Olympiad problems

1985 Tournament Of Towns, (100) 4

Tags: combinatorics , time

Two chess players play each other at chess using clocks (when a player makes a move , the player stops his clock and starts the clock of his opponent) . It is known that when both players have just completed their $40$th move , both of their clocks read exactly $2$ hr $30$ min . Prove that there was a moment in the game when the clock of one player registered $1$ min $51$ sec less than that of the other . Furthermore , can one assert that the difference between the two clock readings was ever equal to $2$ minutes? (S . Fomin , Leningrad)

2014 NIMO Problems, 3

Tags: trigonometry , analytic geometry , symmetry , geometry , number theory , relatively prime , similar triangles

Let $ABCD$ be a square with side length $2$. Let $M$ and $N$ be the midpoints of $\overline{BC}$ and $\overline{CD}$ respectively, and let $X$ and $Y$ be the feet of the perpendiculars from $A$ to $\overline{MD}$ and $\overline{NB}$, also respectively. The square of the length of segment $\overline{XY}$ can be written in the form $\tfrac pq$ where $p$ and $q$ are positive relatively prime integers. What is $100p+q$? [i]Proposed by David Altizio[/i]

2015 Bosnia And Herzegovina - Regional Olympiad, 4

Tags: combinatorics , set

Alice and Mary were searching attic and found scale and box with weights. When they sorted weights by mass, they found out there exist $5$ different groups of weights. Playing with the scale and weights, they discovered that if they put any two weights on the left side of scale, they can find other two weights and put on to the right side of scale so scale is in balance. Find the minimal number of weights in the box

1997 Pre-Preparation Course Examination, 1

Tags: polynomial , greatest common divisor , algebra

Let $n$ be a positive integer. Prove that there exist polynomials$f(x)$and $g(x$) with integer coefficients such that \[f(x)\left(x + 1 \right)^{2^n}+ g(x) \left(x^{2^n}+ 1 \right) = 2.\]

2007 Tournament Of Towns, 5

Tags: combinatorics

Attached to each of a number of objects is a tag which states the correct mass of the object. The tags have fallen off and have been replaced on the objects at random. We wish to determine if by chance all tags are in fact correct. We may use exactly once a horizontal lever which is supported at its middle. The objects can be hung from the lever at any point on either side of the support. The lever either stays horizontal or tilts to one side. Is this task always possible?

2023 CMIMC Geometry, 5

Tags: geometry

In trapezoid $ABCD, AB=3, BC=2, CD=5,$ and $\angle B = \angle C = 90^{\circ}.$ The angle bisectors of $\angle A$ and $\angle D$ intersect at a point $P$ in the interior of $ABCD.$ Compute $BP^2+CP^2.$ [i]Proposed by Kyle Lee[/i]

2013 Balkan MO, 1

Tags: geometry , incenter , trigonometry , perimeter , circumcircle

In a triangle $ABC$, the excircle $\omega_a$ opposite $A$ touches $AB$ at $P$ and $AC$ at $Q$, while the excircle $\omega_b$ opposite $B$ touches $BA$ at $M$ and $BC$ at $N$. Let $K$ be the projection of $C$ onto $MN$ and let $L$ be the projection of $C$ onto $PQ$. Show that the quadrilateral $MKLP$ is cyclic. ([i]Bulgaria[/i])

2010 Princeton University Math Competition, 1

Tags:

Let the operation $\bigstar$ be defined by $x\bigstar y=y^x-xy$. Calculate $(3\bigstar4)-(4\bigstar3)$.

2021 European Mathematical Cup, 2

Tags: trisector , geometry , circumcircle

Let $ABC$ be an acute-angled triangle such that $|AB|<|AC|$. Let $X$ and $Y$ be points on the minor arc ${BC}$ of the circumcircle of $ABC$ such that $|BX|=|XY|=|YC|$. Suppose that there exists a point $N$ on the segment $\overline{AY}$ such that $|AB|=|AN|=|NC|$. Prove that the line $NC$ passes through the midpoint of the segment $\overline{AX}$. \\ \\ (Ivan Novak)

2017 China Team Selection Test, 6

Tags: combinatorics

Every cell of a $2017\times 2017$ grid is colored either black or white, such that every cell has at least one side in common with another cell of the same color. Let $V_1$ be the set of all black cells, $V_2$ be the set of all white cells. For set $V_i (i=1,2)$, if two cells share a common side, draw an edge with the centers of the two cells as endpoints, obtaining graphs $G_i$. If both $G_1$ and $G_2$ are connected paths (no cycles, no splits), prove that the center of the grid is one of the endpoints of $G_1$ or $G_2$.

Today's calculation of integrals, 884

Tags: calculus , integration , trigonometry , inequalities , calculus computations , integral inequality

Prove that : \[\pi (e-1)<\int_0^{\pi} e^{|\cos 4x|}dx<2(e^{\frac{\pi}{2}}-1)\]

2025 CMIMC Combo/CS, 3

Tags: combinatorics , computer science

There are $34$ friends are sitting in a circle playing the following game. Every round, four of them are chosen at random, and have a rap battle. The winner of the rap battle stays in the circle and the other three leave. This continues until one player remains. Everyone has equal rapping ability, i.e. every person has equal probability to win a round. What is the probability that Michael and James end up battling in the same round?

