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A biologist catches and tags 60 fish in a lake on May 1 and releases them. On September 1, she catches a random sample of 70 fish and finds that 3 of them are tagged. Assuming that $25\%$ of the fish from May 1 are no longer in the lake on September 1, and $40\%$ of the fish in the September 1 sample were not in the la... | 840 |
Find the maximum value of $\frac{xy}{4x+(x+y-1)^{2}}$ for positive numbers $x$ and $y$. | \(\frac{1}{4}\) |
Consider the sequence \( x_n \) defined by \( x_{n+1} = \frac{6x_n}{7} + \frac{5491}{7x_n} \) for \( n \geq 1 \) with \( x_1 = 11 \). As \( n \) increases without bound, the limiting value of \( x_n \) is \( k \). Find \( k \). | \(\sqrt{5491}\) |
How many kites are there such that all of its four vertices are vertices of a given regular icosagon (20-gon)? | 85 |
Find a function \( f\colon \mathbb{R}\to\mathbb{R} \) that is continuous at \( x=0 \) and satisfies the equation \( f(x) + f\left(\frac{2}{3}x\right) = x \). | \( f(x) = \frac{3x}{5} \) |
Evaluate the integral \(\int_0^\infty \frac{\arctan(x)}{x} \, dx\). | \(\infty\) |
Evaluate the product $\prod_{k=0}^{2^{1999}}\left(4\sin^{2}\frac{k\pi}{2^{2000}}-3\right)$. | 3 |
Three numbers in the interval \([0,1]\) are chosen independently and at random. What is the probability that the chosen numbers are the side lengths of a triangle with positive area? | \(\frac{1}{2}\) |
A sequence of integers \(a_1, a_2, \ldots, a_{2010}\) satisfies the conditions:
- \(a_2 = 1001\)
- \(a_{n+2} = a_{n+1} - a_n\) for \(1 \leq n \leq 2008\)
What is the value of \(a_1 + a_2 + a_3 + \cdots + a_{2010}\)? | 0 |
Let \(a, b, c\) and \(x, y, z\) satisfy \(a + b + c = 0\), \(x + y + z = 0\), and \(ax + by + cz = 0\). Calculate
\[ T = \frac{(a+b)^2}{a^2+b^2+c^2} + \frac{(x+y)^2}{x^2+y^2+z^2}. \] | \(\frac{2}{3}\) |
For all \( a \in \mathbb{R} \), find all possible values of \( \sqrt{a^2 + a + 1} - \sqrt{a^2 - a + 1} \). | \((-1, 1)\) |
In the system of orthogonal axes \(xOy\), the points \(A(a,b)\) and \(B(c,d)\) are considered, where \(a \ne b \ne c \ne d \ne a\). What is the necessary and sufficient condition that
\[
\min_{M \in Oy}(MA + MB) = \min_{N \in Ox}(NA + NB)?
