ Those which are logical consequences of I,II, and III are: $ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 2,3\qquad\textbf{(D)}\ 2,4\qquad\textbf{(E)}\ 1,2 $

2013 Estonia Team Selection Test, 1

Tags: number theory , system of equations , diophantine , diophantine equation

Find all prime numbers $p$ for which one can find a positive integer $m$ and nonnegative integers $a_0,a_1,...,a_m$ less than $p$ such that $$\begin{cases} a_0+a_1p+...+a_{m-1}p^{m-1}+a_{m}p^{m} = 2013 \\ a_0+a_1+...+a_{m-1}+a_{m} = 11\end{cases}$$

1975 AMC 12/AHSME, 12

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If $ a \neq b$, $ a^3 \minus{} b^3 \equal{} 19x^3$, and $ a\minus{}b \equal{} x$, which of the following conclusions is correct? $ \textbf{(A)}\ a\equal{}3x \qquad \textbf{(B)}\ a\equal{}3x \text{ or } a \equal{} \minus{}2x \qquad$ $ \textbf{(C)}\ a\equal{}\minus{}3x \text{ or } a \equal{} 2x \qquad \textbf{(D)}\ a\equal{}3x \text{ or } a\equal{}2x \qquad \textbf{(E)}\ a\equal{}2x$

1993 All-Russian Olympiad, 4

Tags: function , ceiling function , combinatorics

Thirty people sit at a round table. Each of them is either smart or dumb. Each of them is asked: "Is your neighbor to the right smart or dumb?" A smart person always answers correctly, while a dumb person can answer both correctly and incorrectly. It is known that the number of dumb people does not exceed $F$. What is the largest possible value of $F$ such that knowing what the answers of the people are, you can point at at least one person, knowing he is smart?

2004 Purple Comet Problems, 18

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Find the number of addition problems in which a two digit number is added to a second two digit number to give a two digit answer, such as in the three examples: \[\begin{tabular}{@{\hspace{3pt}}c@{\hspace{3pt}}}23\\42\\\hline 65\end{tabular}\,,\qquad\begin{tabular}{@{\hspace{3pt}}c@{\hspace{3pt}}}36\\36\\\hline 72\end{tabular}\,,\qquad\begin{tabular}{@{\hspace{3pt}}c@{\hspace{3pt}}}42\\23\\\hline 65\end{tabular}\,.\]

2002 AMC 12/AHSME, 23

Tags: geometry , trigonometry , perpendicular bisector , trig identities , law of sines , angle bisector , area of a triangle

In triangle $ ABC$, side $ AC$ and the perpendicular bisector of $ BC$ meet in point $ D$, and $ BD$ bisects $ \angle ABC$. If $ AD \equal{} 9$ and $ DC \equal{} 7$, what is the area of triangle $ ABD$? $ \textbf{(A)}\ 14 \qquad \textbf{(B)}\ 21 \qquad \textbf{(C)}\ 28 \qquad \textbf{(D)}\ 14\sqrt5 \qquad \textbf{(E)}\ 28\sqrt5$

2008 Thailand Mathematical Olympiad, 7

Tags: diophantine , diophantine equation , lcm , gcd

Two positive integers $m, n$ satisfy the two equations $m^2 + n^2 = 3789$ and $gcd (m, n) + lcm (m, n) = 633$. Compute $m + n$.

1936 Moscow Mathematical Olympiad, 027

Tags: algebra , system of equations , parameter

Solve the system $\begin{cases} x+y=a \\ x^5 +y^5 = b^5 \end{cases}$

2023 Belarus Team Selection Test, 4.3

Tags: inequalities , algebra

Let $n \geqslant 3$ be an integer, and let $x_1,x_2,\ldots,x_n$ be real numbers in the interval $[0,1]$. Let $s=x_1+x_2+\ldots+x_n$, and assume that $s \geqslant 3$. Prove that there exist integers $i$ and $j$ with $1 \leqslant i<j \leqslant n$ such that \[2^{j-i}x_ix_j>2^{s-3}.\]

2009 Belarus Team Selection Test, 2

Tags: pentagon , equal segments , geometry

Does there exist a convex pentagon $A_1A_2A_3A_4A_5$ and a point $X$ inside it such that $XA_i=A_{i+2}A_{i+3}$ for all $i=1,...,5$ (all indices are considered modulo $5$) ? I. Voronovich

2022 Thailand Online MO, 9

Tags: number theory

The number $1$ is written on the blackboard. At any point, Kornny may pick two (not necessary distinct) of the numbers $a$ and $b$ written on the board and write either $ab$ or $\frac{1}{a}+\frac{1}{b}+\frac{1}{ab}$ on the board as well. Determine all possible numbers that Kornny can write on the board in finitely many steps.

1973 Czech and Slovak Olympiad III A, 4

Tags: algebra , floor function , sum , sum of roots , function

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 $S$.