2010 All-Russian Olympiad Regional Round, 11.6

Tags: geometry , 3d geometry , volume , tetrahedron

At the base of the quadrangular pyramid $SABCD$ lies the parallelogram $ABCD$. Prove that for any point $O$ inside the pyramid, the sum of the volumes of the tetrahedra $OSAB$ and $OSCD$ is equal to the sum of the volumes of the tetrahedra $OSBC$ and $OSDA$ .

1996 Denmark MO - Mohr Contest, 3

Tags: 3d geometry , sphere , ratio , cube , geometry

This year's gift idea from BabyMath consists of a series of nine colored plastic containers of decreasing size, alternating in shape like a cube and a sphere. All containers can open and close with a convenient hinge, and each container can hold just about anything next in line. The largest and smallest container are both cubes. Determine the relationship between the edge lengths of these cubes.

2020 Princeton University Math Competition, A3/B4

Tags: combinatorics

Katie has a chocolate bar that is a $5$-by-$5$ grid of square pieces, but she only wants to eat the center piece. To get to it, she performs the following operations: i. Take a gridline on the chocolate bar, and split the bar along the line. ii. Remove the piece that doesn’t contain the center. iii. With the remaining bar, repeat steps $1$ and $2$. Determine the number of ways that Katie can perform this sequence of operations so that eventually she ends up with just the center piece.

2023 AIME, 10

Tags:

There exists a unique positive integer $a$ for which the sum \[U=\sum_{n=1}^{2023}\left\lfloor\dfrac{n^{2}-na}{5}\right\rfloor\] is an integer strictly between $-1000$ and $1000$. For that unique $a$, find $a+U$. (Note that $\lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $x$.)

2021 MMATHS, 7

Tags:

Let $P_k(x) = (x-k)(x-(k+1))$. Kara picks four distinct polynomials from the set $\{P_1(x), P_2(x), P_3(x), \ldots ,$ $P_{12}(x)\}$ and discovers that when she computes the six sums of pairs of chosen polynomials, exactly two of the sums have two (not necessarily distinct) integer roots! How many possible combinations of four polynomials could Kara have picked? [i]Proposed by Andrew Wu[/i]

1977 Kurschak Competition, 1

Tags: composite , prime , number theory

Show that there are no integers $n$ such that $n^4 + 4^n$ is a prime greater than $5$.

2020 IOM, 2

Tags: number theory

Does there exist a positive integer $n$ such that all its digits (in the decimal system) are greather than 5, while all the digits of $n^2$ are less than 5?

2018 AIME Problems, 6

Tags: complex number

Let $N$ be the number of complex numbers $z$ with the properties that $|z|=1$ and $z^{6!}-z^{5!}$ is a real number. Find the remainder when $N$ is divided by $1000$.

1996 All-Russian Olympiad, 7

Tags: polynomial , inequalities , algebra

Does there exist a finite set $M$ of nonzero real numbers, such that for any natural number $n$ a polynomial of degree no less than $n$ with coeficients in $M$, all of whose roots are real and belong to $M$? [i]E. Malinnikova[/i]

1992 AMC 12/AHSME, 4

Tags: analytic geometry , graphing lines , slope

If $m > 0$ and the points $(m,3)$ and $(1,m)$ lie on a line with slope $m$, then $m = $ $ \textbf{(A)}\ 1\qquad\textbf{(B)}\ \sqrt{2}\qquad\textbf{(C)}\ \sqrt{3}\qquad\textbf{(D)}\ 2\qquad\textbf{(E)}\ \sqrt{5} $

2022 Germany Team Selection Test, 1

Tags: number theory , algebra , factorization , combinatorics

Let $a_1, a_2, \ldots, a_n$ be $n$ positive integers, and let $b_1, b_2, \ldots, b_m$ be $m$ positive integers such that $a_1 a_2 \cdots a_n = b_1 b_2 \cdots b_m$. Prove that a rectangular table with $n$ rows and $m$ columns can be filled with positive integer entries in such a way that * the product of the entries in the $i$-th row is $a_i$ (for each $i \in \left\{1,2,\ldots,n\right\}$); * the product of the entries in the $j$-th row is $b_j$ (for each $i \in \left\{1,2,\ldots,m\right\}$).

1994 Chile National Olympiad, 7

Tags: geometry , reflection , rectangle

Let $ABCD$ be a rectangle of length $m$ and width $n$, with $m, n$ positive integers. Consider a ray of light that starts from $A$, reflects with an angle of $45^o$ on an opposite side and continues reflecting away at the same angle. $\bullet$ For any pair $(m,n)$, show that the ray meets a vertex at some point. $\bullet$ Suppose $m$ and $n$ are coprime. Determine the number of reflections made by the ray of light before encountering a vertex for the first time.

2015 Lusophon Mathematical Olympiad, 5

Tags: geometry , isosceles , tangent circle

Two circles of radius $R$ and $r$, with $R>r$, are tangent to each other externally. The sides adjacent to the base of an isosceles triangle are common tangents to these circles. The base of the triangle is tangent to the circle of the greater radius. Determine the length of the base of the triangle.