\] | \(ac = bd\) |
Find all integers \( n \) such that \(\frac{8n-25}{n+5}\) is the cube of a rational number. | \( n = 3 \) |
Farmer Georgia has a positive integer number $c$ cows (which have four legs), and zero ostriches, on her farm on day $1$. On each day thereafter, she adds a combination of cows and ostriches to her farm, so that on each day $n \ge 2$, the number of animals on the farm is equal to exactly half the number of legs that we... | \( c \equiv 2 \pmod{3} \) |
Find the maximum number of natural numbers that can be chosen from the set \( A = \{1, 2, 3, \ldots, 2021\} \) such that the sum of any three of the chosen numbers is divisible by 18. | 112 |
Given a list of 10 distinctive natural numbers, you can continue the list by choosing any two numbers and writing their least common multiple (LCM) if it is not already in the list. The list is considered closed when no more LCMs can be added. What is the maximum number of numbers that can be written on a closed list? | \(2^{10} - 1\) |
Let \( x, y, z > 0 \) and \( x + y + z = 1 \). Find the minimum value of the expression:
\[ P = \frac{x^2y}{x+y} + \frac{y^2z}{z+y} + \frac{z^2x}{x+z} \] | 0 |
For the partition \( P = \{0, \pi/6, 5\pi/6, \pi\} \) of the interval \([0, \pi]\), find the lower sum \( L(f, P) \) for the function \( f(x) = \sqrt{\sin x} \). Specifically, determine the values \( m_i \) for each subinterval \( I_i \) where:
- \( I_1 = [0, \pi/6] \), \( d_1 = \pi/6 \)
- \( I_2 = [\pi/6, 5\pi/6] \), ... | \(\frac{\sqrt{2} \pi}{3}\) |
Find the value of \( k \) for which the equations \((x - 2)^4 - (x - 2) = 0\) and \(x^2 - kx + k = 0\) have two roots in common. | 3 |
Find the sum of all variables for all possible solutions to the system of equations $u + v + w = xyz$ and $x + y + z = uwv$, where $u, v, w, x, y, z$ are natural numbers. (Symmetries should be taken into account to avoid counting the same solution multiple times.) | 60 |
Trapezoid \(ABCD\) has \(AB \parallel CD\), \(AD = 17\), \(AB = 7\), \(BC = 18\), and \(CD = 14\). The median of \(ABCD\) is \(XY\), with \(X\) on side \(AD\) and \(Y\) on side \(BC\). The diagonals of the trapezoid intersect the median at points \(S\) and \(T\). What is the length of \(ST\)? | \(\frac{7}{2}\) |
Find the best constant \( k \) such that the inequality
\[
\sqrt{3a+s} + \sqrt{3b+t} + \sqrt{3c+u} \geq k \sqrt{a+b+c+s+t+u}
\]
holds for all non-negative real numbers \( a, b, c, s, t, \) and \( u \). | 1 |
Business sales are at \$5,000,000 in March. Sales are expected to increase by 10 percent in April, 15 percent in May, and 30 percent in June. What would be the expected total earnings in June? | 8,222,500 |
Define a domino to be an ordered pair of distinct positive integers. A proper sequence of dominos is a list of distinct dominos in which the first coordinate of each pair after the first equals the second coordinate of the immediately preceding pair, and in which (i, j) and (j, i) do not both appear for any i and j. Le... | 761 |
You are 10 miles off a straight highway and your destination is 100 miles ahead, also 10 miles off the highway. You can drive 40 miles per hour off-road and 80 miles per hour on the highway. What is the fewest number of minutes in which you can get to your destination? Round your answer to the nearest minute. | 101 |
A fair die with 12 sides numbered 1 through 12 inclusive is rolled \( n \) times. The probability that the sum of the rolls is 2012 is nonzero and is equivalent to the probability that a sum of \( k \) is rolled. Find the minimum value of \( k \). | 172 |
The number $x$ satisfies $5x^2 + 4 = 3x + 9$. Find the value of $(10x - 3)^2$. | 109 |
What is the product of all real solutions \( x \) of the equation \[\log_{7x}2023 \cdot \log_{289x}2023 = \log_{2023x}2023?\] | 1 |
Let \( P \) be a point inside triangle \( ABC \). Let \( D \) be on \( AC \) such that \( DP \parallel AB \). Let \( E \) be on \( AB \) such that \( EP \parallel BC \). Let \( F \) be on \( BC \) such that \( CP \parallel AC \). Find the value of:
\[
\frac{AD}{AC} + \frac{BE}{AB} + \frac{CF}{BC}
\] | 1 |
A function $f$ is defined on the complex numbers by $f(z) = (a + bi)z$, where $a$ and $b$ are positive numbers. This function has the property that the image of each point in the complex plane is equidistant from that point and the origin. Given $|a + bi| = 8$, find the value of $b^2$. | 63.75 |
Find the sum to infinity of the series $5 + \frac{2 \cdot 6}{1!} + \frac{3 \cdot 7}{2!} + \frac{4 \cdot 8}{3!} + \cdots$. | \(13e\) |
Evaluate \[\lim_{x \to 0} \frac{|2x-1|-|2x+1|}{x}.\] | -4 |
Consider the circle $\gamma$ with center at $(0,3)$ and radius $3$, and a line $r$ parallel to the $Ox$-axis at a distance $3$ from the origin. A variable line through the origin intersects $\gamma$ at point $M$ and $r$ at point $P$. Find the locus of the intersection point of the lines through $M$ and $P$ that are par... | \( y = \frac{54}{x^2 + 9} \) |
What is the area of the largest circle that does not contain any lattice point in its interior? | \(\frac{\pi}{2}\) |
If $c$ and $d$ are the roots of the equation $x^2 - 10ax - 11b = 0$ and $a$ and $b$ are the roots of the equation $x^2 - 10cx - 11d = 0$, then find the value of $a + b + c + d$. | 1210. |
Let \( S_n \) be the number of Latin squares of order \( n \) and let \( R_n \) be the number of reduced Latin squares of order \( n \) (Latin squares for which the first column and first row both equal \( 1\ 2\ 3\ \cdots\ n \) in that order). Find \( S_n / R_n \) as a function of \( n \). | \( n!(n-1)! \) |
Find the sum of all integers $n$ for which $n - 3$ and $n^2 + 4$ are both perfect cubes. | 13 |
Solve the equation $\log_{3x+4}(4x^{2}+4x+1)+\log_{2x+1}(6x^{2}+11x+4)=4$. | \(\frac{3}{4}\) |
How many ways can a good salad be made using 2 ingredients out of the following: oranges, peaches, grapes, or bananas, given that oranges and grapes together are gross and bananas don't mix well with everything? | 3 |
Let \(ABC\) be a triangle with \(\angle ABC = 76^\circ\) and \(\angle ACB = 72^\circ\). Points \(P\) and \(Q\) lie on sides \(AB\) and \(AC\), respectively, such that \(\angle ABQ = 22^\circ\) and \(\angle ACP = 44^\circ\). Find the measure of \(\angle APQ\). | 36 |
Let $n$ be an odd integer with exactly 11 positive divisors. Find the number of positive divisors of $8n^3$. | 124 |
Anderson writes an infinitely long decimal number by randomly appending one of his favorite numbers ($1$, $7$, or $33$) to the end of the decimal he has already written. If the expected value of the number Anderson wrote down is of the form $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers, find $... | 392 |
Determine the maximum possible value of \( a \) such that the graph of \( y = mx + 2 \) passes through no lattice point with \( 0 < x \le 100 \) for all \( m \) such that \( \frac{1}{2} < m < a \). | \(\frac{50}{99}\) |
Find the maximum value of \(8 \cdot 27^{\log_6 x} + 27 \cdot 8^{\log_6 x} - x^3\). | 216 |
Given a rhombus with side length $a$ and an acute angle of $45^{\circ}$, determine the length of the sum of the two diagonals. | \( a\sqrt{4 + \sqrt{8}} \) |
Evaluate the limit $\lim_{x\rightarrow \frac{\pi}{3}} \frac{\sin(3x)}{1-2\cos(x)}$ without using L'Hôpital's rule or Taylor series. | $-\sqrt{3}$ |
A box contains $N > 1$ white and black balls, of which $N_B (0 < N_B \leq N)$ are black. Balls are taken randomly from the box without replacement. Find the probability $P(B_n)$ of taking a black ball at the $n$th withdrawal for $1 \leq n \leq N$. | \(\frac{N_B}{N}\) |
Let \( l_1 \) and \( l_2 \) be the tangent lines at points \( P(a, a^2) \) and \( Q(b, b^2) \) (\( a < b \)) on the curve \( C: y = x^2 \), respectively. Let \( R \) be the point of intersection of \( l_1 \) and \( l_2 \). Denote by \( S \) the area of the figure enclosed by the segments \( PR \), \( QR \), and the cur... | \(\frac{1}{12}\) |
Calculate the limit \(\lim_{n \to \infty} \sum_{k=1}^n \frac{k^2 + 1}{k(k + 1)}\). | \(\infty\) |
Find the average value of \( |a_1 - a_2| + |a_3 - a_4| + |a_5 - a_6| + |a_7 - a_8| + |a_9 - a_{10}| \) for all permutations \( a_1, a_2, \ldots, a_{10} \) of \( 1, 2, \ldots, 10 \). | \(\frac{55}{3}\) |
Find all functions \( f: \mathbb{Z} \to \mathbb{R} \) such that \( f(0) = 2 \), \( f(1) = \frac{5}{2} \), and \( f(x)f(y) = f(x+y) + f(x-y) \). | \( f(x) = 2^x + 2^{-x} \) |
How many possible rational zeros does the function \(4x^4 - 17x^2 + 4\) have? | 10 |
Let \( a_1 = 5 \) and \( a_{n+1} = a_n^2 - 2 \) for any \( n = 1, 2, \ldots \). Find \( \lim_{n \rightarrow \infty} \frac{a_{n+1}}{a_1 a_2 \cdots a_n} \). | \(\sqrt{21}\) |
Find a function \( f: \mathbb{N} \to \mathbb{N} \) that satisfies the following conditions:
1. \( f(mf(n)) = n^6 f(mn) \) for any integers \( m \) and \( n \).
2. \( \gcd(f(m), f(n)) = 1 \) for any integers \( m \) and \( n \) such that \( \gcd(m, n) = 1 \). | \( f(n) = n^3 \) |
If $4\sin x\cos y + 2\sin x + 2\cos y + 1 = 0$ where $x, y \in [0, 2\pi]$, find the largest possible value of $x + y$. | \(\frac{23}{6}\pi\) |
\(A, B, C\) are playing a backgammon tournament. At first, \(A\) plays with \(B\). Then the winner plays with \(C\). As the tournament goes on, the last winner plays with the player who did not play in the previous game. When a player wins two successive games, he wins the tournament. If each player has an equal chance... | \(\frac{2}{7}\) |
Two circles, each of which passes through the centre of the other, intersect at points \( M \) and \( N \). A line from \( M \) intersects the circles at \( K \) and \( L \). If \( KL = 6 \), find the area of \( \triangle KLN \). | \( 9\sqrt{3} \) |
Find the volume of a Rhombic Dodecahedron if its side length is $1$. | \(\frac{16}{9}\sqrt{3}\) |
Let \( m \ge 3 \) be an integer and let \( S = \{3, 4, 5, \ldots, m\} \). Find the smallest value of \( m \) such that for every partition of \( S \) into two subsets, at least one of the subsets contains integers \( a \), \( b \), and \( c \) (not necessarily distinct) such that \( ab = c \). | 243 |
How many ways are there to rearrange the letters of the word RAVEN such that no two vowels are consecutive? | 72 |
Point P is given in square ABCD such that AP:BP:CP = 1:2:3. Find the measure of angle APB without using trigonometry. | 135^\circ |
A bag contains 10 red balls, 10 green balls, and 10 white balls. Five balls are drawn at random from the bag without replacement. What is the probability that there are balls of all three different colors among the five? | 0.6789 |
Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that:
\[ (f(x) + f(y))(f(x) - y) \geq x^2 - y^2 \] | \( f(x) = x \) |
Define a finite sequence \( \left( s_i \right)_{1\le i\le 2004} \) with initial conditions \( s_0 + 2 = s_1 + 1 = s_2 = 2 \) and the recurrence relation
\[ s_n = 1 + s_{n-1} + s_{n-2} - s_{n-3}. \]
Calculate the value of \( s_{2004} \). | 1005006 |
Given the sequence defined by \( u_0 > 1 \) and \( u_{n+1} = \frac{u_n + 1 + \sqrt{2(u_n^2 + 1)}}{u_n - 1} \) for \( n = 0, 1, 2, \ldots \), find \( \lim_{n \to +\infty} u_n \). | \( 2 + \sqrt{3} \) |
Plot the graph of the function \( y = -\frac{2}{x} \) for positive real numbers and identify the range. | \((-\infty, 0)\) |
Find the smallest positive angle $\theta$ in degrees satisfying the equation \[\sin^2(2004\theta) + \cos^2(2005\theta) = 1.\] | \(\frac{180}{4009}\) |
Find the maximum value of the expression \(\frac{a}{a+3b} + \frac{b}{b+3c} + \frac{c}{c+3a}\) where \(a\), \(b\), and \(c\) are positive real numbers. | 2 |
There are 20 points inside a square. They are connected by non-intersecting segments with each other and with the vertices of the square, in such a way that the square is dissected into triangles. How many triangles are formed? | 42 |
A group of 100 students from different countries meet at a mathematics competition. Each student speaks the same number of languages, and for every pair of students \(A\) and \(B\), student \(A\) speaks some language that student \(B\) does not speak, and student \(B\) speaks some language that student \(A\) does not s... | 9 |
Let \(ABCD\) be a trapezoid such that \(AD \parallel BC\) and \(|AB| = |BC|\). Let \(E\) and \(F\) be the midpoints of \([BC]\) and \([AD]\), respectively. If the internal angle bisector of \(\triangle ABC\) passes through \(F\), find the ratio \(|BD| / |EF|\). | 2 |
Find the smallest positive integer \( x \) such that \( 1260x = N^3 \), where \( N \) is an integer. | 7350 |
In a triangle $\triangle ABC$, it is given that $(\sin A+\sin B):(\sin B+\sin C):(\sin C+\sin A)=9:10:11$. Find the value of $480\cos A$. | 270 |
In triangle \( \triangle ABC \) with a perimeter of 60, points \( D \), \( E \), and \( F \) are chosen on sides \( BC \), \( AC \), and \( AB \) respectively. The circumcircles of triangles \( \triangle AFE \), \( \triangle BFD \), and \( \triangle CED \) all pass through the orthocenter of \( \triangle DEF \). What i... | 28 |
Solve the equation $\sin x = x^2 + x + 1$. | No real solutions. |
Calculate the limit:
\[
\lim_{n\to \infty}\left[n-\frac{n^2+1^2-1}{\sqrt{(n^2+1^2)(n^2+(1-1)^2)}}-\frac{n^2+2^2-2}{\sqrt{(n^2+2^2)(n^2+(2-1)^2)}}-\cdots-\frac{n^2+n^2-n}{\sqrt{(n^2+n^2)(n^2+(n-1)^2)}}\right]
\] | 0 |
Find the length and location of the shortest straight cut which divides an isosceles right triangle with sides 1, 1, and √2 into two parts with equal area. | \( \frac{\sqrt{2}}{2} \) |
In a convex pentagon $ABCDE$, the areas of triangles $ABC$, $ABD$, $ACD$, and $ADE$ are equal and each have the value $F$. What is the area of triangle $BCE$? | $2F$ |
If \( x \) is an odd number, find the largest integer that always divides the expression \((10x+2)(10x+6)(5x+5)\). | 960 |
The perimeter of triangle $APM$ is $152$, and the angle $PAM$ is a right angle. A circle of radius $19$ with center $O$ on $\overline{AP}$ is drawn so that it is tangent to $\overline{AM}$ and $\overline{PM}$. Given that $OP=m/n$ where $m$ and $n$ are relatively prime positive integers, find $m+n$.
| 98 |
Let \( P \) be the midpoint of side \( AB \) of quadrilateral \( ABCD \), and let \( C' \) and \( D' \) denote the centroids of triangles \( ADB \) and \( ABC \), respectively. Find \( 81 \frac{CD}{C'D'} \). | 243 |
Find the minimum value of the expression \( x^4y^2 + x^2y^2 + 8x^2y + x^2 + 2008 \) where \( x, y \in \mathbb{R} \). | 1999 |
A number of unit squares are placed in a line. Let $O$ be the bottom left corner of the first square and let $P$ and $Q$ be the top right corners of the 2012th and 2013th squares, respectively. The lines $OP$ and $OQ$ intersect the right side of the first square at $X$ and $Y$, respectively. Determine the area of trian... | \(\frac{1}{8100312}\) |
Given that \( x \) is a positive real number such that \( x + \frac{1}{x} = 5 \) and \( x^2 + \frac{1}{x^3} = 8 \), find the value of \( x^3 + \frac{1}{x^2} \). | 125 |
Find all triplets $(x, y, p)$ of positive integers such that $p$ is a prime number and $\frac{xy^3}{x+y}=p.$ | (14, 2, 7) |
From a square of side length 5, four corners are cut off to form a regular octagon. Round off the removed area to the nearest integer. | 4 |
If \(a, b, c\) are real numbers such that \(a^2 + 2b = 7\), \(b^2 + 4c = -7\), and \(c^2 + 6a = -14\), find \(a^2 + b^2 + c^2\). | 14 |
Find the remainder when $0! + 1! + 2! + 3! + 4! + 5! + \ldots + 2006!$ is divided by $60$. | 34 |
In a field, grass grows at a constant rate. It is known that 60 cows can graze the field in 24 days, and 30 cows can graze it in 60 days. How many cows will be needed to graze the field in 100 days? | 22 |
If \( S = \sum_{k=3}^{2020} \binom{k}{3} \binom{2022-k}{2} \), what is the value of \( S \mod 2017 \)? | 1 |
Find the number of ordered triples $(x, y, z)$ of real numbers that satisfy the system of equations:
\[ x + y + z = 7, \]
\[ x^2 + y^2 + z^2 = 27, \]
\[ xyz = 5. \] | 3 |
Let \(a, b, c, d, e\) be real numbers such that \(a^2 + b^2 + c^2 + d^2 + e^2 = 1\). Find the maximum value of \(|a-b| + |b-c| + |c-d| + |d-e| + |e-a|\). | 4 |
Write the expression \(6x^2 - xy + 23x - 2y^2 - 6y + 20\) as a product of two trinomials with integer coefficients. | \((4 + 3x - 2y)(5 + 2x + y)\) |
Determine all real solutions to the system of equations:
\[ 2x + x^2 y = y \]
\[ 2y + y^2 z = z \]
\[ 2z + z^2 x = x \] | \( (x, y, z) = (0, 0, 0) \) |
Let $\mathbb X$ be the set of all bijective functions from the set $S=\{1,2,\cdots, n\}$ to itself. For each $f\in \mathbb X,$ define
\[T_f(j)=\left\{\begin{aligned} 1, \ \ \ & \text{if} \ \ f^{(12)}(j)=j,\\ 0, \ \ \ & \text{otherwise}\end{aligned}\right.\]
where $f^{(k)}(x)=f(f^{(k-1)}(x))$ for all $k\geq 2.$ Determi... | \(6n!\) |
In rectangle \(PQRS\) with \(PQ = 8\) and \(QR = 6\), points \(A\) and \(B\) lie on \(\overline{PQ}\), points \(C\) and \(D\) lie on \(\overline{QR}\), points \(E\) and \(F\) lie on \(\overline{RS}\), and points \(G\) and \(H\) lie on \(\overline{SP}\) such that \(AP = BQ < 4\) and the convex octagon \(ABCDEFGH\) is eq... | 7 |
Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that
\[ f(xy) + f(x-y) + f(x+y+1) = xy + 2x + 1 \]
for all \( x, y \in \mathbb{R} \). | \( f(x) = x \) |
Find the remainder when $1021^{1022}$ is divided by $1023$. | 4 |
Calculate \( S_n \) where
\[ S_n = \binom{n}{1} - \left(1 + \frac{1}{2}\right) \binom{n}{2} + \left(1 + \frac{1}{2} + \frac{1}{3}\right) \binom{n}{3} - \cdots + (-1)^{n-1} \left(1 + \frac{1}{2} + \cdots + \frac{1}{n}\right) \binom{n}{n} \] | \(\frac{1}{n}\) |
Find all functions \( f: \mathbb{R}^+ \to \mathbb{R}^+ \) such that
\[ f(f(x)) + y^2 = f(x + yf(y)), \ \ \ \forall x, y \in \mathbb{R}^+. \] | \( f(x) = x \) |
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